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Journal of Functional Analysis 257 (2009) 1–19 www.elsevier.com/locate/jfa Hardy type inequality and application to the...

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Journal of Functional Analysis 257 (2009) 1–19 www.elsevier.com/locate/jfa

Hardy type inequality and application to the stability of degenerate stationary waves Shuichi Kawashima a,∗ , Kazuhiro Kurata b a Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan b Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-03, Japan

Received 6 August 2008; accepted 7 April 2009

Communicated by C. Villani

Abstract This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t −α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2α = L2 (R+ ; (1 + x)α ) for α < αc (q) := 3 + 2/q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in L2α for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant. © 2009 Elsevier Inc. All rights reserved. Keywords: Viscous conservation laws; Degenerate stationary waves; Asymptotic stability; Hardy inequality

1. Introduction We study the stability problem of degenerate stationary waves for viscous conservation laws in the half space x > 0: ut + f (u)x = uxx , u(0, t) = −1,

u(x, 0) = u0 (x).

* Corresponding author.

E-mail addresses: [email protected] (S. Kawashima), [email protected] (K. Kurata). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.003

(1.1)

2

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

Here the initial function is assumed to satisfy u0 (x) → 0 as x → ∞, and f (u) is a smooth function of the form f (u) =

1 (−u)q+1 1 + g(u) , q

f (u) > 0 for −1 u < 0,

(1.2)

where q is a positive integer (degeneracy exponent) and g(u) = O(|u|) for u → 0. Since f (0) = f (0) = 0 and f (u) is strictly convex for −1 u < 0, we see that f (u) > 0 for −1 u < 0. In particular, we have 1 + g(u) > 0 for −1 u 0. In this situation, the corresponding stationary problem admits a unique solution φ(x) (called degenerate stationary wave), which verifies φx = f (φ), φ(0) = −1,

φ(x) → 0 as x → ∞.

(1.3)

We see easily that φ(x) behaves like φ(x) ∼ −(1 + x)−1/q . In particular, we have φ(x) = −(1 + x)−1/q when g(u) ≡ 0. To discuss the stability of the degenerate stationary wave φ(x), we introduce the perturbation v by u(x, t) = φ(x) + v(x, t) and rewrite the problem (1.1) as vt + f (φ + v) − f (φ) x = vxx , v(0, t) = 0,

v(x, 0) = v0 (x),

(1.4)

where v0 (x) = u0 (x) − φ(x), and v0 (x) → 0 as x → ∞. The stability of degenerate stationary waves was first studied in [15]. It was proved in [15] that if the initial perturbation v0 (x) is in the weighted space L2α , then the perturbation v(x, t) decays in L2 at the rate t −α/4 as t → ∞, provided that α < α∗ (q), where α∗ (q) := q + 1 + 3q 2 + 4q + 1 /q. The decay rate t −α/4 obtained in [15] would be optimal but the restriction α < α∗ (q) was not very sharp. The main purpose of this paper is to relax this restriction. Indeed, by employing the space–time weighted energy method in [15] and by applying a Hardy type inequality with the best possible constant (see Proposition 2.3), we show the same decay rate t −α/4 under the weaker restriction α < αc (q) := 3 + 2/q (see Theorem 4.1). Notice that α∗ (q) < αc (q). It is interesting to note that a similar restriction on the weight is imposed also for the stability of degenerate shock profiles (see [10]). We remark that our stability result for degenerate stationary waves is completely different from those for non-degenerate case. In fact, for non-degenerate stationary waves, we have the better decay rate t −α/2 for the perturbation without any restriction on α. See [4–6,14,16] for the details. See also [2,7,9,11] for the related stability results for stationary waves. In this paper we also discuss the dissipativity of the following linearized operator associated with (1.4): Lv = vxx − f (φ)v x .

(1.5)


3

In a simpler situation including the case g(u) ≡ 0 in (1.2), we show that the operator L is uniformly dissipative in L2α for α < αc (q) but cannot be dissipative for α > αc (q) (see Theorem 3.5). This suggests that the exponent αc (q) is the critical exponent of the stability problem of degenerate stationary waves. This result on the characterization of the dissipativity of L is an improvement on the previous one in [15] and is established by using a Hardy type inequality with the best possible constant (see Proposition 2.3). The contents of this paper are as follows. In Section 2 we introduce several Hardy type inequalities and discuss the best possibility of their constants. In Section 3 we discuss the dissipativity of the operator L in (1.5) in weighted L2 spaces. Finally in Section 4, we study the nonlinear stability of degenerate stationary waves. Notations. For 1 p ∞, Lp = Lp (R+ ) denotes the usual Lebesgue space on R+ = (0, ∞) with the norm · Lp . Let s be a nonnegative integer. Then the Sobolev space W s,p = W s,p (R+ ) is defined by W s,p = {u ∈ Lp ; ∂xk u ∈ Lp for k s} with the norm · W s,p . When p = 2, we write H s = W s,2 . Next we introduce weighted spaces. Let w = w(x) > 0 be a weight function defined on [0, ∞) such that w ∈ C 0 [0, ∞). Then, for 1 p < ∞, we denote by Lp (w) the weighted Lp space on R+ equipped with the norm ∞ 1/p p u(x) w(x) dx . uLp (w) :=

(1.6)

0

The corresponding weighted Sobolev space W s,p (w) is defined by W s,p (w) = {u ∈ Lp (w); 1,p ∂xk u ∈ Lp (w) for k s} with the norm · W s,p (w) . Also, we denote by W0 (w) the completion of C0∞ (R+ ) with respect to the norm ∞ 1/p p ∂x u(x) w(x) dx uW 1,p (w) := ∂x uLp (w) = .

(1.7)

0

0

When p = 2, we write H s (w) = W s,2 (w) and H01 (w) = W01,2 (w). In the special case where w = (1 + x)α with α ∈ R, these weighted spaces are abbreviated as Lpα = Lp (1 + x)α , 1,p 1,p Wαs,p = W s,p (1 + x)α , Wα,0 = W0 (1 + x)α , 1 Hα,0 = H01 (1 + x)α . Hαs = H s (1 + x)α , Let k be a nonnegative integer. Then, for an interval I ⊂ [0, ∞) and a Banach space X on R+ , C k (I ; X) denotes the space of k-times continuously differentiable functions on I with values in X. Finally, letters C and c in this paper denote positive generic constants which may vary from line to line.

4


2. Hardy type inequality Hardy’s inequality was first introduced by Hardy [1] and its best possible constant was given by Landau [8]. Here we introduce several Hardy type inequalities which will be used in this paper. The following first one is found in [13] but its best possible constant is not given explicitly there. Proposition 2.1. Let ψ ∈ C 1 [0, ∞) and assume either (1) ψ > 0, ψx > 0 and ψ(x) → ∞ for x → ∞; or (2) ψ < 0, ψx > 0 and ψ(x) → 0 for x → ∞. Then we have ∞

∞ v ψx dx 4 2

0

vx2 ψ 2 /ψx dx

(2.1)

0

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w) with w = ψ 2 /ψx . Here 4 is the best possible constant, and there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.1). Proof. Let v ∈ C0∞ (R+ ). A simple calculation gives 2 1 1 v ψ x = v 2 ψx + 2vvx ψ = v 2 ψx + (v + 2vx ψ/ψx )2 ψx − 2vx2 ψ 2 /ψx . 2 2

(2.2)

Integrating (2.2) in x, we obtain ∞

∞ ∞ 2 v ψx dx + (v + 2vx ψ/ψx ) dx = 4 vx2 ψ 2 /ψx dx, 2

0

0

(2.3)

0

which gives the desired inequality (2.1). It follows from (2.3) that the equality in (2.1) holds if and only if v + 2vx ψ/ψx ≡ 0. This gives v = C1 |ψ|−1/2 for some constant C1 . But, if C1 = 0, this v is not in H01 (w) with w = ψ 2 /ψx . In fact, in the case (1), we have vx = − 12 C1 ψ −3/2 ψx and hence ∞ vx2 w dx 0

1 = C12 4

∞

∞ 1 ψ −1 ψx dx = C12 log ψ(x) x=0 = ∞. 4

0

The case (2) can be treated similarly. Thus we conclude that there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.1).


5

Finally, we show the best possibility of the constant 4 in (2.1). The following proof is based on the computation in [8]. First, we consider the case (1). Let us fix a > 0. Let > 0 be a small parameter and put ⎧ ⎨ 0, 0 x < a, v (x) = (x − a)ψ(x)−1/2− , a < x < a + 1, ⎩ ψ(x)−1/2− , a + 1 < x.

(2.4)

Then we have ∞ a+1 ∞ 2 2 −1−2 v ψx dx = (x − a) ψ ψx dx + ψ −1−2 ψx dx a

0

a+1

=: I1 + I2 . Here we see that I1 = O(1) for → 0 and ∞ 1 1 −2 I2 = − ψ(x) = ψ(a + 1)−2 . 2 2 x=a+1 On the other hand, we have ⎧ ⎨ 0, 0 x < a, vx (x) = ψ(x)−1/2− − (1/2 + )(x − a)ψ(x)−3/2− ψx (x), ⎩ −(1/2 + )ψ(x)−3/2− ψx (x), a + 1 < x.

a < x < a + 1,

Therefore we get ∞ a+1 2 2 −1/2− 2 vx ψ /ψx dx = ψ − (1/2 + )(x − a)ψ −3/2− ψx ψ 2 /ψx dx a

0

∞ + (1/2 + )

2

ψ −1−2 ψx dx =: J1 + J2 .

a+1 1 Here we find that J1 = O(1) for → 0 and J2 = (1/2 + )2 2 ψ(a + 1)−2 . Consequently, we obtain

∞ 0

1 ψ(a + 1)−2 (v )2 ψ 2 /ψx dx O(1) + (1/2 + )2 2 ∞x = 1 2 O(1) + 2 ψ(a + 1)−2 0 (v ) ψx dx

=

O() + (1/2 + )2 ψ(a + 1)−2 1 −→ 4 O() + ψ(a + 1)−2

for → 0. This shows that 4 in (2.1) is the best possible constant.

6


In the case (2), to show the best possibility of the constant in (2.1), we may take a test function v (x) as ⎧ ⎨ 0, 0 x < a, v (x) = (x − a)(−ψ(x))−1/2− , a < x < a + 1, ⎩ (−ψ(x))−1/2− , a + 1 < x, 2

but we omit the details. This completes the proof of Proposition 2.1. We have the Lp version of Proposition 2.1.

Proposition 2.2. Let ψ be the same as in Proposition 2.1. Let 1 < p < ∞. Then we have ∞

∞ |v| ψx dx p p

0

p−1

|vx |p |ψ|p /ψx

p

(2.5)

dx

0

for v ∈ C0∞ (R+ ) and hence for v ∈ W0 (w) with w = |ψ|p /ψx . Here p p is the best possible 1,p constant, and there is no function v ∈ W0 (w), v = 0, which attains the equality in (2.5). 1,p

p−1

Proof. Let 1 < p < ∞ and v ∈ C0∞ (R+ ). A simple calculation gives p |v| ψ x = |v|p ψx + p|v|p−2 vvx ψ =

1 p p−1 |v| ψx − p p |vx |p |ψ|p /ψx + R, p

(2.6)

where 1 1 p−1 |v|p ψx + p p |vx |p |ψ|p /ψx + p|v|p−2 vvx ψ. R= 1− p p Integrating (2.6) in x, we obtain ∞

∞ |v| ψx dx + p

0

∞ R dx = p

p

0

p−1

|vx |p |ψ|p /ψx

p

dx.

0

Here we see that −p|v|p−2 vvx ψ p|v|p−1 |vx ||ψ| (p−1)/p (p−1)/p p|vx ||ψ|/ψx = |v|p−1 ψx 1 1 p−1 1− |v|p ψx + p p |vx |p |ψ|p /ψx , p p

(2.7)


7

where we have used the Young inequality AB (1 − 1/p)Ap/(p−1) + (1/p)B p for A = (p−1)/p (p−1)/p and B = p|vx ||ψ|/ψx . Thus we have R 0. This together with (2.7) |v|p−1 ψx gives the desired inequality (2.5). It follows from (2.7) that the equality in (2.5) holds if and only if R ≡ 0. This is the case p−1 where vvx ψ 0 and |v|p ψx ≡ p p |vx |p |ψ|p /ψx . This is equivalent to pvx ψ ≡ −vψx and −1/p for some constant C1 . A simple computation shows that when hence we have v = C1 |ψ| 1,p p−1 C1 = 0, this v is not in W0 (w) with w = |ψ|p /ψx . Thus we conclude that there is no 1,p function v ∈ W0 (w), v = 0, which attains the equality in (2.5). The best possibility of the constant p p is proved in the same way as in the proof of Proposition 2.1. For example, in the case (1), we take the test function as ⎧ ⎨ 0, 0 x < a, v (x) = (x − a)ψ(x)−1/p− , a < x < a + 1, ⎩ ψ(x)−1/p− , a + 1 < x,

(2.8)

where a > 0 is fixed and > 0 is a small parameter. Then we see that ∞ 0

p−1 p 1 −p |vx |p |ψ|p /ψx dx O(1) + (1/p + ) p ψ(a + 1) ∞ = 1 p O(1) + p ψ(a + 1)−p 0 |v | ψx dx

=

O() + (1/p + )p ψ(a + 1)−p 1 −→ p O() + ψ(a + 1)−p p

for → 0. This shows that p p in (2.5) is the best possible constant. Thus the proof of Proposition 2.2 is complete. 2 The following variant of Proposition 2.1 is useful in our application. Proposition 2.3. Let φ ∈ C 1 [0, ∞), φ < 0, φx > 0, and φ(x) → 0 for x → ∞. Let σ ∈ R with σ = 0, and define the weight functions w and w1 by w = (−φ)−σ +1 /φx ,

w1 = (−φ)−σ −1 φx .

(2.9)

Then we have ∞

4 v w1 dx 2 σ

∞

2

0

vx2 w dx

(2.10)

0

for v ∈ H01 (w). Here 4/σ 2 is the best possible constant, and there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.10).

8


Proof. Let σ > 0. In this case we put ψ = (−φ)−σ > 0. Then we have ψx = σ (−φ)−σ −1 φx > 0 and ψ(x) → ∞ as x → ∞. This corresponds to the case (1) of Proposition 2.1. Since ψ 2 /ψx = (1/σ )(−φ)−σ +1 /φx , by applying Proposition 2.1, we have ∞ 2

σ

−σ −1

v (−φ)

4 φx dx σ

0

∞

vx2 (−φ)−σ +1 /φx dx.

0

This gives (2.10) and hence the proof of Proposition 2.3 is complete for σ > 0. When σ < 0, we put ψ = −(−φ)−σ < 0. Then, applying the case (2) of Proposition 2.1, we get the desired conclusion also for σ < 0. This completes the proof of Proposition 2.3. 2 As a simple corollary of Proposition 2.3, we have: Corollary 2.4. Let α ∈ R with α = 1. Then we have vL2

α−2

2 vx L2α |α − 1|

(2.11)

1 = H 1 ((1 + x)α ). Here the constant 2/|α − 1| is the best possible, and there is no for v ∈ Hα,0 0 1 , v = 0, which attains the equality in (2.11). function v ∈ Hα,0

Proof. Let φ = −(1 + x)−1/q with q > 0. Then we see that φ < 0, φx = (1/q)(1 + x)−1/q−1 = (1/q)(−φ)q+1 > 0, and φ(x) → 0 as x → ∞. Now we apply Proposition 2.3. Since w = (−φ)−σ +1 /φx = q(−φ)−σ −q = q(1 + x)σ/q+1 , 1 1 (−φ)−σ +q = (1 + x)σ/q−1 , q q

w1 = (−φ)−σ −1 φx =

(2.12)

we have from (2.10) that 1 q

∞ v (1 + x) 2

σ/q−1

4q dx 2 σ

0

∞ vx2 (1 + x)σ/q+1 dx 0

1 for v ∈ Hσ/q+1,0 . Thus we have

v2L2

σ/q−1

4q 2 vx 2L2 , σ2 σ/q+1

(2.13)

1 for v ∈ Hσ/q+1,0 . This together with σ = (α − 1)q gives the desired inequality (2.11). This completes the proof. 2


9

3. Dissipativity of the linearized operator We discuss the dissipativity of the operator L defined by (1.5) in the weighted space L2 (w). For this purpose, we first review the basic properties of the degenerate stationary wave φ(x) (see [9] for the details). Lemma 3.1. Suppose that f (u) satisfies (1.2). Then the stationary wave φ(x), which is a solution of (1.3), verifies the following properties: φ ∈ C ∞ [0, ∞), and −1 φ(x) < 0,

φ(x) → 0 for x → ∞.

φx (x) > 0,

(3.1)

Moreover, we have c(1 + x)−1/q −φ(x) C(1 + x)−1/q .

(3.2)

Now, let w > 0 be a weight function depending only on x such that w ∈ C 2 [0, ∞) and we calculate the inner product Lv, v L2 (w) for v ∈ C0∞ (R+ ), where ∞ u, v L2 (w) :=

uvw dx.

(3.3)

0

We multiply (1.5) by v. Then a simple computation gives 1 1 (Lv)v = vvx − f (φ)v 2 − vx2 − f (φ)φx v 2 . 2 2 x Multiplying by w, we obtain 1 2 1 2 (Lv)vw = vvx − f (φ)v w − v wx 2 2 x 1 − vx2 w + v 2 wxx + wx f (φ) − wf (φ)φx . 2

(3.4)

Now we choose the weight function w and the corresponding w1 in terms of the degenerate stationary wave φ by (2.9), where σ ∈ R. Then we have w = (−φ)−σ +1 /f (φ) by φx = f (φ). Differentiating this expression with respect to x and using φx = f (φ) several times, we find by direct computations that wx = (σ − 1)(−φ)−σ − (−φ)−σ +1 f (φ)/f (φ), wxx = σ (σ − 1)(−φ)−σ −1 f (φ) − (σ − 1)(−φ)−σ f (φ) − (−φ)−σ +1 f (φ) − f (φ)2 /f (φ) . Consequently, we arrive at the expression

10


wxx + wx f (φ) − wf (φ)φx = σ (σ − 1)(−φ)−σ −1 f (φ) − 2(−φ)−σ +1 f (φ) = σ (σ − 1) − 2(−φ)2 f (φ)/f (φ) (−φ)−σ −1 f (φ) = 2 c1 (σ ) − r(φ) w1 , (3.5) where w1 is given in (2.9) and c1 (σ ) := σ (σ − 1)/2 − q(q + 1), r(u) := (−u)2 f (u)/f (u) − q(q + 1).

(3.6)

Substituting (3.5) into (3.4) and integrating with respect to x, we get the following conclusion. Claim 3.2. Let φ be the degenerate stationary wave and define the weight functions w and w1 by (2.9) with σ ∈ R. Then the operator L defined in (1.5) verifies ∞ Lv, v L2 (w) = −vx 2L2 (w)

+ c1 (σ )v2L2 (w ) 1

−

v 2 r(φ)w1 dx

(3.7)

0

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w), where c1 (σ ) and r(φ) are given in (3.6). To discuss the dissipativity of L, we need to estimate the term r(φ) in (3.7). By straightforward computations, using (3.6) and (1.2), we see that r(u) = (−u) (−u)g (u) − 2(q + 1)g (u) 1 + g(u) . (3.8) This shows that r(u) = O(|u|) for u → 0. In particular, we have r(u) ≡ 0 if g(u) ≡ 0. With these preparations, we have the following result on the characterization of the dissipativity of L. Theorem 3.3. Assume (1.2). Let φ be the degenerate stationary wave and L be the operator defined in (1.5). Let w and w1 be the weight functions in (2.9) with the parameter σ ∈ R. Then we have: (1) Let −2q < σ < 2(q + 1). Then, under the additional assumption that r(u) 0 for −1 u 0, the operator L is uniformly dissipative in L2 (w). Namely, there is a positive constant δ such that Lv, v L2 (w) −δ vx 2L2 (w) + v2L2 (w ) (3.9) 1

for v ∈ H01 (w). (2) Let σ > 2(q + 1) or σ < −2q. Then the operator L cannot be dissipative in L2 (w). Namely, we have Lv, v L2 (w) > 0 for some v ∈ H01 (w) with v = 0. Remark 3.4. In (1) of this theorem, we have assumed that r(u) 0 for −1 u 0. In view of (3.8), this additional condition is satisfied if g(u) in (1.2) is of the form g(u) = G(−u), where G (η) 0 and G (η) 0 for 0 η 1. The simplest example of such a g(u) is g(u) = (−u)m with a nonnegative integer m.


11

Proof. The proof is based on the equality (3.7) in Claim 3.2 and the Hardy type inequality (2.10) in Proposition 2.3. Let −2q < σ < 2(q + 1). This is equivalent to c1 (σ ) < σ 2 /4. Therefore we can choose δ > 0 so small that δ(1 + σ 2 /4) σ 2 /4 − c1 (σ ). Since r(φ) 0 by the additional assumption on r(u) and since (σ 2 /4)v2L2 (w ) vx 2L2 (w) by the Hardy type inequality (2.10), we have from (3.7) 1 that Lv, v L2 (w) −vx 2L2 (w) + c1 (σ )v2L2 (w

1)

= −δvx 2L2 (w)

− (1 − δ)vx 2L2 (w)

+ c1 (σ )v2L2 (w ) 1 2 2 2 −δvx L2 (w) − (1 − δ)σ /4 − c1 (σ ) vL2 (w ) 1 2 2 −δ vx L2 (w) + vL2 (w ) 1

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w), where we used the fact that (1 − δ)σ 2 /4 − c1 (σ ) δ. This completes the proof of the uniform dissipative case (1). Next we consider the case where σ > 2(q + 1); the case σ < −2q can be treated similarly and we omit the argument in this latter case. When σ > 2(q + 1), we have c1 (σ ) > σ 2 /4. Then we choose δ > 0 so small that c1 (σ ) σ 2 /4 + 3δ. Since r(u) = O(|u|) for u → 0 and φ(x) → 0 for x → ∞, we take a = a(δ) > 0 so large that |r(φ)| δ for x a. For this choice of a and for > 0, we take a test function v as in (2.4): ⎧ ⎨ 0, 0 x < a, v (x) = (x − a)(−φ(x))σ (1/2+) , a < x < a + 1, (3.10) ⎩ (−φ(x))σ (1/2+) , a + 1 < x. Then we have ∞ ∞ 2 2 2 v r(φ)w1 dx δ v w1 dx = δ v L2 (w ) , 1 a

0

so that we have from (3.7) that

Lv , v

L2 (w)

2 2 −vx L2 (w) + c1 (σ ) − δ v L2 (w ) . 1

Here, a direct computation shows that 2 v 2

L (w1

a+1 ∞ 2 2σ −1 = (x − a) (−φ) φx dx + (−φ)2σ −1 φx dx ) a

a+1

2σ 1 = O(1) + −φ(a + 1) 2σ for → 0, where the term denoted by O(1) depends on δ. Similarly, we have 2 v 2

x L (w)

= O(1) + σ 2 (1/2 + )2

2σ 1 −φ(a + 1) 2σ

(3.11)

12


for → 0. Consequently, we obtain vx 2L2 (w) v 2L2 (w

=

O(1) + σ 2 (1/2 + )2 2σ1 (−φ(a + 1))2σ O(1) +

1)

=

1 2σ (−φ(a

+ 1))2σ

O() + σ 2 (1/2 + )2 (−φ(a + 1))2σ σ2 −→ 2σ 4 O() + (−φ(a + 1))

for → 0. Thus we have vx 2L2 (w) /v 2L2 (w ) σ 2 /4 + δ for a suitably small = (δ) > 0. 1 Consequently, we have from (3.11) that Lv , v L2 (w) v 2L2 (w

1)

−

vx 2L2 (w)

+ c1 (σ ) − δ v 2L2 (w ) 1 − σ 2 /4 + δ + c1 (σ ) − δ δ.

This completes the proof of the non-dissipative case (2). Thus the proof of Theorem 3.3 is complete. 2 Finally in this section, we consider the special case where g(u) ≡ 0 so that f (u) = q1 (−u)q+1 . In this case the degenerate stationary wave is given explicitly by φ(x) = −(1 + x)−1/q and the operator L in (1.5) is reduced to L0 below: L0 v = vxx +

v q +1 . q 1+x x

(3.12)

For this simplest case, we have the complete characterization of the dissipativity of the operator L0 . Theorem 3.5. Let αc (q) := 3 + 2/q. Then we have the complete characterization of the dissipativity of the operator L0 given in (3.12): (1) Let −1 < α < αc (q). Then L0 is uniformly dissipative in L2α . Namely, there is a positive constant δ such that L0 v, v L2α −δ vx 2L2 + v2L2 α

(3.13)

α−2

1 . for v ∈ Hα,0 (2) Let α = αc (q) or α = −1. Then L0 is strictly dissipative in L2α . Namely, we have 1 with v = 0. L0 v, v L2α < 0 for v ∈ Hα,0 (3) Let α > αc (q) or α < −1. Then L0 cannot be dissipative in L2α . Namely, we have 1 with v = 0. L0 v, v L2α > 0 for some v ∈ Hα,0


13

Proof. Consider the case where f (u) = q1 (−u)q+1 with g(u) ≡ 0. In this case we have φ(x) = −(1 + x)−1/q and L = L0 . Moreover, noting that r(u) ≡ 0, we have as a counterpart of (3.7), L0 v, v L2 (w) = −vx 2L2 (w) + c1 (σ )v2L2 (w ) , 1

(3.14)

where w and w1 are the weight functions defined in (2.9) with σ ∈ R, and c1 (σ ) is given in (3.6). In our special case, these weight functions are given explicitly by (2.12), so that we have L0 v, v L2 (w) = q L0 v, v L2

σ/q+1

vx L2 (w) = qvx 2L2

σ/q+1

,

,

vL2 (w1 ) =

1 v2L2 . q σ/q−1

(3.15)

Now we put σ = (α − 1)q. First, let −1 < α < αc (q). This corresponds to the case where −2q < σ < 2(q + 1), for which we have c1 (σ ) < σ 2 /4. Therefore, applying to (3.14) the same arguments as in (1) of Theorem 3.3, we obtain L0 v, v L2 (w) −δ vx 2L2 (w) + v2L2 (w ) 1

for some δ > 0. This inequality together with the relations in (3.15) (with σ = (α − 1)q) shows the uniform dissipativity of L0 in L2α . Second, let α = αc (q) or α = −1. This corresponds to the case where σ = 2(q + 1) or σ = −2q. In this case we have c1 (σ ) = σ 2 /4. On the other hand, we have (σ 2 /4)v2L2 (w ) vx 2L2 (w) by the Hardy type inequality (2.10). Consequently, we get from (3.14) that

1

L0 v, v L2 (w) 0. Here the equality holds if and only if (σ 2 /4)v2L2 (w

1)

= vx 2L2 (w) . However, we know

from Proposition 2.3 that such a v = 0 does not exist in H01 (w). Thus we conclude that L0 v, v L2 (w) < 0 for v ∈ H01 (w) with v = 0, which together with (3.15) (with σ = (α − 1)q) proves the strict dissipativity of L0 in L2α . Finally, let α > αc (q) or α < −1. Then we have σ > 2(q + 1) or σ < −2q and hence c1 (σ ) > σ 2 /4. Therefore, applying to (3.14) the same arguments as in (2) of Theorem 3.3, we find that L0 v, v L2 (w) > 0 for some v ∈ H01 (w) with v = 0. This together with (3.15) (with σ = (α − 1)q) gives the desired conclusion of (3) of Theorem 3.5. This completes the proof. 2 4. Nonlinear stability The aim of this section is to prove the following stability result for the nonlinear problem (1.4) that is a refinement of the result in [15]. Theorem 4.1. Assume (1.2). Suppose that v0 ∈ L2α ∩ L∞ for some α with 1 α < αc (q) := 3 + q/2. Then there is a positive constant δ1 such that if v0 L2 δ1 , then the problem (1.4) has 1

a unique global solution v ∈ C 0 ([0, ∞); L2α ∩ Lp ) for each p with 2 p < ∞. Moreover, the solution verifies the decay estimate v(t) p C v0 2 + v0 L∞ (1 + t)−α/4−ν (4.1) Lα L for t 0, where 2 p < ∞, ν = (1/2)(1/2 − 1/p), and C is a positive constant.

14


A key to the proof of Theorem 4.1 is to show the following space–time weighted energy inequality. Proposition 4.2. Assume the same conditions as in Theorem 4.1. Let v be a solution to the problem (1.4) with the initial data v0 ∈ L2α ∩ L∞ , where 1 α < αc (q) := 3 + 2/q. Then there is a positive constant δ2 such that if v0 L2 δ2 , then we have 1

v(t)

L21

Cv0 L2 .

(4.2)

1

Moreover, we have the following space–time weighted energy inequality: 2 v(t)L2 +

γ

(1 + t)

t

β

2 2 (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t Cv0 2L2 β

+γC

2 (1 + τ )γ −1 v(τ )L2 dτ

(4.3)

β

0

for any γ 0 and β with 0 β α, where the constant C is independent of γ and β. Proof. The main part of the proof of this proposition is to derive the following space–time weighted energy inequality: 2 v(t)L2 +

γ

(1 + t)

t

β

2 2 (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t Cv0 2L2 β

+γC

2 γ (1 + τ )γ −1 v(τ )L2 dτ + CSβ (t)

(4.4)

β

0

for any γ 0 and β with 0 β α, where 1 α < αc (q) := 3 + 2/q, C is a constant independent of γ and β, and

γ Sβ (t) =

t

3 (1 + τ )γ v(τ )L3

(4.5)

dτ.

β−1

0

Once (4.4) is obtained, we can show the desired estimates (4.2) and (4.3) as follows. We observe that γ Sβ (t) C

t

2 2 (1 + τ )γ v(τ )L2 vx (τ )L2 + v(τ )L2 dτ, 1

0

β

β−2

(4.6)


15

which is an easy consequence of the following inequality (see [15] for the details): v3L3

β−1

CvL2 vL2 1

β−2

vx L2 + vL2 , β

β−2

where β ∈ R. Now we put γ = 0 and β = 1 in (4.4) and define V (t) 0 by 2 V (t) = sup v(τ )L2 +

t

2

0τ t

1

vx (τ )2 2 + v(τ )2 2 dτ. L L 1

−1

0

Since S10 (t) CV (t)3 by (4.6), we get the inequality V (t)2 Cv0 2 2 + CV (t)3 . This can be L1

solved as V (t) Cv0 L2 , provided that v0 L2 is suitably small. Thus we obtain 1

1

v(t)2 2 +

t

L1

vx (τ )2 2 + v(τ )2 2 dτ Cv0 2 2 , L L L −1

1

(4.7)

1

0

which gives the uniform estimate (4.2). Consequently, we have t Sγβ (t) Cv0 L2 1

2 2 (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ. β

β−2

0

Substituting this estimate into (4.4) and assuming that v0 L2 is suitably small (say, v0 L2 δ2 ), 1 1 we arrive at the desired energy inequality (4.3). It remains to prove the inequality (4.4). To this end, we recall the following uniform estimate: v(t)

L∞

M∞ ,

(4.8)

where M∞ = v0 L∞ + 2. This is an easy consequence of the maximum principle (see [5] for the details). Proof of (4.4) for β = 0. The proof is based on the time weighted L2 energy method. We multiply Eq. (1.4) by v. This yields

1 2 v 2

+ (F − vvx )x + vx2 + G = 0,

(4.9)

t

where F = f (φ + v) − f (φ) v −

v

f (φ + η) − f (φ) dη,

0

v G= 0

f (φ + η) − f (φ) dη · φx .

(4.10)

16


We note that 1 F = f (φ)v 2 + O |v|3 , 2

1 G = f (φ)φx v 2 + φx O |v|3 2

(4.11)

for v → 0. Also, we observe that 1 (−φ)q+1 1 + O |φ| , q f (φ) = (q + 1)(−φ)q−1 1 + O |φ|

φx = f (φ) =

for |φ| → 0 and that f (φ) > 0 by (1.2). Therefore, noting (3.2) and (4.8), we have from (4.11) that G c(1 + x)−2 v 2 − C(1 + x)−1−1/q |v|3

(4.12)

for any x ∈ R+ . We integrate (4.9) over R+ and substitute (4.12) into the resulting equality. This gives 1 d v2L2 + vx 2L2 + cv2L2 Cv3L3 . 2 dt −2 −1 We multiply this inequality by (1 + t)γ and integrate with respect t. This yields the desired inequality (4.4) for β = 0. Proof of (4.4) for β > 0. We apply the space–time weighted energy method employed in [15] (see also [3]). Let w > 0 be a smooth weight function depending only on x, which will be specified later. We multiply (4.9) by w, obtaining

1 2 v w 2

t

1 2 1 2 2 v wxx + F wx − Gw = 0. + (F − μvvx )w + v wx + vx w − 2 2 x

(4.13)

Here, using (4.11), we have 1 2 1 v wxx + F wx − Gw = v 2 wxx + wx f (φ) − wf (φ)φx + R, 2 2

(4.14)

where R = wx O(|v|3 ) − wφx O(|v|3 ) for v → 0. Notice that the coefficient wxx + wx f (φ) − wf (φ)φx in (4.14) is just the same as that appeared in (3.4). Now we choose the weight function w and the corresponding w1 by (2.9) with σ = (β − 1)q, where 0 β α and 1 α < αc (q) := 3 + 2/q. Then we have (3.5) with σ = (β − 1)q. Substituting these expressions into (4.13) and integrating over R+ , we obtain 1 d v2L2 (w) + vx 2L2 (w) − c1 (σ )v2L2 (w ) + 1 2 dt

∞

∞ v r(φ)w1 dx = 2

0

R dx, 0

(4.15)


17

where c1 (σ ) and r(φ) are given in (3.6) with σ = (β − 1)q. Here our weight functions in (2.9) are w = q(−φ)−σ −q / 1 + g(φ) ,

w1 =

1 (−φ)−σ +q 1 + g(φ) . q

Therefore, noting (3.2), we see that w ∼ (1 + x)σ/q+1 = (1 + x)β ,

w1 ∼ (1 + x)σ/q−1 = (1 + x)β−2 ,

(4.16)

where the symbol ∼ means the equivalence. This implies that the norms · L2 (w) and · L2 (w1 ) are equivalent to · L2 and · L2 , respectively. β β−2 We estimate (4.15) similarly as in (1) of Theorem 3.3. To this end, we note that σ1 σ σ2 , where σ1 = −q and σ2 = (α − 1)q. Since c1 (σ ) < σ 2 /4 for −2q < σ < 2(q + 1) and since −2q < σ1 < σ2 < 2(q + 1), we can choose δ > 0 so small that δ

min

σ1 σ σ2

σ 2 /4 − c1 (σ ) . 2 + σ 2 /4

Notice that this δ is independent of β. For this choice of δ, we take a = a(δ) > 0 so large that |r(φ)| δ for x a. Then we have ∞ 2 v r(φ)w1 dx δv2L2 (w1 ) + Cv2L2 , −2 0

where C is a constant satisfying C (1 + x)2 |r(φ)|w1 for 0 x a. Also, using the Hardy type inequality (σ 2 /4)v2L2 (w ) vx 2L2 (w) in (2.10), we have 1

vx 2L2 (w) − c1 (σ )v2L2 (w ) = δvx 2L2 (w) + (1 − δ)vx 2L2 (w) − c1 (σ )v2L2 (w ) 1 1 2 2 2 δvx L2 (w) + (1 − δ)σ /4 − c1 (σ ) vL2 (w ) 1

δvx 2L2 (w)

+ 2δv2L2 (w ) , 1

where we used the fact that (1 − δ)σ 2 /4 − c1 (σ ) 2δ. On the other hand, using (4.8), we see that |R| C(|wx | + wφx )|v|3 . Moreover, a straightforward computation shows that |wx | + wφx C(1 + x)β−1 . Substituting all these estimates into (4.15), we obtain 1 d v2L2 (w) + δ vx 2L2 (w) + v2L2 (w ) Cv2L2 + Cv3L3 , 1 2 dt −2 β−1

(4.17)

where δ and C are independent of β. We multiply this inequality by (1 + t)γ and integrate with respect to t. By virtue of (4.16), we have

18


t

2 v(t)L2 +

γ

(1 + t)

β

2 2 (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t Cv0 2L2 β

+γC

2 (1 + τ )γ −1 v(τ )L2 dτ β

0

t +C

2 γ (1 + τ )γ v(τ )L2 dτ + CSβ (t),

(4.18)

−2

0

where the constant C is independent of γ and β. Here the third term on the right-hand side of (4.18) was already estimated by (4.4) with β = 0. Hence we have proved (4.4) also for 0 < β α. This completes the proof of Proposition 4.2. 2 Proof of Theorem 4.1. The proof is essentially the same as that of Theorem 3.3 of [15] so that we only give an outline of the proof of the decay estimate (4.1). First, we show the following L2 decay estimate by applying the induction argument to the space–time weighted energy inequality (4.3). 2 (1 + t) v(t)L2

j

α−2j

t +

2 (1 + τ )j vx (τ )L2

α−2j

2 + v(τ )L2

α−2j −2

dτ Cv0 2L2

α

(4.19)

0

for each integer j with 0 j [α/2]. To see this, we put γ = j and β = α − 2j in (4.3), obtaining t

2 (1 + t) v(t)L2

j

α−2j

+

2 (1 + τ )j vx (τ )L2

α−2j

2 + v(τ )L2

α−2j −2

dτ

0

t Cv0 2L2 α−2j

+ jC

2 (1 + τ )j −1 v(τ )L2

(4.20)

dτ.

α−2j

0

Then, applying the induction with respect to the integer j with 0 j [α/2], we obtain (4.19). On the other hand, when α/2 is not an integer, we have 2 (1 + t)γ v(t)L2 +

t

2 2 (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ −2

0

Cv0 2L2 (1 + t)γ −α/2 . α

(4.21)

for any γ with γ > α/2. This can be shown by using (4.3) with γ > α/2, β = 0 and (4.19) with j = [α/2] together with Nishikawa’s technique in [12]. For the details, see the proof of Proposition 2.6 of [15].


Thus we have shown the following L2 decay estimate: v(t) 2 Cv0 2 (1 + t)−α/4 . Lα L

19

(4.22)

The desired Lp decay estimate (4.1) is then obtained by the time weighted Lp energy method. In fact, under the additional smallness condition on vL2 , we have 1

p v(t)Lp +

γ

(1 + t)

t

p 2 (1 + τ )γ |v|p/2 x (τ )L2 + v(τ )Lp dτ −2

0

Cv0 Lp + Cv0 L2 (1 + t)γ −(α/4+ν)p , p

p

(4.23)

α

where 2 p < ∞, ν = (1/2)(1/2 − 1/p) and γ > (α/4 + ν)p. We omit the details and refer the reader to the proof of Theorem 2.3 of [15]. This completes the proof of Theorem 4.1. 2 References [1] G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920) 314–317. [2] Y. Kagei, S. Kawashima, Stability of planar stationary solutions to the compressible Navier–Stokes equation on the half space, Comm. Math. Phys. 266 (2006) 401–430. [3] S. Kawashima, A. Matsumura, Asymptotic stability of traveling waves solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127. [4] S. Kawashima, S. Nishibata, M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane, Discrete Contin. Dyn. Syst. Suppl. (2003) 469–476. [5] S. Kawashima, S. Nishibata, M. Nishikawa, Lp energy method for multi-dimensional viscous conservation laws and application to the stability of planar waves, J. Hyperbolic Differ. Equ. 1 (2004) 581–603. [6] S. Kawashima, S. Nishibata, M. Nishikawa, Asymptotic stability of stationary waves for multi-dimensional viscous conservation laws in half space, preprint, 2004. [7] S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible NavierStokes equations in the half space, Comm. Math. Phys. 240 (2003) 483–500. [8] E. Landau, A note on a theorem concerning series of positive terms, J. London Math. Soc. 1 (1926) 38–39. [9] T.-P. Liu, A. Matsumura, K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves, SIAM J. Math. Anal. 29 (1998) 293–308. [10] A. Matsumura, K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994) 83–96. [11] T. Nakamura, S. Nishibata, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier–Stokes equation in a half line, J. Differential Equations 241 (1) (2007) 94–111. [12] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac. 41 (1998) 107–132. [13] B. Opic, A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes in Math. Ser., vol. 219, Longman Scientific & Technical, 1990. [14] Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl. 18 (1) (2008) 329–343. [15] Y. Ueda, T. Nakamura, S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., in press. [16] Y. Ueda, T. Nakamura, S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinetic Related Models 1 (2008) 49–64.


The planar algebra of group-type subfactors ✩ Dietmar Bisch, Paramita Das, Shamindra Kumar Ghosh ∗ Vanderbilt University, Department of Mathematics, SC 1326, Nashville, TN 37240, USA Received 11 August 2008; accepted 25 March 2009

Communicated by D. Voiculescu

Abstract If G is a countable, discrete group generated by two finite subgroups H and K and P is a II1 factor with an outer G-action, one can construct the group-type subfactor P H ⊂ P K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones’ planar operad is determined explicitly. © 2009 Elsevier Inc. All rights reserved. Keywords: Planar algebra; Planar operad; Subfactor; IRF model

1. Introduction The technique of composing subfactors pioneered in [4] led to a zoo of exotic subfactors of the hyperfinite II1 factor. In particular, the first examples of irreducible, amenable subfactors that are not strongly amenable (in the sense of [16]) were constructed in this way in [4]. The idea is simple: Let H and K be two finite groups with an outer action on a II1 factor P (e.g. the hyperfinite II1 factor) and consider the composition of the fixed-point subfactor P H ⊂ P with the crossed product subfactor P ⊂ P K to obtain, what we will call here, a group-type subfactor P H ⊂ P K. If P is hyperfinite, analytical properties of this subfactor, such as amenability ✩

The authors were supported by NSF under Grant No. DMS-0301173.


E-mail addresses: [email protected] (D. Bisch), [email protected] (P. Das), [email protected] (S.K. Ghosh). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.014

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

21

and property (T) in the sense of Popa [16,18], were proved to be equivalent to the corresponding properties (amenability, property (T) in the sense of Kazhdan) of the group G generated by H and K in the outer automorphism group of P [4,7]. Note that a group-type subfactor is obtained from the two fixed-point subfactors P H ⊂ P and P K ⊂ P by performing the basic construction of [10] to one of them. A group-type subfactor is therefore an invariant for the relative position of the two fixed-point subfactors. For more on this, see [12]. The main invariant for a subfactor is the so-called standard invariant (see for instance [9,8,13]). It is a very sophisticated mathematical object that can be portrayed in a number of seemingly quite different ways. For example, it has descriptions as a certain category of bimodules ([14], see also [3]), as a lattice of algebras [17], or as a planar algebra [11]. Jones’ planar algebra technology has become a very efficient tool to capture and analyze the standard invariant of a subfactor. Composition of subfactors was the motivating idea that led to the results in [5]. It was proved there that two standard invariants without extra structure, i.e. consisting of just the Temperley– Lieb algebras, can always be composed freely to form a new standard invariant – namely the one generated by the Fuss–Catalan algebras of [5]. This concept of free composition was then pushed much further in [6], where it is shown that any two planar algebras arising from subfactors can be composed freely to form a new subfactor planar algebra. The principal graphs of a subfactor encode the algebraic information contained in the standard invariant [9], and their structure determines the growth properties of the invariant. It was shown in [4] that subfactors whose standard invariants have very exotic growth properties exist by constructing concrete group-type subfactors. The principal graphs of these subfactors were computed there. In this paper we go a step further and give a concrete description of the standard invariant (or equivalently the planar algebra) of the group-type subfactors. We concentrate on the case when the group G, generated by H and K in the automorphism group of P , has an outer action on P . Note that any group G generated by two finite subgroups H and K has such an action on the hyperfinite II1 factor (for instance by a Bernoulli shift action). We find a description of the planar algebra of these group-type subfactors that is reminiscent of an IRF (interaction round a face) model in statistical mechanics. The general case, where G is generated by H and K in the outer automorphism group of P , and hence may or may not lift to an action of G on P , is much more elaborate and will be treated in a separate paper. Here is a more detailed outline of the sections of this paper. We review in Section 2 the basic notions of planar algebras. In Section 3, we define an abstract planar algebra P associated to a countable, discrete group G and two of its finite subgroups H and K which generate G. The vector spaces underlying the planar algebra are spanned by alternating words in H and K that multiply to the identity element. The action of Jones’ planar operad is given explicitly by a particular labelling of planar tangles that can be viewed as an IRF-like model. It takes some work to show that the action of planar tangles is well defined and preserves composition. The latter is achieved by showing that the action respects composition with certain elementary tangles that generate any annular tangle. We then analyze the filtered ∗-algebra structure of P and determine the action of Jones projection and conditional expectation tangles. In Section 4 we compute the basic construction and higher relative commutants of a grouptype subfactor P H ⊂ P K. We exhibit a nice basis which is used in Section 5 to prove that the abstract group-type planar algebra of Section 3 is indeed isomorphic to the concrete one computed in Section 4. Moreover, we prove in Section 5 that if the standard invariant of an

22


arbitrary subfactor is isomorphic to a group-type planar algebra, then the subfactor is indeed of group-type. This is proved using results on intermediate subfactors from [2,6,1]. 2. Planar algebra basics In this section, we will give a very brief overview of planar algebras; the reader is encouraged to see [11] for details. Let us first describe the key ingredients that constitute a planar tangle. • There is an external disc, several (possibly none) internal discs and a collection of disjoint smooth curves. • To each disc – internal or external, we attach a non-negative integer. This integer will be referred to as the ‘color’ of the disc. If a disc has color k > 0, there will be 2k points on the boundary of the disc marked 1, 2, . . . , 2k counted clockwise, starting with a distinguished marked point, which is decorated with ‘∗’. A disc having color 0 will have no marked points on its boundary. • Each of the curves is either closed, or joins a marked point on the boundary of a disc to another such point, meeting the boundary of the disc transversally. Each marked point must be the endpoint of exactly one curve. • The whole picture has to be planar, in the sense that there should be no crossing of curves or overlapping of discs. • Finally, we will not distinguish between pictures obtained from one another by planar isotopy preserving the ∗’s. The data of such a picture will be termed as a planar k-tangle, where k refers to the color of its external disc. Remark 2.1. We can induce a black-and-white shading on the complement of the union of the internal discs and curves in the external disc by specifying that the region between the last and first marked points be left unshaded. This leaves a scope for ambiguity in the case of 0-discs, thus, 0-discs may be of two types depending on whether the region surrounding their boundary is shaded or unshaded. Given a planar k-tangle T – one of whose internal discs have color ki – and a ki -tangle S, one can define the k-tangle T ◦i S by isotoping S so that its boundary, together with the marked points and the ‘∗’ coincides with that of Di and then erasing the boundary of Di . The collection of tangles – along with the composition defined thus – is called the colored operad of planar tangles. A planar algebra is a collection of vector spaces {Pk }k0 such that every k-tangle T that has b internal discs D1 , D2 , . . . , Db having colors k1 , k2 , . . . , kb respectively gives rise to a multilinear map ZT : Pn1 × Pn2 × · · · × Pnb → Pn0 . The collection of maps is required to be compatible with substitution of tangles and renumbering of internal discs. 3. An abstract planar algebra In this section we will abstractly define a planar algebra which will be identified in Section 5 to be the one corresponding to the group-type subfactor P H ⊂ P K of [4].


23

Fig. 1. Example of faces in a tangle.

Let G be a group generated by two of its finite subgroups H and K and e denote the identity element of G. Let us define ⎧ if n = 0, ⎨ {e} K × H × K × H × · · · Sn = if n 1, ⎩ S=

(n factors)

Sn ,

n0

Ln =

K, H,

if n is even, otherwise.

Let μ : S → G be the multiplication map. With the above notation, we are ready to define the planar algebra but first we would need some terminology. Terminology. By a face in an unlabelled tangle T , we will mean a connected component of D0 \ [( bi=1 Di ) ∪ S] where D0 is the external disc, Di is the ith internal disc for i = 1, 2, . . . , b and S is the set of strings of (an element in the isotopy class of) T . By an opening in a tangle, we will mean the subset of the boundary of a disc lying between two consecutive marked points. An opening will be called internal (resp., external) if it is a subset of the boundary of the internal (resp., external) disc. Note that the boundary of a generic face may be disconnected due to the presence of loops or networks inside it (see Fig. 1). The set of connected components of the boundary of each face will have a single outer component and several (possibly none) inner component(s). Definition 3.1. A state f on a tangle T is a function f : {all openings in T } → H K such that following holds:

(i)

f (α) ∈

K, H,

if the face containing α is shaded, otherwise.

(ii) Triviality on the outer component of the boundary of a face: Let α1 , α2 , . . . , αm be the openings on the outer component of the boundary of a face counted clockwise, then we must have

24


Fig. 2. An example of a state on an unlabelled tangle.

f (α1 )η1 f (α2 )η2 · · · f (αm )ηm = e where

ηi =

+1, if αi is an external opening, −1, otherwise.

(iii) Triviality on internal discs: f induces a map ∂f : {D0 , D1 , . . . , Db } → n0 Sn defined by

(i)

(i) (i) ∂f (Di ) = f α1 , f α2 , . . . , f α2ni ∈ S2ni (i)

(i)

(i)

(i)

where α1 , α2 , . . . , α2ni are consecutive openings counted clockwise such that α1 is the opening between the first and the second marked points of ∂Di . We demand that μ(∂f (Di )) = e, for all i = 1, 2, . . . , b. Note that triviality on the external disc and every boundary component of every face follows (see Remark 3.2 for a proof of this fact). The above definition also applies to networks (positively or negatively oriented). Fig. 2 illustrates conditions (ii) and (iii) of Definition 3.1; triviality on internal discs gives b1 b2 b3 b4 = c1 c2 c3 c4 = d1 d2 d3 d4 d5 d6 d7 d8 = f1 f2 = e and triviality on the outer component of the boundaries of faces gives a1 b1−1 = a2 b2−1 = a3 d5−1 d1−1 b3−1 = c4−1 f2−1 = c2 = d2−1 d4−1 = d3 = a4 a6 b4−1 d8−1 d6−1 = d7 = a5 = e. Note that the above relations also imply a1 a2 a3 a4 a5 a6 = e, and c1−1 c3−1 f1−1 = e.


25

As the computation involving Fig. 2 suggests, triviality on inner boundaries of a face and on the external disc are actually consequences of the definition of a state. This is made precise in the following remark. Remark 3.2. Let f be a state on a tangle or a network. Then the following conditions hold: (ii) Triviality on every inner component of the boundary of a face: For every inner component of the boundary of a face with openings α1 , α2 , . . . , αm counted clockwise, we have f (α1 )f (α2 ) · · · f (αm ) = e. (iii) Triviality on the external disc: This is just a restatement of condition (iii) in Definition 3.1 applied to the external disc only if its color is greater than zero. We prove the above remark using planar graphs. Without loss of generality, we may start with a tangle or a network which is connected. By a network or a tangle (with non-zero color on its external disc) being connected, we mean that the union of the boundaries of all the discs and strings is a connected set; a 0-tangle is said to be connected if the network obtained after removing its external disc is connected. To each tangle or network, we associate a planar graph whose vertex set is the set of all marked points on the internal and external discs, and the edges are the openings and the strings. Further, we make this graph a directed one in such a way that the directions on the edges arising from the openings are induced by clockwise orientation on the boundary of the discs, and the remaining edges (coming from the strings) are free to have any direction. Any state f assigns group elements to edges arising from openings; we label each of the remaining edges by e and that is why we did not put any restriction on the direction of such edges. Note that the definition of the state implies the following condition on the group labelled planar directed graph: Triviality on the boundary each bounded face of the graph1 : If g1 , g2 , . . . , gm are the group elements assigned to consecutive edges around a face read clockwise, then η

η

ηm =e g1 1 g2 2 · · · gm

where

ηi =

+1, −1,

if ith edge induces clockwise orientation in the face, otherwise.

To establish Remark 3.2, it is enough to prove triviality on the boundary of the unbounded face. To see this, we first consider a pair of bounded faces which have at least one vertex or edge in common. If this pair is considered as a separate graph, then using triviality on each face, it is easy to check triviality on the boundary of the unbounded face of this pair. One can then use this fact inductively to deduce the desired result for the whole graph. Next, we give the definition of the planar algebra. Let the set of states on a tangle T be denoted by S(T ). 1 Since we started with a connected tangle or network, the associated planar graph will be connected; in particular, boundary of each face of the graph will be connected.

26


Fig. 3. Assigning scalars to maxima or minima.

The vector spaces. For n 0, define Pn = C{s ∈ S2n : μ(s) = e}. Action of tangles. Let T be an unlabelled tangle with (possibly zero) internal disc(s) D1 , D2 , . . . , Db and external disc D0 where the color of Di is ni . Then T defines a multilinear map, denoted by ZT : Pn1 × Pn2 × · · · × Pnb → Pn0 . We will define ZT (s1 , s2 , . . . , sb ) ∈ Pn0 , where si ∈ S2ni such that μ(si ) = e. In fact, we will just prescribe the coefficient of s0 ∈ S2n0 (such that μ(s0 ) = e) in the expansion of ZT (s1 , s2 , . . . , sb ) in terms of the canonical basis mentioned above. We choose and fix a representative in the isotopy class of T and call it the standard form of T , denoted by T˜ . It is assumed to satisfy the following properties: • T˜ is in rectangular form – meaning that – all of its discs are replaced by boxes and it is placed in R2 in such a way that the boundaries of the boxes are parallel to the co-ordinate axes. • The first marked point on the boundary of each box is on the top left corner. • The collection of strings have finitely many local maxima or minima. • The external box can be sliced by horizontal lines in such a way that each maxima, minima, internal box is in a different slice. To each local maximum or minimum of a string with end-points, we assign a scalar according to Fig. 3. Let p(T ) denote the product of all scalars arising from the local maxima or minima and n+ (T ) (resp., n− (T )) be the number of non-empty connected positively (resp., negatively) oriented network(s) in the tangle T˜ . Then, the coefficient ZT (s1 , s2 , . . . , sb )|s0 of s0 in ZT (s1 , s2 , . . . , sb ) is given by p(T )|H |n+ (T ) |K|n− (T ) f ∈ S(T ): ∂f (Di ) = si for all i = 0, 1, . . . , b . Note that there could be several standard form representatives for T . However, one standard form representative for T can be transformed into another by a finite sequence of moves of the following three types:


27

I. Horizontal or vertical sliding of boxes. II. Wiggling of the strings. III. Rotation of an internal box by a multiple of 360◦ . It is easy to check that the above three moves do not alter the number of connected networks and keep |{f ∈ S(T ): ∂f (Di ) = si for all i = 0, 1, . . . , b}| unaltered. So, it remains to show that p(T ) is unchanged under the moves. Type I moves are the easiest because they do not generate any new local maxima or minima. In each of the moves of type II and III, there arises pair(s) of local maximum and minimum in such a way that the two scalars assigned to each pair are inverses of each other; as a result, p(T ) remains unchanged. Action of the tangles preserve composition. For S an n0 -tangle with internal discs D1 , D2 , . . . , Db having colors n1 , n2 , . . . , nb respectively, and T an nj -tangle for some j ∈ {1, 2, . . . , b}, we would like to show ZS◦Dj T = ZS ◦ (idPn1 × · · · ZT · · · × idPnb ). For this, we first identify a set E of annular tangles (with the distinguished internal disc as D1 ) which we call elementary tangles, namely: (i) Capping tangles:

with col(D1 ) = n 1, col(D0 ) = n − 1, and 1 i 2n − 1 (col(D1 ) = 1 just means that there are no strings connecting the internal disc to the external disc). (ii) Cap inclusion tangles:

with col(D1 ) = n 0, col(D0 ) = n + 1, and 1 i 2n + 1.

28


If i = 1 in (i) or (ii) then the pictures should be interpreted as simply not having the bunch of (i − 1) straight strings joining the marked points of the internal disc and the corresponding points of the external disc. (iii) Left inclusion tangles:

with col(D1 ) = n 0, col(D0 ) = n + 2. (iv) Disc inclusion tangle:

(iv) Disc inclusion tangle:

with col(D1 ) = col(D0 ) = n 0, col(D2 ) = q, where p 0, q 0, r 0 such that p + q + r = n and p is even. Note that any annular tangle can be expressed as a composition of the elementary tangles. To see this, express the annular tangle in standard form and cut it into horizontal strips each of which contains at most one internal disc or one local maxima or minima. Now, the strip containing the distinguished internal disc inside the annular tangle can be obtained by composition of elementary tangles of type (iii) and type (ii) (more specifically, the inclusion tangles); one can then glue the other strips consecutively one after the other along the lines of cutting to get back the original tangle. Each such gluing operation is given by composition of an elementary tangle of type (i), (ii), (iv) or (iv) . So, to prove that the action of the tangles preserve composition, it is enough to prove ZE◦D1 T = ZE ◦ ZT (resp., ZE◦D1 T = ZE ◦ (ZT × idPq )) for any tangle T and any E ∈ E of type (i), (ii) or (iii) (resp. (iv) or (iv) ) whenever the composition makes sense.


29

We fix an n-tangle T with internal discs D1 , D2 , . . . , Db with colors n1 , n2 , . . . , nb respectively, and an n0 -tangle E ∈ E such that both T and E are in standard forms and color of D1 in E is n. Let us consider the standard form on E ◦D1 T induced by the standard forms of E and T . Our goal is to show: ZE◦D1 T (s1 , s2 , . . . , sb )|s0 =

ZE (s)|s0 ZT (s1 , s2 , . . . , sb )|s

s∈S2n s.t. μ(s)=e

if E is of type (i), (ii) or (iii), and ZE◦D1 T (s1 , s2 , . . . , sb , t)|s0 =

ZE (s, t)|s0 ZT (s1 , s2 , . . . , sb )|s


if E is of type (iv) or (iv) where sj ∈ S2nj for 0 j b and t ∈ Sq such that μ(sj ) = e = μ(t) for all j . An interesting situation arises when we pick elementary tangles of type (i), since composition of tangles in this case may lead to a change in the number of connected networks. The reasoning in the other cases is either similar or straightforward. For E being type (i) elementary tangle, the above equation is equivalent to: ⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ p(E ◦D1 T )|H |n+ (E◦D1 T ) |K|n− (E◦D1 T ) f ∈ S(E ◦D1 T ) for 1 j b, ∂f (D0 ) = s0 ⎭ ⎩ = p(E)p(T )|H |n+ (E)+n+ (T ) |K|n− (E)+n− (T ) ×


f ∈ S(E) ∂f (D1 ) = s ∂f (D0 ) = s0

⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ f ∈ S(T ) for 1 j b, ⎩ ∂f (D ) = s ⎭ 0

where D0 denotes the external disc of T . First, observe that p(E ◦D1 T ) = p(E)p(T ). To show the equality of the remaining scalars, we consider the following two cases. Case 1. The string which connects the ith and the (i + 1)th points on D1 in E, does not produce any new network in E ◦D1 T other than those that are already present in T . Clearly, n (E ◦D1 T ) = n (T ) (since no new network appears in E ◦D1 T ) and n (E) = 0 for ∈ {+, −}. A typical example of such a case can be viewed in the following picture where we label the openings on the internal discs of T by group elements coming from the coordinates of sj for 1 j b, and the openings on D0 of E by coordinates of s0 = (g1 , g2 , . . . , g2n−2 ). Using triviality on the boundary of faces in the definition of a state, we get

f ∈ S(E) ∂f (D1 ) = s = 0 ⇔ f ∈ S(E) ∂f (D1 ) = s = 1 ∂f (D0 ) = s0 ∂f (D0 ) = s0

⇔ s = g1 , . . . , gi−2 , (gi−1 g), e, g −1 , gi , . . . , g2n−2 ∈ S2n for some g ∈ Li .

30


Fig. 4.

Define s g = (g1 , . . . , gi−2 , (gi−1 g), e, g −1 , gi , . . . , g2n−2 ) for g ∈ Li . So, it is enough to check ⎫ ⎧ ⎫ ⎧ ∂f (Dj ) = sj ⎬ ∂f (Dj ) = sj ⎬ ⎨ ⎨ f ∈ S(T ) for 1 j b, . f ∈ S(E ◦D T ) for 1 j b, = 1 ⎩ ∂f (D ) = s g ⎭ ∂f (D0 ) = s0 ⎭ g∈Li ⎩ 0 Carefully observing Fig. 4 and using triviality on the boundary of faces once again, we get ⎧ ⎫ ⎨ ∂f (Dj ) = sj ⎬ f ∈ S(E ◦D T ) for 1 j b, = 0, equivalently, equals to 1 1 ⎩ ∂f (D0 ) = s0 ⎭ ⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ ⇒ f ∈ S(T ) for 1 j b, = δg,bη1 bη2 ···bηl l 1 2 ∂f (D ) = s g ⎭ ⎩ 0 where ηj = ±1 according as the corresponding opening is external or internal. Conversely, if ⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ f ∈ S(T ) for 1 j b, ⎩ ∂f (D ) = s g ⎭ g∈Li 0

η

η

η

is non-zero, then it has to equal to 1 since g must be b1 1 b2 2 · · · bl l by triviality on the boundary of faces in T ; from the unique state on T which makes the above sum non-zero, one can easily induce a well-defined state on E ◦D1 T , and hence ⎫ ⎧ ∂f (Dj ) = sj ⎬ ⎨ f ∈ S(E ◦D T ) for 1 j b, = 1. 1 ⎩ ∂f (D0 ) = s0 ⎭ This completes the proof of Case 1.


31

Fig. 5.

Case 2. The string which connects the ith and the (i + 1)th points on D1 in E, produces a new (connected) network in E ◦D1 T other than those that are already present in T . First, let us assume that the new network is positively oriented, equivalently, i is odd. Clearly, n− (E ◦D1 T ) = n− (T ) (since no new negatively oriented network appears in E ◦D1 T ), and n+ (E ◦D1 T ) = n+ (T ) + 1. Further, assume that col(D0 ) 1. In this case, n (E) = 0 for ∈ {+, −}. A typical example of this case can be viewed in the following picture where we label the openings on the internal discs of T by group elements coming from the coordinates of sj for 1 j b, and the openings on D0 of E by coordinates of s0 = (g1 , g2 , . . . , g2n−2 ). So, in this case, it is enough to check ⎧ ⎫ ⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ ∂f (Dj ) = sj ⎬ ⎨ f ∈ S(T ) for 1 j b, . |H | f ∈ S(E ◦D1 T ) for 1 j b, = ⎩ ⎩ ⎭ ⎭ ∂f (D ) = s g ∂f (D0 ) = s0 g∈H 0

If ⎧ ⎨

⎫ ∂f (Dj ) = sj ⎬ f ∈ S(E ◦D1 T ) for 1 j b, ⎩ ∂f (D0 ) = s0 ⎭ η

η

is non-empty (equivalently, singleton), from Fig. 5, we have a1 a2 · · · ak = e and gi−1 b1 1 b2 2 η · · · bl l = e where ηj = ±1 according as the corresponding opening is external or internal. For any g ∈ H , define f g by setting ∂f (Dj ) = sj for 1 j b and ∂f (D0 ) = s g . To check whether f g is a state, we consider the face in T appearing in Fig. 5; triviality on the boundary of this face η η η is given by the equation g −1 b1 1 b2 2 · · · bl l (gi−1 g)a1 a2 · · · ak = e which indeed holds. Triviality on all other discs or faces is induced by the existence of the state on E ◦D1 T . Thus, ⎧ ⎫ ∂f (Dj ) = sj ⎬ ⎨ f ∈ S(T ) for 1 j b, = 1 ⎩ ∂f (D ) = s g ⎭ 0

32


for all g ∈ H . Conversely, if the right-hand side is non-zero, then there exists g ∈ H such that η η η f g (defined earlier) is a state on T . Analyzing Fig. 5, we get g −1 b1 1 b2 2 · · · bl l (gi−1 g)ak−1 · · · a2−1 a1−1 = e. Note that the opening between the ith and the (i + 1)th points of the disc D0 of T , is assigned e. Now, if we consider the network appearing in Fig. 5 separately, then we have triviality on each of its internal faces and discs (induced by f g being a state); by Remark 3.2(ii) , we also have a1 a2 · · · ak = e (triviality on the internal boundary of the external face of the network). This η η η implies b1 1 b2 2 · · · bl l gi−1 = e; as a result, f h is a state for every h ∈ H . So, if right-hand side is η η η non-zero, then it has to be |H |; moreover, we get b1 1 b2 2 · · · bl l gi−1 = e which plays an important role in showing that ⎧ ⎫ ⎨ ∂f (Dj ) = sj ⎬ f ∈ S(E ◦D T ) for 1 j b, = 1 ⎩ 0 (equivalently, equals to 1). ⎭ ∂f (D0 ) = s0 This finishes the proof for the case where i is odd. For i even, the proof is exactly similar, except that one has to interchange |H | and |K|. The subcase that deserves separate treatment is when col(D0 ) = 0. In this case, n+ (E) = 1 and |{f ∈ S(E): ∂f (D1 ) = s}| = δs,(e,e) . Therefore, it is enough to show

f ∈ S(E ◦D T ) ∂f (Dj ) = sj 1 for 1 j b

⎧ ⎫ ∂f (Dj ) = sj ⎨ ⎬ = f ∈ S(T ) for 1 j b, . ⎩ ∂f (D ) = (e, e) ⎭ 0

The proof of the equality of the two sides is similar and is left to the reader. We now analyze the filtered ∗-algebra structure of P and the action of Jones projection tangles and conditional expectation tangles which will be useful in Section 5 to show that P is isomorphic to the planar algebra arising from a group-type subfactor. We start with laying some notations. Define ⎧ if n = 0, ⎨ {e} K × H × K if n 1, Sn = ·· · × H × ⎩ (n factors)

⎧ if n = 0, ⎨ {e} H × K × H × K × · · · if n 1. Tn = ⎩ (n factors)

Define : Sn → Sn by (s1 , s2 , . . . , sn )= (sn−1 , . . . , s2−1 , s1−1 ) for (s1 , s2 , . . . , sn ) ∈ Sn and let : Sn → Sn denote its inverse. Remark 3.3. We describe below the main structural features of the planar algebra P . (i) Identity:

e 1Pn =

s, e) s∈Sn−1 (s, e,

if n = 0, if n 1.


33

(ii) ∗-structure: Define ∗ on P by defining on the basis as

−1 −1 s ∗ = s2n−1 , . . . , s2−1 , s1−1 , s2n where s = (s1 , s2 , . . . , s2n ) ∈ S2n such that μ(s) = e for n 1, and then extend conjugate linearly. Clearly, ∗ is an involution. One also needs to verify whether the action of a tangle T preserves ∗, that is, ZT ∗ ◦ (∗ × · · · × ∗) = ∗ ◦ ZT ; in particular, ZT ∗ (s1∗ , . . . , sb∗ )|s0∗ =

ZT (s1 , . . . , sb )|s0 . It is enough to check this equation for the cases when T has no internal disc or closed loops, and when T is an elementary tangle. The actual verification in each of these cases is completely routine and is left to the reader. (iii) Multiplication: (a1 , l1 , b1 , h1 ) · (a2 , l2 , b2 , h2 ) = δb1 ,a2 (a1 , l1 l2 , b2 , h2 h1 ) where ai ∈ Sn−1 , bi ∈ S n−1 , li ∈ Ln−1 , hi ∈ H such that μ(ai , li , bi , hi ) = e for i = 1, 2 and n 1 (where we consider the elements ai and bi to be void in the case of n = 1). (iv) Inclusion:

Pn s →

(s1 , s2 , . . . , sn−1 , l1 , e, l2 , sn+1 , . . . , s2n ) ∈ Pn+1

l1 ,l2 ∈Ln−1 such that l1 l2 =sn

where s = (s1 , s2 , . . . , s2n ) ∈ S2n such that μ(s) = e for n 1. (v) Jones Projection Tangle: For P2 ,

=

|K| e, h, e, h−1 |H | h∈H

and for Pn+1 for n > 1,

=

|Ln−1| |Ln |

s ∈ Sn−2 l1 , l2 , l3 ∈ Ln s.t. l1 l2 l3 = e

(s, l1 , e, l2 , e, l3 , s˜ , e).

34


(vi) Conditional Expectation Tangle from Pn+1 onto Pn : For n 1, let s1 ∈ Sn−1 , s2 ∈ S n−1 , m1 , m2 ∈ Ln−1 , l ∈ Ln , h ∈ H such that μ(s1 , m1 m2 , s2 , h) = e. Then,

= δl,e

|Ln | (s1 , m1 m2 , s2 , h) |Ln−1|

and for n = 0,

= δh,e δk,e

|K| e. |H |

(vii) Conditional Expectation Tangle from Pn onto P1,n : For n 2 let k1 , k2 ∈ K, t ∈ T2n−3 , h ∈ H such that μ(k1 , t, k2 , h) = e. Then,

= δh,e

|H | |K|

k ,k ∈K s.t. k k =k2 k1

and for n = 1,

= δh,e δk,e

|H | e. |K|

(k , t, k , e)


35

4. Iterated basic construction and higher relative commutants of group-type subfactors Let P be a II1 factor and G a countable discrete group with an outer action on P , and suppose G is generated by two of its finite subgroups H and K. Consider the associated group-type subfactor P H ⊂ P K. In this section we will give a concrete realization of the Jones tower associated to this subfactor and compute its higher relative commutants. See also [19] for related results. First, let us recall the following characterization of the basic construction of a finite index subfactor [15]. Lemma 4.1. Let N ⊂ M be a finite index subfactor with E : M → N being the trace-preserving conditional expectation, B be a II1 factor containing M as a subfactor and f be a projection in B satisfying: (i) f xf = E(x)f for all x ∈ M, (ii) B is the algebra generated by M and f . Then, B is isomorphic to the basic construction M1 of N ⊂ M. Second, we recall some basic facts and notations for the crossed product construction. Unless otherwise specified, we will reserve the symbol e for the identity element of a group. The crossed product P K can be realized as the von Neumann subalgebra of L(l 2 (K) ⊗ L2 (P )) (∼ = MK (C) ⊗ L(L2 (P ))) generated by the images of P and K in the following way:

P x → Ek,k ⊗ k −1 (x) ∈ MK (C) ⊗ L L2 (P ) , (4.1) k∈K

K k → λk ⊗ 1 ∈ MK (C) ⊗ L L2 (P )

(4.2)

where we set the convention of considering k(x) as the element of P obtained by applying the automorphism k on x (in P ) and λk is the matrix in MK (C) corresponding to left multiplication by k. Consequently, the following commutation relation holds in P K: kxk −1 = k(x)

for all x ∈ P , k ∈ K.

(4.3)

However, P K can also be viewed as the vector space generated by elements of the form x k, k∈K k xk ∈ P , where the multiplication structure is given by the relation (4.3). The unique trace on P K is given by tr xk k = tr(xe ) k∈K

and the unique trace-preserving conditional expectation is given by P K EP xk k = xe . k∈K

If P K denotes the fixed point subalgebra of P , then P K is isomorphic to the basic construction of P K ⊂ P where the Jones projection is given by

36


e1 =

1 k∈P K |K| k∈K

implementing the conditional expectation: 1 k(x) ∈ P K |K|

EPP K (x) =

for all x ∈ P .

k∈K

The basic construction of P ⊂ P K is isomorphic to MK (C) ⊗ P where the inclusion P K → MK (C) ⊗ P is given by the maps (4.1), (4.2), and the corresponding Jones projection is given by e2 = Ee,e ⊗ 1 ∈ MK (C) ⊗ P implementing the conditional expectation EPP K . The next element in the tower of basic construction is given by MK (C) ⊗ (P K) where the inclusion MK (C) ⊗ P → MK (C) ⊗ (P K) is induced by the inclusion P ⊂ P K and the Jones projection is given by e3 =

1 ρk −1 ⊗ k ∈ MK (C) ⊗ (P K) |K|

(4.4)

k∈K

implementing the conditional expectation: M (C)⊗P

K EP K

(Ek1 ,k2 ⊗ x) =

1 k1 xk2 −1 ∈ P K |K|

for all x ∈ P , k1 , k2 ∈ K,

(4.5)

where ρk is the matrix in MK (C) corresponding to right multiplication by k. Coming back to the context of group-type subfactors we consider the unital inclusions P H → P K → MK (C) ⊗ (P H ) where the second inclusion factors through MK (C) ⊗ P in the obvious way. Lemma 4.2. MK (C) ⊗ (P H ) is the basic construction for P H ⊂ P K with Jones projection 1 e1 = Ee,e ⊗ |H | h∈H h. Proof. We need to show that conditions (i) and (ii) of Lemma 4.1 are satisfied. To show (i), let us assume that x˜ = k∈K xk k denotes a typical element of P K,

1 1 ˜ 1 = Ee,e ⊗ h x˜ Ee,e ⊗ h e1 xe |H | |H | h∈H h ∈H 1 1 −1 = Ee,e ⊗ h Ek ,k λk ⊗ k (xk ) Ee,e ⊗ h |H | |H | =

Ee,e λk Ee,e ⊗

k∈K

= Ee,e ⊗

1 |H |2

h∈H h ∈H

h ∈H

k ∈K k∈K

h∈H

1 |H |2

h∈H h ∈H

h(xe )hh

hxk h


37

whereas 1 −1 1 Ek,k ⊗ k h(xe ) Ee,e ⊗ h E(x)e ˜ 1= |H | |H | k∈K h∈H h ∈H 1 . h(x )h = Ee,e ⊗ e |H |2 h∈H

h ∈H

Therefore, LHS = RHS. To show (ii), it is enough to show that elements of the form Ek1 ,k2 ⊗ xh for x ∈ P , h ∈ H , Let us denote the Jones projection in P H k1 , k2 ∈ K are in the algebraic span of P K and e1 . corresponding to the inclusion P H ⊂ P by f = |H1 | h∈H h. Thus, e1 = Ee,e ⊗ f ∈ MK (C) ⊗ P H. This implies P e1 P = Ee,e ⊗ Pf P = Ee,e ⊗ P H ⊂ MK (C) ⊗ P H where P in the left-hand side is identified with its image inside MK (C) ⊗ P H (namely, the prescription given by (4.1)). Thus the algebraic span of P K and e1 contains elements of the type Ee,e ⊗ xh for x ∈ P , h ∈ H . To obtain elements of the form Ek1 ,k2 ⊗ xh, note that the relation λk1 Ee,e λk −1 = Ek1 ,k2 holds in MK (C). 2 2

Lemma 4.3. MK (C) ⊗ MH (C) ⊗ (P K) is the basic construction for P K ⊂ MK (C) ⊗ 1 (P H ) where the Jones projection is given by e2 = |K| k∈K ρk −1 ⊗ Ee,e ⊗ k. M (C)⊗(P H )

K Proof. The conditional expectation EP K

M (C)⊗P

K EP K

is the composition

M (C)⊗(P H )

◦ EMKK (C)⊗P

.

1 k1 xk2 −1 . Therefore, E(Ek1 ,k2 ⊗ xh) = δh,e |K| To show condition (i) of Lemma 4.1,

e2 (Ek1 ,k2 ⊗ xh)e2 =

1

ρk −1 ⊗ Ee,e ⊗ k

|K|2 k ∈K ρk −1 ⊗ Ee,e ⊗ k ×

Ek1 ,k2 ⊗ Eh ,h λh ⊗ h −1 (x)

h ∈H

k ∈K

= δh,e = δh,e

1 |K|2 1 |K|2

ρk −1 Ek1 ,k2 ρk −1 ⊗ Ee,e λh Ee,e ⊗ k xk

k ,k ∈K

k ,k ∈K

Ek1 k −1 ,k2 k ⊗ Ee,e ⊗ k xk

38


whereas E(Ek1 ,k2 ⊗ xh)e2

−1 1 −1 k (x) = δh,e λk1 Ek,k λk −1 ⊗ Eh ,h ⊗ h ρk −1 ⊗ Ee,e ⊗ k 2 |K|2 h ∈H k∈K k ∈K 1 −1 λk1 Ek,k λk −1 ρk −1 ⊗ Ee,e ⊗ k (x)k = δh,e 2 |K|2 k∈K 1 −1 Ek1 k,k2 kk ⊗ Ee,e ⊗ k xkk = δh,e |K|2 k,k ∈K and the two sides are the same after renaming the indices. To show condition (ii), note that MK (C) ⊗ (P K) is algebraically generated by MK (C) ⊗ P 1 and |K| k∈K ρk −1 ⊗ k (by the remarks preceding Eq. (4.4)). The following holds in MK (C) ⊗ sits inside MH (C) ⊗ P (since in the MH (C) ⊗ (P K) because of this fact and the way P second tensor component we get expressions of the form h,h ∈H Eh,h Ee,e Eh ,h which reduces to Ee,e ):

MK (C) ⊗ P e2 MK (C) ⊗ P = MK (C) ⊗ Ee,e ⊗ (P K)

⇒ MK (C) ⊗ P H e2 MK (C) ⊗ P H = MK (C) ⊗ MH (C) ⊗ (P K) where the last implication is again due to the relation λh1 Ee,e λh−1 = Eh1 ,h2 in MH (C). 2

2

Thus, we have the first two levels in the tower of basic construction: P H ⊂ P K ⊂ MK (C) ⊗ (P H ) ⊂ MK×H (C) ⊗ (P K) where we identify MK×H (C) with MK (C) ⊗ MH (C). The next levels in the tower are obvious generalizations and we gather everything in the following proposition. Proposition 4.4. Let G be a group acting outerly on the II1 factor P and assume G is generated by two of its finite subgroups H and K. Then the nth element of the tower of basic construction of the group-type subfactor N = P H ⊂ P K = M is given by Mn ∼ = MSn (C) ⊗ (P Ln ) where the inclusion of Mn inside Mn+1 is as follows: Mn Es,t ⊗ x →

Es,t ⊗ El,l ⊗ l −1 (x) ∈ Mn+1

for all x ∈ P , s, t ∈ Sn ,

l∈Ln

Mn Es,t ⊗ l → Es,t ⊗ λl ⊗ e ∈ Mn+1

for all l ∈ Ln , s, t ∈ Sn ,


and the nth Jones projection is: 1 Mn e n =

l∈Ln IMSn−2 ⊗ ρl −1 |Ln | 1 h∈H Ee,e ⊗ h, |H |

39

⊗ Ee,e ⊗ l, if n > 1, if n = 1.

Proof. We use induction. The case of n = 1 is a little different from the rest and is proved in Lemma 4.2 and the n = 2 case is proved in Lemma 4.3. Suppose the statement of the above proposition holds upto a level n (> 2). Now, the subfactor Mn−1 ⊂ Mn is isomorphic to: MSn−1 ⊗ P Ln−1 ⊂ MSn−1 ⊗ MLn−1 ⊗ P Ln where we identify MSn−1 ⊗ MLn−1 with MSn and the inclusion is induced by identity over MSn−1 tensored with the inclusion of the subfactor P Ln−1 ⊂ MLn−1 ⊗ P Ln . Using Lemma 4.3 for K = Ln−1 and H = Ln , it is clear that the statement of the proposition holds for level n + 1. 2 Remark 4.5. The formula for the unique trace-preserving conditional expectation is: n EM Mn−1 (Es1 ,s2 ⊗ Em1 ,m2 ⊗ xl) = δl,e

1 Es ,s ⊗ m1 xm−1 2 |Ln | 1 2

where s1 , s2 ∈ Sn−1 , m1 , m2 ∈ Ln−1 , l ∈ Ln and x ∈ P and the unique trace on Mn is given by trMn (Es1 ,s2 ⊗ xl) =

1 δl,e δs1 ,s2 trM (x) |Sn |

where s1 , s2 ∈ Sn , l ∈ Ln and x ∈ P . We will now compute the higher relative commutants using the above model of the Jones tower. To this end, we need the following two lemmas, where we denote the set of automorphisms of M ⊃ N that fixes elements of N pointwise by Gal(N ⊂ M). Lemma 4.6. Let N ⊂ M be a finite index subfactor and θ ∈ Gal(N ⊂ M), then the bimodule 2 2 M L (θ )M (where the module is L (M) with usual left action of M but right action is twisted by θ ) is a 1-dimensional irreducible sub-bimodule of M L2 (M1 )M . Proof. Define uθ : L2 (M) → L2 (M) by uθ (xΩ) = θ (x)Ω for x ∈ M where Ω is the cyclic and separating vector in the GNS construction with respect to the canonical trace tr. Note that uθ (n1 · xΩ · n2 ) = uθ (n1 xn2 Ω) = n1 · θ (x)Ω · n2 for n1 , n2 ∈ N , x ∈ M. This implies uθ ∈ N ∩ M1 . Now define T : M L2 (θ )M → M L2 (M1 )M by T (xΩ) = xuθ Ω1 for x ∈ M. It is completely routine to check that T is a well-defined M-M linear isometry and we leave this to the reader. 2 Corollary 4.7. H = Gal(P H ⊂ P ). Proof. Clearly H ⊂ Gal(P H ⊂ P ). Let θ ∈ Gal(P H ⊂ P ). Note that P L2 (P H )P ∼ = 2 (h) . Thus by Lemma 4.6, L2 (θ ) ∼ L2 (h) for some h ∈ H . This implies L = P P P P P h∈H P θ h−1 ∈ Inn(P ) ∩ Gal(P H ⊂ P ) = {idP } since P H ⊂ P is irreducible. Hence, θ ∈ H . 2

40


Lemma 4.8. Let N ⊂ M be an irreducible subfactor, i.e. N ∩ M ∼ = C and θ ∈ Aut(M). For x ∈ M, the following are equivalent: (i) x = 0 and xθ (y) = yx for all y ∈ N , x ∈ U(M) and Adx0 ◦ θ ∈ Gal(N ⊂ M). (ii) x0 := x Proof. (ii) ⇒ (i) part is easy. For (i) ⇒ (ii), note that we also have θ (y)x ∗ = x ∗ y for all y ∈ N . Thus, x ∗ x ∈ θ (N ) ∩ M and ∗ xx ∈ N ∩ M. Since N ∩ M ∼ = C, xx ∗ = x ∗ x = x2 . Hence, x0 ∈ U(M) and x0 θ (y)x0 ∗ = y for all y ∈ N . This implies Adx0 ◦ θ ∈ Gal(N ⊂ M). 2 Proposition 4.9. For the group-type subfactor N = P H ⊂ P K = M, the relative commutants N ∩ Mn and M ∩ Mn are given by ⎧ C, ⎧ ⎪ ⎫ if n = −1, ⎪ ⎨ s , s ∈ S ⎨ ⎬ 1 2 n N ∩ Mn ∼ = span E l ∈ Ln ⊗ l , if n 0, ⎪ s ,s ⎪ ⎩ 1 2 ⎩ μ(s1 )lμ(s2 )−1 ∈ H ⎭ ⎧ C, ⎧ ⎪ ⎫ if n = 0, ⎪ ⎨ t , t ∈ T ⎨ ⎬ n−1 1 2 M ∩ Mn ∼ = span Ek,kk0 ⊗ Et1 ,t2 ⊗ l μ(t1 )lμ(t2 )−1 ∈ K , if n 1. ⎪ k∈K ⎪ ⎩ ⎩ k0 = μ(t1 )lμ(t2 )−1 ⎭ Proof. We compute the relative commutants in relation to the concrete model of the basic construction described in Proposition 4.4. Consider the inclusion Es,s ⊗ μ(s)−1 (x) ∈ MSn (C) ⊗ (P Ln ) = Mn . N = P H x → s∈Sn

Let w=

s1 ,s2 ∈Sn l∈Ln

Es1 ,s2 ⊗ xsl1 ,s2 l ∈ N ∩ Mn ,

for some xsl1 ,s2 ∈ P .

Then, wy = yw ⇔

for all y ∈ N

Es1 ,s2 ⊗ xsl1 ,s2 l μ(s2 )−1 (y)

s1 ,s2 ∈Sn l∈Ln

=

s1 ,s2 ∈Sn l∈Ln

⇔ ⇔

Es1 ,s2 ⊗ μ(s1 )−1 (y)xsl1 ,s2 l

for all y ∈ N

xsl1 ,s2 lμ(s2 )−1 (y) = μ(s1 )−1 (y)xsl1 ,s2 for all y ∈ N, s1 , s2 ∈ Sn , l ∈ Ln

for all y ∈ N, s1 , s2 ∈ Sn , l ∈ Ln . μ(s1 ) xsl1 ,s2 μ(s1 )lμ(s2 )−1 (y) = yμ(s1 ) xsl1 ,s2


41

Now, by Lemma 4.8 and Corollary 4.7, for s1 , s2 ∈ Sn and l ∈ Ln , xsl1 ,s2

= 0

Adx0 ◦ μ(s1 )lμ(s2 )−1 ∈ H

⇔

where x0 =

μ(s1 )(xsl1 ,s2 ) μ(s1 )(xsl1 ,s2 )

.

Moreover, xsl1 ,s2 = 0 ⇒ Adx0 ∈ G ∩ Inn(P ) = {idP }. Since P is a factor, x0 ∈ C1. Thus,

xsl1 ,s2 will be non-zero only if μ(s1 )lμ(s2 )−1 ∈ H and in such cases xsl1 ,s2 will be a scalar multiple of identity. Hence, N ∩ Mn is spanned by the linearly independent set {Es1 ,s2 ⊗ l: s1 , s2 ∈ Sn , l ∈ Ln , μ(s1 )lμ(s2 )−1 ∈ H }. For M ∩ Mn where n 1, we consider the inclusion M = P K ⊃ P x →

Es,s ⊗ μ(s)−1 (x) ∈ MSn (C) ⊗ (P Ln ) = Mn ,

s∈Sn

M = P K ⊃ K k → λk ⊗ ITn−1 ⊗ 1 ∈ MSn (C) ⊗ (P Ln ) = Mn . Let w=

s1 ,s2 ∈Sn l∈Ln

Es1 ,s2 ⊗ xsl1 ,s2 l ∈ M ∩ Mn ,

for some xsl1 ,s2 ∈ P .

Using arguments similar to those for the calculations for the case of N ∩ Mn it follows that wy = yw ⇔

for all y ∈ P

μ(s1 ) xsl1 ,s2 μ(s1 )lμ(s2 )−1 (y) = yμ(s1 ) xsl1 ,s2

for all y ∈ P , s1 , s2 ∈ Sn , l ∈ Ln ,

and hence xsl1 ,s2 = 0 ⇔ μ(s1 )lμ(s2 )−1 = e since P is a factor; further, in such cases, xsl1 ,s2 is a scalar multiple of 1. Now, wk = kw ⇔ ⇔

for all k ∈ K

k −1 wk = w for all k ∈ K l λ−1 k ⊗ ITn−1 ⊗ 1 Es1 ,s2 ⊗ xs1 ,s2 l (λk ⊗ ITn−1 ⊗ 1) s1 ,s2 ∈Sn l∈Ln

=

s1 ,s2 ∈Sn l∈Ln

⇔

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

Es1 ,s2 ⊗ xsl1 ,s2 l

for all k ∈ K

l λ−1 k Ek1 ,k2 λk ⊗ Et1 ,t2 ⊗ x(k1 ,t1 ),(k2 ,t2 ) l

42


=

l Ek1 ,k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

for all k ∈ K

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

⇔

l Ek −1 k1 ,k −1 k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

=

l Ek1 ,k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

for all k ∈ K

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

l l = x(kk x(k 1 ,t1 ),(k2 ,t2 ) 1 ,t1 ),(kk2 ,t2 )

⇔

for all k, k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln .

Finally, combining the conditions that we get from considering commutation of w with elements of P and K, we can express w as a linear combination of elements of the form:

Ekk1 ,kk2 ⊗ Et1 ,t2 ⊗ l

k∈K

where k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln such that k1 μ(t1 )lμ(t2 )−1 k2−1 = e. Equivalently, w can be realized as a linear combination of k∈K

Ekk −1 ,k ⊗ Et1 ,t2 ⊗ l = 0

Ek,kk0 ⊗ Et1 ,t2 ⊗ l

k∈K

where t1 , t2 ∈ Tn−1 , l ∈ Ln such that μ(t1 )lμ(t2 )−1 ∈ K and k0 = μ(t1 )lμ(t2 )−1 .

2

Remark 4.10. (i) The set ⎫ s1 , s2 ∈ Sn , ⎬ Es1 ,s2 ⊗ l l ∈ Ln , ⎩ μ(s1 )lμ(s2 )−1 ∈ H ⎭ ⎧ ⎨

⎫# t1 , t2 ∈ Tn−1 , ⎬ Ek,kk0 ⊗ Et1 ,t2 ⊗ l μ(t1 )lμ(t2 )−1 ∈ K, resp. ⎩ k0 = μ(t1 )lμ(t2 )−1 ⎭ k∈K

"

⎧ ⎨

forms a basis of N ∩ Mn (resp. M ∩ Mn ). (ii) The unique trace-preserving conditional expectation from N ∩ Mn onto M ∩ Mn is given by

∩Mn EN M ∩Mn (Ek1 ,k2 ⊗ Et1 ,t2 ⊗ l) = δk1 μ(t1 )l,k2 μ(t2 )

1 Ek,kk1 −1 k2 ⊗ Et1 ,t2 ⊗ l |K| k∈K

where k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln such that k1 μ(t1 )lμ(t2 )−1 k2−1 ∈ H . To see (ii), we need to check that for t1 , t2 ∈ Tn−1 and k0 = μ(t1 )l μ(t2 )−1 ∈ K,


tr

$ k ∈K

% Ek ,k k0 ⊗ Et1 ,t2 ⊗ l (Ek1 ,k2 ⊗ Et1 ,t2 ⊗ l)

= δl l,e δt2 ,t1 δt1 ,t2 δk2 k0 ,k1

1 |Sn |

= δk1 μ(t1 )l,k2 μ(t2 ) δl l,e δt1 ,t2 δt2 ,t1 δe,k0 k1 −1 k2 = tr

43

$ k ∈K

Ek ,k k0 ⊗ Et1 ,t2 ⊗ l

1 |Sn |

% 1 δk1 μ(t1 )l,k2 μ(t2 ) Ek,kk1 −1 k2 ⊗ Et1 ,t2 ⊗ l |K| k∈K

which is a routine computation using the second part of Remark 4.5. 5. The planar algebra of group-type subfactors In this section, our main goal is to show that the planar algebra defined in Section 3, is isomorphic to the one arising from the group-type subfactor P H ⊂ P K. Conversely, we will show that any subfactor whose standard invariant is given by such planar algebra, is indeed of this type. We will use the following well-known fact regarding isomorphism of two planar algebras and Theorem 4.2.1 of [11] describing the planar algebra arising from an extremal finite index subfactor. Fact. Let P 1 and P 2 be two planar algebras. Then P 1 ∼ = P 2 if and only if there exists a vector 1 2 space isomorphism ψ : P → P such that: (i) ψ preserves the filtered algebra structure, (ii) ψ preserves the actions of all possible Jones projection tangles and the (two types of) conditional expectation tangles. If P1 and P2 are ∗-planar algebras, then we consider such ψ that are ∗-preserving to be a ∗-planar algebra isomorphism. Let us denote the planar algebra in Section 3 by P BH and the one arising from the group-type subfactor N = P H ⊂ P K = M by P N ⊂M . Theorem 5.1. P N ⊂M ∼ = P BH . Proof. By Theorem 4.2.1 of [11] we have that PnN ⊂M = N ∩ Mn where the nth Jones projection N ∩M

N ⊂M is δEN ∩Mn+1 tangle is given by δen , the conditional expectation tangle from PnN ⊂M onto Pn−1 n N ∩M

N ⊂M and the conditional expectation tangle from PnN ⊂M onto P1,n is δEM ∩Mn+1 . n+1 Define the map

ψ : P N ⊂M → P BH ∪

∪

by ψn (Es1 ,s2 ⊗ l) = (s1 , l, s˜2 , h)

ψn : PnN ⊂M → PnBH where n 0, s1 , s2 ∈ Sn , l ∈ Ln such that μ(s1 )lμ(s2 )−1 ∈ H and μ(s1 )lμ(s2 )−1 h = e.

44


Clearly, ψ is a vector space isomorphism by definition. In order to check that ψ is a filtered ∗-algebra isomorphism, we use Remark 3.3(i), (ii), (iii) and (iv). For instance, to show that ψn is an algebra homomorphism, we need to show ψn (Es1 ,s2 ⊗ l1 · Es3 ,s4 ⊗ l2 ) = ψn (Es1 ,s2 ⊗ l1 ) · ψn (Es3 ,s4 ⊗ l2 )

⇐ δs2 ,s3 s1 , l1 l2 , s˜4 , μ(s4 )l2−1 l1−1 μ(s1 )−1

= s1 , l1 , s˜2 , μ(s2 )l1−1 μ(s1 )−1 · s3 , l2 , s˜4 , μ(s4 )l2−1 μ(s3 )−1 which indeed holds by Remark 3.3(iii). Now, it remains to be shown that ψ preserves the action of Jones projection tangles and the two types of conditional expectation tangles. For this, we use Remark 3.3(v), (vi) and (vii). Proof of each of the three kinds of tangles is completely routine; however, we will discuss the action of conditional expectation tangle in details. Let us consider the conditional expectation tangle from n + 1 (color of the internal disc) to color n (color of the external disc) applied to the element N ⊂M Es1 ,s2 ⊗ Em1 ,m2 ⊗ l ∈ Pn+1 = N ∩ Mn ; by Jones’s theorem (4.2.1 of [11]) and Remark 4.5, the output should be

|Ln | Es ,s ⊗ m1 m−1 2 |Ln+1| 1 2 |Ln | ψn s1 , m1 m−1 −→ δl,e , s˜2 , μ(s2 )m2 m−1 μ(s1 )−1 . 2 1 |Ln+1|

δl,e

On the other hand, by Remark 3.3(vi), the conditional expectation tangle applied to −1 −1 ψn+1 (Es1 ,s2 ⊗ Em1 ,m2 ⊗ l) = (s1 , m1 , l −1 , m−1 2 , s˜2 , μ(s2 )m2 lm1 μ(s1 ) ) is given by & −1 −1 n| δl,e |L|Ln+1| (s1 , m1 m−1 2 , s˜2 , μ(s2 )m2 m1 μ(s1 ) ). This completes the proof.

2

Corollary 5.2. Given any countable discrete group G generated by two of its finite subgroups, there exists a hyperfinite subfactor with standard invariant described by P BH . Moreover, P BH is a spherical C ∗ planar algebra. Proof. The proof follows from the fact that any countable discrete group G has an outer action on the hyperfinite II1 factor. 2 Theorem 5.3. Given a subfactor N ⊂ M with standard invariant isomorphic to P BH , there exists an intermediate subfactor N ⊂ P ⊂ M and outer actions of H and K on P such that (N ⊂ M) (P H ⊂ P K). Proof. Let P N ⊂M denote the planar algebra of N ⊂ M formed by its relative commutants and φ : P N ⊂M → P BH be a planar algebra isomorphism. Consider the element q = φ −1 (e, e, e, e) ∈ N ∩ M1 . Clearly, (i) q is a projection, (ii) qe1 = e1 , and (iii) EM (q) = |K|−1 1. Using action of tangles and the planar algebra isomorphism φ, it also follows that


45

(iv)

The conditions (i)–(iv) assert that q is an intermediate subfactor projection as described in [2]. Define P = M ∩ {q} . To show P is a factor, first note that

P ∩ P ⊂ N ∩ P = N ∩ M ∩ {q} = φ P1BH ∩ (e, e, e, e) . So, it is enough to show that P1BH ∩ {(e, e, e, e)} = C1. If x = {(e, e, e, e)} , then (e, e, e, e) · x = x · (e, e, e, e)

⇒

g∈K∩H (g, g

−1 )

∈ P1BH ∩

λg e, e, g, g −1 = λg g, e, e, g −1 .

g∈K∩H

g∈K∩H

Hence, λg = δg,e λe for all g ∈ K ∩ H and x ∈ C1. It remains to establish that N (resp. M) is the fixed-point subalgebra (resp. crossed-product algebra) of P with respect to an outer action of the group H (resp. K). It is easy to prove (see [11]) that if the standard invariant of a subfactor N˜ ⊂ M˜ is given by the planar algebra corresponding to the fixed-point subfactor (resp. crossed-product subfactor) with respect to a ˜ then there exists an outer action of G ˜ on M˜ (resp. N˜ ) such that N˜ (resp. M) ˜ finite group G, ˜ is isomorphic to fixed-point subalgebra (crossed-product algebra) of the action. Again, if N ⊂ P˜ ⊂ M˜ is an intermediate subfactor and q˜ is its corresponding intermediate subfactor projection, ˜ is given by the range of the idempotent tangle then the planar algebra of N˜ ⊂ P˜ (resp. P˜ ⊂ M)

or

(resp.

according as n is even or odd ([6], see also [1]).

or

)

46


Getting back to our context, to get the planar algebra of N ⊂ P (resp. P ⊂ M), the elements in the image of the above idempotent tangles are given by the words with letters coming from K and H alternately where every element coming from K (respectively H ) must necessarily be e. Such a planar algebra is the same as P BH with K = {e} (resp. H = {e}); by Theorem 5.1 this is indeed the planar algebra corresponding to fixed-point subfactor (resp. crossed-product subfactor) with respect to H (resp. K). 2 References [1] B. Bhattacharyya, Z. Landau Intermediate standard invariants and intermediate planar algebras, preprint. [2] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (2) (1994) 201–216. [3] D. Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, in: Operator Algebras and Their Applications, in: Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13– 63. [4] D. Bisch, U. Haagerup, Composition of subfactors: New examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. 29 (1996) 329–383. [5] D. Bisch, V. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1) (1997) 89–157. [6] D. Bisch, V. Jones, The free product of planar algebras, and subfactors, in preparation. [7] D. Bisch, S. Popa, Examples of subfactors with Property (T) standard invariant, Geom. Funct. Anal. 9 (1999) 215– 225. [8] D. Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, New York, 1998, xvi+829 pp. [9] F. Goodman, P. de la Harpe, V. Jones, Coxeter Graphs and Towers of Algebras, Math. Sci. Res. Inst. Publ., vol. 14, Springer-Verlag, New York, 1989, x+288 pp. [10] V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. [11] V. Jones, Planar algebras, arxiv:math/9909027v1. [12] V. Jones, Two subfactors and the algebraic decomposition of bimodules over II1 factors, preprint. [13] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, ISBN 0-52158420-5, 1997. [14] A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, in: Operator Algebras and Applications, vol. 2, in: London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. [15] M. Pimsner, S. Popa, Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1) (1988) 127–133. [16] S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994) 163–255. [17] S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995) 427–445. [18] S. Popa, Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T, Doc. Math. 4 (1999) 665–744. [19] J.-M. Vallin, Relative matched pairs of finite groups from depth two inclusions of von Neumann algebras to quantum groupoids, J. Funct. Anal. 254 (2) (2008) 2040–2068.


Noncommutative ball maps J. William Helton a,1 , Igor Klep b,c,2 , Scott McCullough d,∗,3 , Nick Slinglend a a Department of Mathematics, University of California, San Diego, United States b Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Slovenia c Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Slovenia d Department of Mathematics, University of Florida, Gainesville, FL, United States

Received 8 October 2008; accepted 10 March 2009

Communicated by N. Kalton

Abstract In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D’Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables. © 2009 Elsevier Inc. All rights reserved.


E-mail addresses: [email protected] (J.W. Helton), [email protected] (I. Klep), [email protected] (S. McCullough), [email protected] (N. Slinglend). 1 Research supported by NSF grants DMS-0700758, DMS-0757212, and the Ford Motor Co. 2 Supported by the Slovenian Research Agency (project No. Z1-9570-0101-06). 3 Research supported by the NSF grant DMS-0758306. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.008

48

J.W. Helton et al. / Journal of Functional Analysis 257 (2009) 47–87

Keywords: Noncommutative analytic function; Complete isometry; Ball map; Linear matrix inequality

1. Introduction In the introduction we will state some of our main results. For this we need to start with the definitions of NC polynomials (Section 1.1) and NC analytic maps (Section 1.2). We then proceed to define NC ball maps in Section 1.3, where we explain what it means for an NC ball map to map ball to ball with boundary to boundary. After that we can and do state our main results classifying NC ball maps in Sections 1.3 and 1.4. Finally, the introduction concludes by considering two types of generalizations, the first being to balls defined by LMIs, the second being to NC analytic maps carrying special sets on the boundary of a ball to the boundary of a ball. Before continuing with the more detailed introduction, we pause to offer some perspective and mention related significant contributions. Ball maps form a distinguished subset of the space of noncommutative analytic functions on a noncommutative domain and we direct the interested reader to the elegant general theory of noncommutative analytic functions developed in the articles [12–14,29,30,16,17] for a more complete account than found here and to the work of Popescu on free analytic functions. Some of these references are [22,27,23,24,26]. Noncommutative rational, Schur class and analytic functions are also intimately related to systems theory. A small sample of the references includes [2,3,10,12,18]. The noncommutative balls that we consider are modeled on g × g matrices, and in the special case that g = 1, they correspond to those studied by Popescu for operator, not just matrix, coefficients. Precomposition by an automorphism of the domain preserves ball maps and such automorphisms are studied at various levels of generality in [28,17]. Linear ball maps are an important special case, and these are identified with completely isometric mappings from one operator space into another. The books [19,20,6] provide comprehensive introductions to the theory of operator systems, spaces, and algebras, and the papers [5] and [4] treat very generally complete isometries into a C-star algebra. 1.1. Words and NC polynomials Let g , g ∈ N. We write x for the monoid freely generated by x, i.e., x consists of words in the g g letters x11 , . . . , x1g , x21 , . . . , xg g (including the empty word ∅ which plays the role of the identity 1). Let Cx denote the associative C-algebra freely generated by x, i.e., the elements of Cx are polynomials in the noncommuting variables x with coefficients in C. Its elements are called NC polynomials. An element of the form aw where 0 = a ∈ C and w ∈ x is called a monomial and a its coefficient. Hence words are monomials whose coefficient is 1. Let ∗ , . . . , x ∗ ) denote another set of g g symbols. We shall also consider the free algebra x ∗ = (x11 g g Cx, x ∗ that comes equipped with the natural involution xij → xij∗ . For example, ∗ 2 2 ∗ ∗ ∗ 1 + ix11 x23 x34 = 1 − ix34 x23 x11 . (Here i denotes the imaginary unit

√

−1.)


49

1.1.1. NC matrix polynomials A matrix-valued NC polynomial is an NC polynomial with matrix coefficients. We shall use the phrase scalar NC polynomial if we want to emphasize the absence of matrix constructions. Often when the context makes the usage clear we drop adjectives such as scalar, 1 × 1, matrix polynomial, matrix of polynomials and the like. 1.1.2. Polynomial evaluations If p is an NC polynomial in x and X ∈ (Cn×n )g ×g , the evaluation p(X) is defined by simply replacing xij by Xij . For example, if p(x) = Ax11 x21 , where

−4 3 2 A= , 2 −1 0 then p

⎡ 0 4 0 −3 0 0 1 1 0 0 1 1 0 3 0 2 ⎢ −4 0 , =A⊗ =⎣ 1 0 0 −1 1 0 0 −1 0 −2 0 1 0 2 0 −1 0 0

⎤ −2 0 ⎥ ⎦. 0 0

On the other hand, if p(x) = A and X ∈ (Cn×n )g ×g , then p(X) = A ⊗ In . The tensor product in the expressions above is the usual (Kronecker) tensor product of matrices. Thus we have reserved the tensor product notation for the tensor product of matrices and have eschewed the strong temptation of using A ⊗ xk in place of Axk when xk is one of the noncommuting variables. 1.2. Definition of NC analytic functions An elegant theory of noncommutative analytic functions is developed in the articles [12–14, 29,30]; see also [22]. What we need in this article are specializations of definitions of these papers. In this section we summarize the definitions and properties needed in the sequel. For d , d ∈ N define Bd ×d :=

∞ d ×d Idn − X ∗ X 0 , X ∈ Cn×n

(1.1)

n=1

int Bd ×d :=

∞ d ×d Idn − X ∗ X 0 , X ∈ Cn×n

(1.2)

n=1

∂Bd ×d :=

∞ d ×d X = 1 , X ∈ Cn×n

(1.3)

n=1

Md ×d :=

∞ n×n d ×d C . n=1

(1.4)

50


We shall occasionally use the notation ,d Xj, ∈ CN ×N , X 1 , Bd ×d (N ) = X = [ Xj, ]dj,=1 ,d Xj, ∈ CN ×N . Md ×d (N ) = X = [ Xj, ]dj,=1

g ×g is the (disjoint) Given g , g ∈ N, the noncommutative (NC) ε-neighborhood of 0 in C union N ∈N {X ∈ Mg ×g (N ) | X < ε}. An open NC domain D containing 0 (in its interior) is a union N DN of open sets DN ⊆ Mg ×g (N ) which is closed with respect to direct sums and such that there is an ε > 0 such that D contains the NC ε-neighborhood of 0. A d × d NC analytic function f on an open NC domain D containing 0 is defined as follows:

(1) f has an NC power series, for which there exists an NC ε > 0 neighborhood of 0 on which it is convergent. That is,

f=

aw w

(1.5)

w∈x

for aw ∈ Cd ×d and for every N ∈ N and every g × g-tuple of square matrices X ∈ Bg ×g with X < ε the series f (X) =

aw ⊗ w(X)

(1.6)

w∈x

converges. We interpret convergence for a given X as conditional convergence of the series ∞

aw ⊗ w(X).

α=0 |w|=α

Thus the order of summation is over the homogeneous parts of the power series expansion. Thus with f (α) equal to the α homogeneous part in the NC power series expansion of f , the series converges for a given X provided ∞

f (α) (X)

(1.7)

α=0

converges. Since both aw and w(X) are matrices, the particular norm topology chosen has no influence on convergence. The radius ε of this ball of convergence (or sometimes, by abuse of notation, the ball itself) will be called the series radius. (2) If a : W → D is a matrix-valued function analytic on a domain W in CN , the composition f ◦ a is a matrix-valued analytic function on W and continuous to ∂W. Remark 1.1. Popescu [22] has a notion of free analytic function in g variables (that is, g = 1) based upon power series expansions like that in (1.7). His definition allows for operator coefficients, but on the other hand requires convergence of the NC power series on all of int Bg (the


51

NC 1-neighborhood of 0 in Cg ). It turns out that for bounded NC analytic functions with matrix coefficients the two notions are the same, see Lemma 6.1. Here we have avoided extending the theory of free analytic functions to Bg ×g , looking forward to working on more general domains in the future. 1.2.1. Properties of NC analytic functions Proposition 1.2. Let D be an NC domain containing 0. (i) The sum of two d × d NC analytic functions on D is a d × d NC analytic function on D. (ii) The product of two d × d NC analytic functions on D is a d × d NC analytic function on D. (iii) The composition of two NC analytic functions is an NC analytic function. More precisely, if f : D → D is a d1 × d1 NC analytic function, where D is an NC domain with 0 ∈ D , and h is a d2 × d2 NC analytic function on D , then h ◦ f is a d1 d2 × d1 d2 NC analytic function on D. Proof. Properties (i) and (ii) are standard and we only consider (iii). The fact that h ◦ f admits an NC power series as in (1.5) was observed e.g. in [14,29]. The composition property (2) of Section 1.2 is easily checked. 2 More is said about properties of NC analytic functions in Section 6. 1.3. NC ball maps f and their classification when f (0) = 0 A function f : int Bg ×g → Md×d which is NC analytic will often be called an NC analytic function on the ball Bg ×g and denoted f : Bg ×g → Md×d . An NC analytic function f : Bg ×g → Bd ×d mapping the boundary to the boundary is called an NC ball map. The notion of f mapping boundary to boundary is a bit complicated (because of convergence issues) so requires explanation. For a given X ∈ Bg ×g (N ), define the function fX : D → Md×d (N ) by z → f (zX). (Here D denotes the unit disc D = {z ∈ C | |z| < 1} in the complex plane.) If lim fX reit r1

exists, denote that limit by f (eit X). The function f maps the boundary to the boundary if whenever X = 1 and f (eit X) exists, then it f e X = 1. Since f is bounded, Fatou’s theorem implies that for each X ∈ Bg ×g the limit fX (eit ) = f (eit X) exists for almost every t. If f is an NC ball map, X ∈ ∂Bg ×g and f (X) is defined, then a (nonzero) vector γ such that f (X)γ = γ is called a binding vector and this property binding.

52


Our main result on NC ball maps which map 0 to 0 is: Theorem 1.3. Let h : Bg ×g → Bd ×d be an NC ball map with h(0) = 0. Then there exist unitaries U : Cd → Cd and V : Cd → Cd such that x 0 h(x) = V U ∗, (1.8) ˜ 0 h(x) ˜ where h˜ : Bg ×g → B(d −g )×(d−g) is an NC analytic contraction-valued map with h(0) = 0. Conversely, every NC analytic h satisfying (1.8) for unitaries U, V and an NC analytic contraction-valued map h˜ fixing the origin, is an NC ball map Bg ×g → Bd ×d sending 0 to 0. The proof of the theorem is completed in Section 4. The general result is built from the linear version of the theorem, Theorem 3.3 which appears as Corollary 3.6 in [5]. See also [4]. As an illustration of Theorem 1.3 we describe a special case. For convenience, we adopt the notation Bg for Bg ×1 . Corollary 1.4. If h : Bg → Bd is an NC ball map with h(0) = 0, then h is linear and there is a unique isometry M ∈ Cd ×g such that h = Mx. In particular, if d < g then no such NC ball maps exist. ˜ column is gone. Moreover, Proof. When h maps Bg to Bd the h(x) I M =V∗ 0 is an isometry.

2

1.4. NC ball maps f when f (0) is not necessarily 0 In the previous section we treated NC ball maps f with f (0) = 0, an assumption we drop in this section. The strategy is to compose f with a bianalytic automorphism of an NC ball to reduce the problem to the f (0) = 0 setting. Section 1.4.1 contains information on bianalytic mappings on an NC ball, while the main results appear in Section 1.4.2. 1.4.1. Linear fractional transformations For a given d × d scalar matrix v with v < 1, define Fv : Bd ×d → Bd ×d by 1/2 −1 1/2 Fv (u) := v − Id − vv ∗ Id − v ∗ v u Id − v ∗ u .

(1.9)

Of course it must be shown that Fv actually takes values in Bd ×d . This is done in Lemma 1.6 below. Linear fractional transformations such as Fv are common in circuit and system theory, since they are associated with energy conserving pieces of a circuit (cf. [31]). Lemma 1.5. Suppose D is an open NC domain containing 0. If u : D → Bd ×d is NC analytic, then Fv (u(x)) is an NC analytic function (in x) on D. Proof. See Section 5.

2


53

Notice that if d = d = 1, then v is a scalar and u is a scalar NC analytic function, hence ¯ −1 = (1 − uv) ¯ −1 (v − u). Fv (u) = (v − u)(1 − uv) Now fix v ∈ D and consider the map D → C, u → Fv (u). This map is a linear fractional map that maps the unit disc to the unit disc, maps the unit circle to the unit circle, and maps v to 0. The geometric interpretation of the map in NC variables in (1.9) is similar. Suppose we fix N ∈ N and V ∈ Bd ×d (N ) with V < 1 and consider the map U → FV (U ).

(1.10)

The first part of Lemma 1.6 tells us that the map defined in (1.10) maps the unit ball of d × dtuples of N ×N matrices to the unit ball of d ×d-tuples of N ×N matrices carrying the boundary to the boundary. The third part of Lemma 1.6 tells us that FV (V ) = 0; that is, the map given in (1.10) takes V to 0. Lemma 1.6. Suppose that N ∈ N and V ∈ Bd ×d (N ) with V < 1. (1) U → FV (U ) maps the Bd ×d (N ) into itself with boundary to the boundary. (2) If U ∈ Bd ×d (N ), then FV (FV (U )) = U . (3) FV (V ) = 0 and FV (0) = V . Proof. See Section 5.

2

1.4.2. Classification of NC ball maps General NC ball maps – those where f (0) is not necessarily 0 – are described using the linear fractional transformation F . / ∂Bd ×d . Then Theorem 1.7. Let f : Bg ×g → Bd ×d be an NC ball map with f (0) ∈ f (x) = Ff (0) ϕ(x) ,

(1.11)

where ϕ(x) = Ff (0) f (x) = V

x 0

0 U∗ ϕ(x) ˜

(1.12)

for some unitaries U : Cd → Cd and V : Cd → Cd and an NC analytic contraction-valued map ϕ˜ with ϕ(0) ˜ < 1. Conversely, every NC analytic f satisfying (1.11) and (1.12) for unitaries U, V and ϕ˜ as / ∂Bd ×d . above, is an NC ball map f : Bg ×g → Bd ×d with f (0) ∈ Proof. Define ϕ(x) := Ff (0) (f (x)). Then ϕ(0) = 0. By Lemma 1.5, ϕ(x) is an NC analytic map. Hence it is an NC ball map sending 0 to 0 and is thus classified by Theorem 1.3. Moreover, Eq. (1.11) is implied by Lemma 1.6(2). The converse easily follows from Lemmas 1.5 and 1.6. 2 The results of Sections 1.3 and 1.4 are treated in Part I of this paper.

54


1.5. More generality In this subsection we extend the main results presented so far in two directions. The first concerns LMIs. Our interest will be in properties of the set of all solutions to a given LMI. In Section 1.5.2 we will define what we mean by an LMI, then show that the set of solutions to a “monic” LMI equals a general type of matrix ball we call a pencil ball. Ultimately we would like to study maps from pencil balls to pencil balls and this paper is a beginning which handles the special case where the domain pencil ball is the ordinary NC ball Bg ×g (see Corollary 1.10). Eventually we hope to understand which NC analytic change of variables takes one LMI to another. Work is in progress on such problems. In the next generalization we do not have applications in mind, but do something that is mathematically natural. A basic notion in several complex variables is the Shilov or distinguished boundary. A natural problem is to classify NC analytic functions mapping the ball to the ball and carrying the distinguished boundary to the boundary. Classification of linear maps of this type proves to be an interesting challenge tackled in Sections 7 and 9. For NC analytic maps we introduce the semi-distinguished boundary (a set larger than the distinguished boundary) and study the NC analytic functions mapping the semi-distinguished boundary to the boundary. All of this we only do for balls of vectors, rather than balls of matrices, that is for g = 1. 1.5.1. Linear pencils Let L(x) := A11 x11 + · · · + Ag g xg g

(1.13)

denote an NC analytic truly linear pencil in x. If the matrices Aij that are used to define it are in Cd ×d , then L(x) is called a d × d linear pencil. As an example, for g = 2 and g = 1,

1 2 , A11 = 3 4

0 1 A21 = , −1 0

the linear pencil is L(x) =

x11 3x11 − x21

2x11 + x21 . 4x11

1.5.2. Linear matrix inequalities and (pencil) balls Let L˜ be a d × d monic symmetric linear pencil. The positivity domain of L˜ is defined to be ˜ 0 . DL˜ := X ∈ Mg ×g L(X) ˜ In other words, it is the set of all solutions to the LMI L(X) 0. We wish to analyze this solution set and we can using results on balls which we have already obtained. Now we describe DL˜ as a type of ball [11]. To do this write L˜ as L˜ = I + L + L∗ where L is a d × d NC analytic truly linear pencil, then to L(x) we associate the (pencil) ball


BL :=

55

∞ X ∈ Mg ×g (n) Idn − L(X)∗ L(X) 0 n=1

=

∞ X ∈ Mg ×g (n) L(X) 1 .

(1.14)

n=1

Observe that Bg ×g = BL for L(x) =

Eij xij

i,j

with Eij being the elementary g × g matrix with 1 located at position (i, j ). Lemma 1.8. For X ∈ Mg ×g ,

0 X ∈ DL˜ 0 0

iff X ∈ BL .

(1.15)

0 X ∈ ∂DL˜ 0 0

iff X ∈ ∂BL .

(1.16)

Furthermore,

Proof. By definition, L˜

0 X 0 0

=

I L(X)∗

L(X) . I

2

1.5.3. Pencil ball maps Now we turn to Bg ×g → BL maps. As a generalization of NC ball map, given a linear pencil L, an NC analytic mapping f : Bg ×g → BL will be called a pencil ball map provided L(f (X)) = 1, whenever X = 1 and f (X) is defined. Lemma 1.8 tells us that understanding pencil ball maps is equivalent to understanding maps on the sets of solutions to certain types of LMIs. Theorem 1.9. Let L be a d × d NC analytic truly linear pencil and f : Bg ×g → BL a pencil ball map with f (0) = 0. Write h := L ◦ f . Then there exist unitaries U : Cd → Cd and V : Cd → Cd such that x 0 (1.17) h(x) = V U ∗, ˜ 0 h(x) where h˜ is an NC analytic contraction-valued map. Proof. Follows easily by applying Theorem 1.3 to h.

2

56


Corollary 1.10. Let L be a d × d NC analytic truly linear pencil and f : Bg ×g → BL a pencil ball map with L ◦ f (0) < 1. Then L ◦ f (x) = FL◦f (0) ϕ(x) ,

(1.18)

where ϕ(x) = FL◦f (0) (L◦f (x)) is an NC ball map Bg ×g → Bd ×d taking 0 to 0 and is therefore completely described by Theorem 1.3. Proof. Apply Theorem 1.7 to L ◦ f (x).

2

1.5.4. Semi-distinguished pencil ball maps Many of our proofs with little extra effort work for a class of functions more general than pencil ball maps. These involve the notion of distinguished boundary which we now define. The Shilov boundary or distinguished boundary of Bg ×g (N ) is the smallest closed subset of Bg ×g (N ) with the following property: For f : Bg ×g (N ) → CK analytic and continuous to the boundary ∂Bg ×g (N ), for any X ∈ Bg ×g (N ) we have f (X) maxf (U ). U ∈

(1.19)

In other words, the maximum of f over Bg ×g (N ) occurs in the distinguished boundary. We refer the reader to [15, p. 145] or [8, Ch. 4] for more details. It is a theorem [1, p. 77] that the distinguished boundary of Bg (N ) is

X ∈ Bg (N ) X ∗ X = I .

Accordingly, we let ∂dist Bg denote the disjoint union of these distinguished boundaries and call this the distinguished boundary of Bg . A further discussion of distinguished boundaries for Bg ×g is in Section 6.3. An NC analytic function f : Bg → BL satisfying f (0) ∈ / ∂BL and f (∂dist Bg ) ⊆ ∂BL

(1.20)

is called a distinguished pencil ball map. Here, (1.20) means that for every isometry X for which limδ1 f (δX) exists, this limit lies in ∂BL . A natural open question is: classify distinguished pencil ball maps. Our proof of Theorem 8.1 does something like this but a little weaker. A key distinction between the semi-distinguished maps and the case treated earlier in Theorems 1.3 and 1.9 occurs with linear distinguished ball maps. These we find much harder to classify than linear NC ball maps, which we leave as an interesting open question. Definition 1.11. The semi-distinguished boundary of Bg is defined to be 1/2

∂dist Bg :=

∞ n=1

1 X ∈ Bg (n) X ∗ X is a projection of dimension n . 2


57

An NC analytic function f : Bg → BL satisfying f (0) ∈ / ∂BL and 1/2 f ∂dist Bg ⊆ ∂BL

(1.21) 1/2

is called a semi-distinguished pencil ball map. Here, (1.21) means that for every X ∈ ∂dist Bg for which limδ1 f (δX) exists, this limit lies in ∂BL . The study of semi-distinguished pencil ball maps is the subject of Part II of this article. For semi-distinguished pencil ball maps we get a weak version of the pencil ball map classification Theorem 1.9 – see Theorem 8.1. Part I. Binding 2. Models for NC contractions Let S denote the (g -tuple of) shift(s) on noncommutative Fock space Fg . The Hilbert space Fg is the Hilbert space with orthonormal basis consisting of words x in g NC variables x = (x1 , . . . , xg ). Then Sj w = xj w for a word w ∈ x and Sj extends by linearity and continuity to Fg . The key properties we need about S are: Sj∗ S = δj I

for j, = 1, . . . , g ,

I−

g

Sj Sj∗ = P0 ,

(2.1)

j =1

where P0 is the (rank one) projection onto the span of the empty word. A column contraction is a g -tuple of square matrices (operators), ⎡

⎤ X1 . X = ⎣ .. ⎦ Xg such that I − X ∗ X = I − Xj∗ Xj 0. If X acts on finite dimensional space, then X is a column contraction if and only if X ∈ Bg . In general, X is a column contraction if and only if X ∗ is a row contraction. Row contractions (and so column contractions too) are well studied – e.g. by Popescu and also Arveson. A strict column contraction is a column contraction X for which there is an ε > 0 such that I − Xj∗ Xj ε. If X is acting on a finite dimensional space, this last condition is equivalent to I − Xj∗ Xj 0, i.e., X ∈ int Bg . Column contractions are modeled by S ∗ , which is the content of Lemma 2.1 below and a major motivation for these definitions. We do not use this property of the Sj until proving Theorem 6.2. Lemma 2.1. (See [7,21].) If X is a strict column contraction acting on a Hilbert space H, then there is a Hilbert space K and an isometry V : H → K ⊗ Fg such that V X = (I ⊗ S ∗ )V ; i.e., for each j , V Xj = (I ⊗ Sj∗ )V and in particular, for each word w ∈ x, V w(X) = (I ⊗ w(S ∗ ))V . Here I is the identity on K. Further, if X ∈ Bg (N ) (so is a tuple of matrices), then the dimension of K can be assumed to be at most N .

58


A natural generalization of the g -tuple of shifts on Fock space to Mg ×g and its (sequence of) ball(s) is g ,g

X = [ Sj∗ ⊗ S ]j,=1 . (A word of caution: we have abused notation by using Sj to denote shifts on both Fg and Fg .) The operator X should be compared to the reconstruction operator in [27]. Though we do not know if X serves as a universal model for Bg ×g in the same way that S does for Bg , it does serve as a type of boundary for NC analytic functions. The statement of the results requires approximating X by matrices. The operator (not matrix) X acts upon Fg ×g – the g ,g

Hilbert space with orthonormal basis consisting of words in g g NC variables x = (xj, )j,=1 . Given a natural number n, let Fg (n) denote the span of words of length at most n in Fg , and set Fg ×g (n) = Fg (n) ⊗ Fg (n). Let Xn denote the compression of X to the (semi-invariant finite dimensional) subspace Fg ×g (n). Lemma 2.2. Let Pn denote the projection onto the complement of the span of ∅ in Fg (n) (and also in Fg (n)) and let Qn denote the projection onto the complement of the span of {w | w is a word of length n} in Fg (n) (and also in Fg (n)). Then: X∗n Xn = Ig ⊗ Pn ⊗ Qn , Xn X∗n = Ig ⊗ Qn ⊗ Pn .

(2.2)

Remark 2.3. In view of the definition of Bg ×g , it is natural to think of an NC analytic function h on Bg ×g as a function of the g g variables xj, , 1 j g and 1 g. In turn, a monomial m in (xj, ) can be viewed as a homogeneous monomial u ⊗ v, where u and v are monomials of the same length (same as the length of m) and u and v monomials in NC variables yj (1 j g ) and z (1 g) respectively. In this way, au⊗v u ⊗ v = h(α) . h= α |u|=|v|=α

α

For instance, the monomial x23 x41 is identified with y2 y4 ⊗ z3 z1 . We want to evaluate NC analytic functions Bg ×g → Md ×d on Xn , which is a norm one matrix thereby causing power series convergence difficulties. However, evaluating NC analytic functions on nilpotent tuples X ∈ Bg ×g behaves especially well. Here a tuple X is called nilpotent of order β if w(X) = 0 for every word w of length β. Lemma 2.4. If f : Bg ×g → Md×d is NC analytic and X ∈ Bg ×g is nilpotent of order β, then f (X) is defined and moreover, f (α) (X). f (X) = α 1. To establish the affine Lp Sobolev inequality for p > 1, Lutwak, Yang and Zhang [25] had to first establish an Lp Petty projection inequality. In this article we establish a new sharp affine Lp Sobolev inequality which strengthens and directly implies the previously known sharp affine Lp Sobolev inequality of Lutwak, Yang, and Zhang. The geometry behind this new Sobolev inequality is an Lp affine isoperimetric inequality, stronger than the Lp Petty projection inequality, which was recently established by the authors in [13]. This crucial geometric inequality was made possible by recent advances in valuation theory by Ludwig [17,19]. We denote by W 1,p (Rn ) the space of real-valued Lp functions on Rn (n 2) with weak Lp partial derivatives. Let | · | denote the standard Euclidean norm on Rn and let f p denote the usual Lp norm of f in Rn . The classical sharp Lp Sobolev inequality states that if f ∈ W 1,p (Rn ), with real p satisfying 1 p < n, then

1/p |∇f | dx

cˆn,p f p∗ ,

p

(1.1)

Rn

where p ∗ = np/(n − p). The optimal constants cˆn,p in this inequality are due to Federer and Fleming [10] and Maz’ya [31] for p = 1 and to Aubin [1] and Talenti [36] for p > 1. The extremal functions for inequality (1.1) are the characteristic functions of balls for p = 1 and for p > 1 equality is attained when p/(p−1) 1−n/p f (x) = a + b(x − x0 ) , with a, b > 0, and x0 ∈ Rn . The sharp affine Lp Sobolev inequality of Zhang [38] and Lutwak, Yang, and Zhang [27] states that if f ∈ W 1,p (Rn ), 1 p < n, then

−1/n

Du f −n p du

cñ,p f p∗ ,

(1.2)

S n−1

where Du f is the directional derivative of f in the direction u ∈ S n−1 . The optimal constants cñ,p in (1.2) were explicitly computed in [38] (for p = 1) and [27]. The determination of cˆn,p and cñ,p in (1.1) and (1.2) is in many situations not as important as the identification of extremal functions. The extremals associated with inequality (1.2) for p = 1 are the characteristic functions of ellipsoids and for p > 1 equality is attained when p/(p−1) 1−n/p , f (x) = a + φ(x − x0 ) with a > 0, φ ∈ GL(n) and x0 ∈ Rn . We emphasize that inequality (1.2) is invariant under affine transformations of Rn , while the classical Lp Sobolev inequality (1.1) is invariant only under rigid motions. That the affine

C. Haberl, F.E. Schuster / Journal of Functional Analysis 257 (2009) 641–658

643

Lp Sobolev inequality is stronger than (1.1) follows from an application of Hölder’s inequality (cf. [27, p. 33]):

1/p |∇f |p dx

an,p

Rn

−1/n Du f −n du cˆn,p f p∗ . p

S n−1

Here, equality in the left inequality holds if and only if Du f p is independent of u ∈ S n−1 . The constant an,p was computed in [27]. For u ∈ S n−1 and f ∈ W 1,p (Rn ), we denote by D+ u f (x) = max Du f (x), 0 the positive part of the directional derivative of f in the direction u. The main result of this article is the following: Theorem 1. If f ∈ W 1,p (Rn ), with 1 p < n, then

−1/n

+ −n

D f du cn,p f p∗ , u

p

(1.3)

S n−1

where p ∗ = np/(n − p). For p > 1, the optimal constant cn,p is given by cn,p = 2

−1/p

n−p p−1

1−1/p Γ n Γ n + 1 − n 1/n p

p

Γ (n + 1)

1/p nΓ n2 Γ p+1 2 , √ πΓ n+p 2

and cn,1 = limp→1 cn,p . If p = 1, equality holds in (1.3) for characteristic functions of ellipsoids and for p > 1 equality is attained when p/(p−1) 1−n/p f (x) = a + φ(x − x0 ) , with a > 0, φ ∈ GL(n) and x0 ∈ Rn . Note that inequality (1.3) is invariant under affine transformations of Rn . We will show in Section 6 that, for p 1, S n−1

−1/n −1/n

+ −n 1/p

D f du Du f −n du 2 . p u p

(1.4)

S n−1

Since cñ,p = 21/p cn,p , the new affine Lp Sobolev inequality (1.3) is stronger than inequality (1.2) of Lutwak, Yang, and Zhang. In particular, inequality (1.3) is also stronger than the classical Lp Sobolev inequality (1.1). It is crucial to observe that while for inequality (1.2) only the even part of the directional derivatives of f contribute, for the new inequality (1.3) also asymmetric parts are accounted for. This is reflected by the fact that equality in (1.4) holds precisely when n−1 . D+ u f p is an even function on S

644


The classical L2 Sobolev inequality has drawn particular attention due to its conformal invariance, see, e.g., [3,6,16]. As noted in [27], the affine L2 Sobolev inequality of Lutwak, Yang, and Zhang is equivalent under an affine transformation to the L2 Sobolev inequality. The case p = 2 of inequality (1.3), however, yields a stronger inequality. While the geometric inequalities behind the affine Zhang–Sobolev inequality and inequality (1.3) for p = 1 are the same, a new affine isoperimetric inequality recently established by the authors [13] is needed to establish inequality (1.3) for p > 1. We will apply this inequality to convex bodies (associated with the given function) which occur as solutions to the Lp Minkowski problem for 1 < p < n. Since the geometric inequality assumes that the convex bodies contain the origin in their interiors, its application is intricate in the asymmetric situation. Here, the origin can lie on the boundary of the convex bodies which occur as a solution to the Lp Minkowski problem. All this geometric background will be discussed in detail in Sections 3 and 4. 2. Background material In the following we state some basic facts about convex bodies and compact domains. General references for the theory of convex bodies are the books by Gardner [12] and Schneider [35]. We will also collect background material from real analysis needed in the proof of Theorem 1. The setting for this article is Euclidean n-space Rn with n 2. A convex body is a compact convex set in Rn with non-empty interior. Let Kn denote the set of convex bodies in Rn endowed with the Hausdorff metric. We write Kon for the set of convex bodies containing the origin in their interiors. A compact convex set K is uniquely determined by its support function h(K, ·), where h(K, x) = max{x · y: y ∈ K}, x ∈ Rn , and where x · y denotes the usual inner product of x and y in Rn . Note that h(K, ·) is positively homogeneous of degree one and subadditive. Conversely, every function with these properties is the support function of a unique compact convex set. If K ∈ Kon , the polar body K ∗ of K is defined by K ∗ = x ∈ Rn : x · y 1 for all y ∈ K . Let ρ(K, x) = max{λ 0: λx ∈ K}, x ∈ Rn \ {0}, denote the radial function of K. It follows from the definitions of support functions and radial functions, and the definition of the polar body of K, that ρ(K ∗ , ·) = h(K, ·)−1

and h(K ∗ , ·) = ρ(K, ·)−1 .

(2.1)

A compact domain is the closure of a bounded open subset of Rn . If M and N are compact domains in Rn , then the Brunn–Minkowski inequality states that V (M + N )1/n V (M)1/n + V (N )1/n , where V denotes the usual n-dimensional Lebesgue measure. For a compact domain M and a convex body K in Rn , define nV1 (M, K) = lim inf ε→0+

V (M + εK) − V (M) . ε


If the boundary ∂M of M is a C 1 submanifold of Rn , then 1 V1 (M, K) = h K, ν(x) dHn−1 (x), n

645

(2.2)

∂M

where ν(x) is the exterior unit normal vector of ∂M at x and Hn−1 denotes (n − 1)-dimensional Hausdorff measure (cf. [38, Lemma 3.2]). We need the following immediate consequence of the Brunn–Minkowski inequality: If M is a compact domain and K is a convex body in Rn , then V1 (M, K)n V (M)n−1 V (K).

(2.3)

We will frequently apply Federer’s co-area formula (see, e.g., [9, p. 258]). For quick reference we state a version which is sufficient for our purposes: If f : Rn → R is locally Lipschitz and g : Rn → [0, ∞) is measurable, then, for any Borel set A ⊆ R, g(x) g(x) dx = (2.4) dHn−1 (x) dy. |∇f (x)| f −1 (A)∩{|∇f |>0}

A f −1 {y}

Finally, we require the following consequence (cf. [2, Proposition 2.18]) of Bliss’ inequality [4]. For an elementary proof we refer to [27, Lemma 4.1]: Let f : (0, ∞) → [0, ∞) be decreasing and locally absolutely continuous and let 1 < p < n. If the integrals exist, then 1/p 1/p∗ ∞ ∞ p n−1 ∗ f (x) x dx bn,p f (x)p x n−1 dx , 0

(2.5)

0

where p ∗ = np/(n − p) and bn,p = n

1/p ∗

n−p p−1

1−1/p Γ n Γ n + 1 − n 1/n p

p

Γ (n)

.

Equality in (2.5) holds if f (x) = (ax p/(p−1) + b)1−n/p , with a, b > 0. 3. Lp projection bodies and the Lp Minkowski problem In this section we collect the material which forms the geometric core in the proof of our main result. The critical ingredients are an Lp affine isoperimetric inequality recently established in [13] and the solution (to the discrete data case) of an Lp extension of the classical Minkowski problem obtained in [7]. The projection body ΠK of K ∈ Kn is the convex body defined by h(ΠK, u) = voln−1 K | u⊥ , where voln−1 (K | u⊥ ) is the (n − 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u.

646


Introduced by Minkowski, projection bodies have become a central notion in convex geometry, see, e.g., [12,13,17,26] and the references therein. A recent result by Ludwig [19] has demonstrated their special place in affine geometry: The projection operator was characterized as the unique valuation which is contravariant with respect to linear transformations. The fundamental affine isoperimetric inequality for projection bodies is the Petty projection inequality: If K ∈ Kn , then V (K)n−1 V (Π ∗ K)

κn κn−1

n ,

with equality if and only if K is an ellipsoid. Here Π ∗ K = (ΠK)∗ and κn denotes the volume of the Euclidean unit ball in Rn . This inequality turned out to be far stronger than the classical isoperimetric inequality. It is the geometric inequality behind the affine Zhang–Sobolev inequality [38]. Projection bodies are part of the classical Brunn–Minkowski theory. In a series of articles [22,23], Lutwak showed that merging the notion of volume with Firey’s Lp addition of convex sets leads to a Brunn–Minkowski theory for each p 1. Since Lutwak’s seminal work, the topic has been much studied, see, e.g., [5,7,18–21,24,26,27,29,30]. For p 1, K, L ∈ Kon and α, β 0 (not both zero), the Lp Minkowski combination α · K +p β · L is the convex body defined by h(α · K +p β · L, ·)p = αh(K, ·)p + βh(L, ·)p . One of the basic notions of the Lp Brunn–Minkowski theory is the Lp mixed volume Vp (K, L) of two bodies K, L ∈ Kon . It was defined in [22] by Vp (K, L) =

V (K +p ε · L) − V (K) p lim . n ε→0+ ε

Clearly, the diagonal form of Vp reduces to ordinary volume, i.e., for K ∈ Kon , Vp (K, K) = V (K).

(3.1)

It was shown in [22] that corresponding to each convex body K ∈ Kon , there exists a positive Borel measure on S n−1 , the Lp surface area measure Sp (K, ·) of K, such that for every L ∈ Kon , Vp (K, L) =

1 n

h(L, u)p dSp (K, u).

(3.2)

S n−1

The measure S1 (K, ·) is just the classical surface area measure S(K, ·) of K. Moreover, it was proved in [22], that the Lp surface area measure is absolutely continuous with respect to S(K, ·): dSp (K, u) = h(K, u)1−p dS(K, u),

u ∈ S n−1 .

(3.3)

Recall that for a Borel set ω ⊆ S n−1 , S(K, ω) is the (n − 1)-dimensional Hausdorff measure of the set of all boundary points of K for which there exists a normal vector of K belonging to ω.


647

From the homogeneity properties of the surface area measure and the support function of K, one obtains that, for every λ > 0, Sp (λK, ·) = λn−p Sp (K, ·).

(3.4)

n−1 , the For a finite Borel measure μ on S n−1 , we define a continuous function C+ p μ on S asymmetric Lp cosine transform of μ, by + p Cp μ (u) = (u · v)+ dμ(v), u ∈ S n−1 , S n−1

where (u · v)+ = max{u · v, 0}. For f ∈ C(S n−1 ), let C+ p f be the asymmetric Lp cosine transform of the absolutely continuous measure (with respect to spherical Lebesgue measure) with density f . The asymmetric Lp projection body Πp+ K of K ∈ Kon , first considered in [23], is the convex body defined by p (3.5) h Πp+ K, · = C+ p Sp (K, ·). For p > 1, Ludwig [19] established the Lp analogue of her classification of the projection operator: She showed that the convex bodies c1 · Πp+ K +p c2 · Πp− K,

K ∈ Kon ,

(3.6)

where Πp− K = Πp+ (−K) and c1 , c2 0 (not both zero), constitute all natural Lp extensions of projection bodies. The (symmetric) Lp projection body Πp K of K ∈ Kon , defined in [26], is Πp K =

1 1 · Πp+ K +p · Πp− K. 2 2

Lutwak, Yang, and Zhang [26] (see also Campi and Gronchi [5]) established an Lp extension of the Petty projection inequality for the (symmetric) Lp projection operator which forms the geometry behind their sharp affine Lp Sobolev inequality: If K ∈ Kon , then n/p κn Γ n+p 2 V (K)n/p−1 V Πp∗ K , (3.7) π (n−1)/2 Γ 1+p 2 with equality if and only if K is an ellipsoid centered at the origin. Recently the authors [13] established the Lp Petty projection inequality for each member of the family (3.6) of Lp projection operators. The geometric core of the asymmetric affine Lp Sobolev inequality (1.3) is the following special case of this result: Theorem 2. If p > 1 and K ∈ Kon , then n/p−1

V (K)

V Πp+,∗ K

κn Γ

n+p

π (n−1)/2 Γ

n/p

2

1+p 2

where equality is attained if K is an ellipsoid centered at the origin.

,

(3.8)

648


Although this inequality was formulated in [13] for dimensions n 3, we remark that it also holds true in dimension n = 2. The proof is verbally the same as the one given in [13]. Since surface area measures have their center of mass at the origin, we have Π1+ K = ΠK. Thus, for p = 1, inequality (3.8) is the classical Petty projection inequality. It was also shown in [13] that inequality (3.8), for p > 1, is stronger than the Lp Petty projection inequality (3.7) of Lutwak, Yang, and Zhang: If K ∈ Kon , then V Πp∗ K V Πp+,∗ K .

(3.9)

If p is not an odd integer, equality holds precisely for origin-symmetric K. We turn now to the second main ingredient in the proof of Theorem 1. The Lp Minkowski problem asks for necessary and sufficient conditions for a Borel measure μ on S n−1 to be the Lp surface area measure of a convex body. A solution to this problem for p > n was given by Chou and Wang [7]. Moreover, Chou and Wang [7] established the solution to the discrete-data case of the Lp Minkowski problem for all p > 1 (see also [15] for an alternate approach). The following solution to the discrete Lp Minkowski problem due to Chou and Wang will be crucial: Theorem 3. If α1 , . . . , αk > 0 and u1 , . . . , uk ∈ S n−1 are not contained in a closed hemisphere, then, for any p > 1, p = n, there exists a unique polytope P ∈ Kon such that k

αj δuj = Sp (P , ·).

j =1

Here, δu denotes the probability measure with unit point mass at u ∈ S n−1 . We will also apply two auxiliary results [28, Lemmas 2.2 and 2.3] concerning the volume normalized Lp Minkowski problem: Let μ be a positive Borel measure on S n−1 , and let K ∈ Kn contain the origin. Suppose that V (K)h(K, ·)p−1 μ = S(K, ·), and that for some constant c > 0,

p

(u · v)+ dμ(v)

n cp

for every u ∈ S n−1 .

S n−1

Then V (K) κn

n μ(S n−1 )

n/p

where Bn denotes the Euclidean unit ball in Rn .

and K ⊂ cBn ,

(3.10)


649

4. A critical lemma A crucial part in the proof of our main result is the construction of a family of convex bodies containing the origin in their interiors from a given function. It is essential that the origin is an interior point in order to apply the critical geometric inequality (3.8) afterwards. In [26], this was done by using the solution to the even Lp Minkowski problem. In our case, we have to deal with the solutions to the general Lp Minkowski problem. Here, the bodies can contain the origin in their boundaries (cf. [15]). Therefore, we will associate a two parametric family of convex polytopes with a given function. These polytopes are obtained from the solution to the discretedata case of the Lp Minkowski problem which ensures that they contain the origin as an interior point. This will allow us to use the relevant geometric inequality. A function f ∈ C ∞ (Rn ) is called smooth. Suppose f is smooth and has compact support. Then the level set [f ]t = x ∈ Rn : f (x) t is compact for every 0 < t f ∞ , where f ∞ denotes the maximum value of |f | over Rn . Lemma 1. Suppose that f : Rn → R is smooth and has compact support. Then, for almost every t ∈ (0, f ∞ ), there exists a sequence of convex polytopes Pkt ∈ Kon , k ∈ N, such that lim Pkt = Kft ∈ Kn

k→∞

and 1 V Kft = n

−1 p h Kft , ∇f (x) ∇f (x) dHn−1 (x).

(4.1)

∂[f ]t

Moreover, there exists a convex body Ltf ∈ Kon such that lim Πp+ Pkt = Ltf .

k→∞

Proof. By Sard’s theorem, for almost every t ∈ (0, f ∞ ), the boundary ∂[f ]t of [f ]t is a smooth (n − 1)-dimensional submanifold with everywhere nonzero normal vector ∇f . Let t be chosen in this way and denote by ν(x) = ∇f (x)/|∇f (x)| the unit normal of ∂[f ]t at x. Let μt be the finite positive Borel measure on S n−1 defined by

g(v) dμt (v) =

S n−1

p−1 g ν(x) ∇f (x) dHn−1 (x),

(4.2)

∂[f ]t

for g ∈ C(S n−1 ). Since ν(x): x ∈ ∂[f ]t = S n−1 ,

(4.3)

650


it follows that for every u ∈ S n−1 , p−1 t u · ν(x) + ∇f (x) (u · v)+ dμ (v) = dHn−1 (x) > 0. ∂[f ]t

S n−1

Therefore, the measure μt cannot be concentrated in a closed hemisphere. As in [35, pp. 392–393], construct a sequence μtk , k ∈ N, of discrete measures on S n−1 whose support is not contained in a closed hemisphere and such that μtk converges weakly to μt as k → ∞. By Theorem 3, for each k ∈ N, there exists a polytope Pkt ∈ Kon such that μtk = Sp Pkt , · .

(4.4)

We want to show that the sequence of polytopes Pkt is bounded. To this end, define for each k ∈ N a new polytope Qtk by −1/p t Qtk = V Pkt Pk . By (3.3) and the homogeneity (3.4) of Lp surface area measures, the polytopes Qtk , k ∈ N, form a solution to the volume normalized Lp Minkowski problem p−1 t V Qtk h Qtk , · μk = S Qtk , · .

(4.5)

Moreover, from definition (3.5), relation (4.4) and the weak convergence of the measures μtk , it follows that for every u ∈ S n−1 , + t p p p t (u · v)+ dμk (v) −→ (u · v)+ dμt (v) > 0. (4.6) h Πp Pk , u = S n−1

S n−1

Since pointwise convergence of support functions implies uniform convergence (see, e.g., [35, Theorem 1.8.12]), there exists a c > 0 such that for all k ∈ N, p (u · v)+ dμtk (v) > c, for every u ∈ S n−1 . (4.7) S n−1

From (4.5), (4.7) and (3.10), we deduce that the sequence Qtk , k ∈ N, is bounded. Moreover, by (3.10) and the weak convergence of the measures μtk , the volumes V (Qtk ) are bounded from below by a constant independent of k. Therefore, the original sequence Pkt = V (Qtk )1/(p−n) Qtk is also bounded. By the Blaschke selection theorem (see, e.g., [35, Theorem 1.8.6]), we can select a subsequence of the Pkt converging to a convex body Kft . After relabeling (if necessary) we may assume that limk→∞ Pkt = Kft . From (3.1), (3.2), and relation (4.4), we obtain 1 V Kft = lim V Pkt = lim k→∞ k→∞ n

S n−1

p h Pkt , v dμtk (v).


651

Thus, the uniform convergence of the support functions h(Pkt , ·), the weak convergence of the measures μtk , and definition (4.2), yield 1 V Kft = n

−1 p h Kft , ∇f (x) ∇f (x) dHn−1 (x).

∂[f ]t

t Finally, we define h(Ltf , ·)p = C+ p μ . By definition (4.2), we have

p h Ltf , u =

−1 p u · ∇f (x) + ∇f (x) dHn−1 (x),

u ∈ S n−1 .

(4.8)

∂[f ]t

From (4.6), we deduce that h(Ltf , ·) is the support function of a convex body Ltf ∈ Kon and that limk→∞ Πp+ Pkt = Ltf . 2 5. Proof of the main result After these preparations, we are now in a position to proof our main result. We want to point out that the approach we use to establish Theorem 1 is based on ideas and techniques of Lutwak, Yang, and Zhang [27]. We will need the decreasing rearrangement f¯ of a function f : Rn → R. It is defined by f¯(x) = inf t > 0: V [f ]t < κn |x|n . Note that the level set [f¯]t is a dilate of the unit ball Bn and its volume is equal to V ([f ]t ). Moreover, for all p 1, f p = f¯p .

(5.1)

We will first reduce the proof of Theorem 1 to the class of smooth functions with compact support. Lemma 2. In order to prove Theorem 1, it is sufficient to verify the following assertion: If f ∈ C ∞ (Rn ) has compact support and 1 p < n, then

−1/n

+ −n

D f du cn,p f p∗ . u

p

(5.2)

S n−1

Proof. Assume that (5.2) holds for smooth functions with compact support and let f ∈ W 1,p . We may assume that the set {x ∈ Rn : f (x) = 0} has positive measure. First, we will show that n−1 . D+ u f p > 0 for every u ∈ S We may assume that u = en is the last canonical basis vector. We denote the indicator function of a set A ⊆ Rn by IA . Since for each N ∈ N, almost all points in Rn are Lebesgue points of f · I[−N,N]n (see, e.g., [34, Theorem 7.7]), there exists an n-box P = [a1 , b1 ] × · · · × [an , bn ] such that P f = 0.

652


If

P

f > 0, then, since f ∈ Lp (Rn ), there exist real a < b < c such that b

c f
0. + Now suppose that Den f p = 0. This implies, by the definition of weak derivatives, that f ∂ φ 0 ( 0) for every smooth and compactly supported φ which is non-negative n Rn (non-positive). This is a contradiction to the above construction. Thus D+ u f p > 0 for every u ∈ S n−1 . Since f ∈ W 1,p , we can find a sequence fk , k ∈ N, of smooth functions with compact support such that fk − f p → 0

and ∂i fk − ∂i f p → 0

for i = 1, . . . , n. By Minkowski’s inequality we have cn,p fl − fm

p∗

−1/n

+

D (fl − fm ) −n du u

p

S n−1

n 1 1/n

ωn

∂i fl − ∂i fm p

i=1

for all l, m ∈ N, where ωn denotes the surface area of the Euclidean unit ball in Rn . Consequently, the sequence fk , k ∈ N, is a Cauchy sequence in Lp∗ (Rn ). By the completeness of Lp∗ (Rn ), there exists a function g such that fk − gp∗ → 0. Since sequences of functions converging in Lq , q > 0, posess a subsequence converging almost everywhere, we can find fkj , j ∈ N, such that fkj → f and fkj → g almost everywhere. We conclude that f = g almost everywhere and hence fk → f also in Lp∗ (Rn ). −n + −n n−1 . By the first part of the proof, limk→∞ D+ u fk p = Du f p for every unit vector u ∈ S Thus an application of Fatou’s lemma yields

+ −n

D f du = u

S n−1

p

−n

lim D+ u fk p du

k→∞

S n−1

lim inf k→∞

+ −n

D fk du u

p

S n−1 −n −n −n fk −n lim cn,p p ∗ = cn,p f p ∗ . k→∞

2


653

Proof of Theorem 1. In the following let p > 1. By Lemma 2 we may assume that f is a smooth function with compact support which is not identically zero. An application of the coarea formula (2.4) shows that

+ p

D f = u p

f ∞

p u · ∇f (x) + dx =

Rn

0

p

∂[f ]t

(u · ∇f (x))+ dHn−1 (x) dt. |∇f (x)|

By Lemma 1 and (4.8), there exists a convex body Ltf ∈ Kon such that

−p/n f −n/p −p/n ∞

+ −n t p

D f du = h L , u dt du . u f p

S n−1

S n−1

0

Since h(Ltf , ·) is positive, we can apply a consequence of Minkowski’s integral inequality (see, e.g., [14, p. 148]), to obtain

−p/n f −p/n ∞

+ −n t −n

D f du h L , u du dt. u f p 0

S n−1

S n−1

Using (2.1) and the polar coordinate formula for volume, we deduce

−p/n f ∞

+ −n t,∗ −p/n

D f du nV Lf dt. u p

(5.3)

0

S n−1

By Lemma 1, there exists a sequence of convex polytopes Pkt ∈ Kon such that limk→∞ Pkt = Kft ∈ Kn and limk→∞ Πp+ Pkt = Ltf . Thus, from an application of Theorem 2, we obtain

−p/n −p/n (n−p)/n nV Lt,∗ = lim nV Πp+,∗ Pkt en,p V Kft , f k→∞

(5.4)

where π (n−1)/2 Γ 1+p 2 en,p = . np/n κn Γ n+p 2 From (5.3) and (5.4), we deduce

f −p/n ∞

+ −n (n−p)/n

D f du en,p V Kt dt. u

S n−1

f

p

0

(5.5)

654


An application of Hölder’s integral inequality to volume formula (4.1), yields (n−p)/np V Kft n1−1/p

∂[f ]t

dHn−1 (x) |∇f (x)|

(1−p)/p

−1/n V Kft V1 [f ]t , Kft ,

where we have used integral representation (2.2). From inequality (2.3), we deduce further that (n−p)/n np−1 V Kft

∂[f ]t

dHn−1 (x) |∇f (x)|

1−p

(n−1)p/n V [f ]t .

(5.6)

Another application of the co-area formula (2.4), yields f ∞

t

∂[f ]s

dHn−1 (x) ds = V [f ]t ∩ |∇f | > 0 . |∇f (x)|

Using Sard’s theorem, it is not hard to show that for almost every t satisfying 0 < t < f ∞ , there exists a neighborhood Ut of t such that V f −1 (Ut ) ∩ |∇f | > 0 = V f −1 (Ut ) . Therefore, we obtain for almost every t with 0 < t < f ∞ , ∂[f ]t

dHn−1 (x) = −V [f ]t . |∇f (x)|

(5.7)

Combining (5.5), (5.6), and (5.7), we obtain S n−1

f −p/n ∞

+ −n en,p V ([f ]t )(n−1)p/n

D f du dt. u p n1−p (−V ([f ]t ) )p−1

(5.8)

0

In order to estimate the right integral in (5.8), define fˆ : (0, ∞) → R, by f¯(x) = fˆ 1/|x| . Since the decreasing rearrangement f¯(x) depends only on the Euclidean norm of x, the function fˆ is well defined and increasing. Noting that fˆ is locally Lipschitz, the substitution rule thus yields f ∞

0

V ([f ]t )(n−1)p/n 1−p/n dt = n1−p κn (−V ([f ]t ) )p−1

∞ 0

fˆ (s)p s 2p−n−1 ds.


655

Hence, we can rewrite (5.8) as

−p/n ∞

+ −n 1−p/n

D f du en,p κn fˆ (s)p s 2p−n−1 ds. u

p

(5.9)

0

S n−1

Using polar coordinates and (5.1), we see that ∗

p f¯p∗ = nκn

∞

∗

∗ p fˆ(s)p s −n−1 ds = f p∗ .

0

The substitution t = 1/s and an application of inequality (2.5), therefore yields ∞

1/p ˆ

p 2p−n−1

f (s) s 0

ds

bn,p ∗ 1/p n1/p κn

∗

f p∗ .

(5.10)

Finally, combine inequalities (5.9) and (5.10), to obtain the desired result

−1/n

+ −n

D f du cn,p f p∗ . u

p

(5.11)

S n−1

In order to see that inequality (1.3) is sharp, take for smooth K ∈ Kon , 1−n/p f (x) = 1 + ρ(K, x)p/(1−p) .

(5.12)

Then, a straightforward (but tedious) calculation shows that inequality (1.3) reduces to the Lp affine isoperimetric inequality (3.8), where equality holds if K is an ellipsoid centered at the origin. Clearly, the case p = 1 of inequality (1.3) can be obtained from a limit of inequality (5.11) as p → 1:

1 n

−1/n

+ −n κn−1

D f du f 1∗ . u 1 κn

(5.13)

S n−1

Noting that Π1+ = Π , one can show (cf. [38]) that for characteristic functions of convex bodies, inequality (5.13) reduces to the Petty projection inequality, where equality is attained for ellipsoids. 2 We remark that for p > 1 the affine Lp Sobolev inequality (1.2) of Lutwak, Yang, and Zhang reduces to the Lp Petty projection inequality (3.7) if we take f as in (5.12). Thus, it follows from (3.9) that the new inequality (1.3) is in general stronger than (1.2). We will make this fact even more explicit in the next section.

656


6. A stronger inequality In this last section we show that Theorem 1 provides a stronger result than the affine Lp Sobolev inequality (1.2) of Zhang and Lutwak, Yang, and Zhang. The basic concept behind this observation is a convex body associated with a given function f . For p 1 and f ∈ W 1,p (Rn ), let Bp+ (f ) be the convex body defined by h Bp+ (f ), u =

+ p Du f (x) dx

1/p

=

Rn

p u · ∇f (x) + dx

1/p .

Rn

From Minkowski’s integral inequality, we deduce that h(Bp+ (f ), ·) is sublinear and therefore the support function of a unique convex body Bp+ (f ). Moreover, by Lemma 2, this body contains the origin in its interior. By (2.1) and the polar coordinate formula for volume, the volume of its polar body is given by 1 V Bp+,∗ (f ) = n

+ −n

D f du. u

p

S n−1

Therefore, we can rewrite our main theorem as follows: Theorem 1 . If f ∈ W 1,p (Rn ), with 1 p < n, then −1/n V Bp+,∗ (f ) kn,p f p∗ . The optimal constant kn,p is given by kn,p = 2

−1/p

n−p p−1

1−1/p Γ n Γ n + 1 − n 1/n p

p

Γ (n)

1/p nΓ n2 Γ p+1 2 . √ πΓ n+p 2

From the definition of Lp Minkowski addition, it follows that h Bp+ (f ) +p Bp+ (−f ), u =

Du f (x)p dx

1/p .

(6.1)

Rn

Thus, the following reformulation of inequality (1.4) shows that Theorem 1 is stronger than inequality (1.2): Theorem 4. If p 1 and f ∈ W 1,p (Rn ), then ∗ V Bp+ (f ) +p Bp+ (−f ) 2−n/p V Bp+,∗ (f ) , with equality if and only if Bp+ (f ) is origin symmetric.


657

In order to prove this theorem, we need a result from the dual Lp Brunn–Minkowski theory. The basis of this theory is the following addition on convex bodies. For α, β 0 (not both zero), p β · L of K, L ∈ Kon is the convex body defined Firey’s Lp harmonic radial combination α · K + by p β · L, · −p = αρ(K, ·)−p + βρ(L, ·)−p . ρ α·K + Firey started investigations of harmonic Lp combinations in the 1960’s which were continued by Lutwak leading to a dual Lp Brunn–Minkowski theory. A cornerstone of this theory is the dual Lp Brunn–Minkowski inequality [23]: If K, L ∈ Kon , then p L −p/n V (K)−p/n + V (L)−p/n , V K+

(6.2)

with equality if and only if K and L are dilates. Proof of Theorem 4. From (2.1), (6.1) and the definition of Lp harmonic radial addition, it follows that + ∗ p Bp+,∗ (−f ). Bp (f ) +p Bp+ (−f ) = Bp+,∗ (f ) + Since V (Bp+,∗ (f )) = V (Bp+,∗ (−f )), an application of (6.2) yields the desired inequality along with its equality conditions. 2 Acknowledgments The authors are grateful for the help of Monika Ludwig, Erwin Lutwak, Christian Steineder, Deane Yang, and Gaoyong Zhang in the presentation of this article. This work was supported by the Austrian Science Fund (FWF), within the project P 18308, “Valuations on convex bodies”. References [1] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976) 573–598. [2] T. Aubin, Nonlinear Analysis on Manifolds: Monge–Ampère Equations, Springer, Berlin, 1982. [3] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math. 138 (1993) 213–242. [4] G.A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930) 40–46. [5] S. Campi, P. Gronchi, The Lp -Busemann–Petty centroid inequality, Adv. Math. 167 (2002) 128–141. [6] E.A. Carlen, M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990) 437–456. [7] K.-S. Chou, X.-J. Wang, The Lp -Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006) 33–83. [8] D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo– Nirenberg inequalities, Adv. Math. 182 (2004) 307–332. [9] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969. [10] H. Federer, W. Fleming, Normal and integral currents, Ann. Math. 72 (1960) 458–520. [11] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, J. Funct. Anal. 244 (2007) 315–341. [12] R. Gardner, Geometric Tomography, second ed., Cambridge Univ. Press, Cambridge, 2006. [13] C. Haberl, F.E. Schuster, General Lp affine isoperimetric inequalities, J. Differential Geom., in press. [14] G. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1952.

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[15] D. Hug, E. Lutwak, D. Yang, G. Zhang, On the Lp Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005) 699–715. [16] E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math. 118 (1983) 349– 374. [17] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002) 158–168. [18] M. Ludwig, Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003) 159–188. [19] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005) 4191–4213. [20] M. Ludwig, M. Reitzner, A classification of SL(n) invariant valuations, Ann. Math., in press. [21] E. Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986) 1–13. [22] E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993) 131–150. [23] E. Lutwak, The Brunn–Minkowski–Firey theory. II: Affine and geominimal surface areas, Adv. Math. 118 (1996) 244–294. [24] E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995) 227–246. [25] E. Lutwak, D. Yang, G. Zhang, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000) 111–132. [26] E. Lutwak, D. Yang, G. Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000) 375–390. [27] E. Lutwak, D. Yang, G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002) 17–38. [28] E. Lutwak, D. Yang, G. Zhang, On the Lp Minkowski problem, Trans. Amer. Math. Soc. 356 (2004) 4359–4370. [29] E. Lutwak, D. Yang, G. Zhang, Optimal Sobolev norms and the Lp Minkowski problem, Int. Math. Res. Not. 65 (2006) 1–21. [30] E. Lutwak, G. Zhang, Blaschke–Santaló inequalities, J. Differential Geom. 47 (1997) 1–16. [31] V.G. Maz’ya, Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR 133 (1960) 527–530. [32] V.G. Maz’ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, in: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 307–340. [33] V.G. Maz’ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings, J. Funct. Anal. 224 (2005) 408–430. [34] W. Rudin, Real and Complex Analysis, McGraw–Hill Book Co., New York, 1987. [35] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993. [36] G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pura Appl. 110 (1976) 353–372. [37] J. Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007) 417–435. [38] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999) 183–202.


On the sum of superoptimal singular values Alberto A. Condori Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Received 23 October 2008; accepted 3 April 2009 Available online 29 April 2009 Communicated by N. Kalton

Abstract In this paper, we study the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an m × n matrix function Φ, when is there a such that matrix function Ψ∗ in the set An,m k

trace Φ(ζ )Ψ∗ (ζ ) dm(ζ ) =

T

sup trace Φ(ζ )Ψ (ζ ) dm(ζ )? n,m

Ψ ∈ Ak

T

The set An,m is defined by k def An,m = Ψ ∈ H01 (Mn,m ): Ψ L1 (Mn,m ) 1, rank Ψ (ζ ) k a.e. ζ ∈ T . k To address this extremal problem, we introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. The main result of this paper is a characterization of the smallest number k for which

trace Φ(ζ )Ψ (ζ ) dm(ζ )

T

equals the sum of all the superoptimal singular values of an admissible matrix function Φ (e.g. a continuous matrix function) for some function Ψ ∈ An,m k . Moreover, we provide a representation of any such function Ψ when Φ is an admissible very badly approximable unitary-valued n × n matrix function. © 2009 Elsevier Inc. All rights reserved. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.002

660

A.A. Condori / Journal of Functional Analysis 257 (2009) 659–682

Keywords: Best and superoptimal approximation; Badly and very badly approximable matrix functions; Hankel and Toeplitz operators

1. Introduction The problem of best analytic approximation for a given m × n matrix-valued bounded function Φ on the unit circle T is to find a bounded analytic function Q such that Φ − QL∞ (Mm,n ) = inf Φ − F L∞ (Mm,n ) : F ∈ H ∞ (Mm,n ) . Throughout, def Ψ L∞ (Mm,n ) = ess supΨ (ζ )M

m,n

ζ ∈T

,

Mm,n denotes the space of m × n matrices equipped with the operator norm · Mm,n (of the space of linear operators from Cn to Cm ), and H ∞ (Mm,n ) denotes the space of bounded analytic m × n matrix-valued functions on T. It is well known that, unlike scalar-valued functions, a polynomial matrix function Φ may have many best analytic approximants. Therefore it is natural to impose additional conditions in order to distinguish a “very best” analytic approximant among all best analytic approximants. To do so here, we use the notion of superoptimal approximation by bounded analytic matrix functions. 1.1. Superoptimal approximation and very badly approximable matrix functions Recall that for an m × n matrix A, the j th-singular value sj (A), j 0, is defined to be the distance from A to the set of matrices of rank at most j under the operator norm. More precisely, sj (A) = inf A − BMm,n : B ∈ Mm,n such that rank B j . Clearly, s0 (A) = AMm,n . Definition 1.1. Let Φ ∈ L∞ (Mm,n ). For k 0, we define the sets Ωk = Ωk (Φ) by Ω0 (Φ) = F ∈ H ∞ (Mm,n ): F minimizes ess supΦ(ζ ) − F (ζ )M

m,n

ζ ∈T

Ωj (Φ) = F ∈ Ωj −1 : F minimizes ess sup sj Φ(ζ ) − F (ζ ) ζ ∈T

and

for j > 0.

Any function F ∈ k0 Ωk = Ωmin{m,n}−1 is called a superoptimal approximation to Φ by bounded analytic matrix functions. In this case, the superoptimal singular values of Φ are defined by tj = tj (Φ) = ess sup sj (Φ − F )(ζ ) ζ ∈T

for j 0.


661

Moreover, if the zero matrix function O belongs to Ωmin{m,n}−1 , we say that Φ is very badly approximable. Notice that any function F ∈ Ω0 is a best analytic approximation to Φ. Also, any very badly approximable matrix function is the difference between a bounded matrix function and its superoptimal approximant. It turns out that Hankel operators on Hardy spaces play an important role in the study of superoptimal approximation. For a matrix function Φ ∈ L∞ (Mm,n ), we define the Hankel operator HΦ by HΦ f = P− Φf

for f ∈ H 2 Cn , def

where P− denotes the orthogonal projection from L2 (Cm ) onto H−2 (Cm ) = L2 (Cm ) H 2 (Cm ). When studying superoptimal approximation, we only consider bounded matrix functions that are admissible. A matrix function Φ ∈ L∞ (Mm,n ) is said to be admissible if the essential norm HΦ e of the Hankel operator HΦ is strictly less than the smallest non-zero superoptimal singular value of Φ. As usual, the essential norm of a bounded linear operator T between Hilbert spaces is defined by def T e = T − K: K is compact . Note that any continuous matrix function Φ is admissible, as the essential norm of HΦ equals zero in this case. Moreover, in the case of scalar-valued functions, to say that a function ϕ is admissible simply means that Hϕ e < Hϕ . It is known that if Φ is an admissible matrix function, then Φ has a unique superoptimal approximation Q by bounded analytic matrix functions. Moreover, the functions ζ → sj ((Φ − Q)(ζ )) equal tj (Φ) a.e. on T for each j 0. These results were first proved in [6] for the special case Φ ∈ (H ∞ + C)(Mm,n ) (i.e. matrix functions which are a sum of a bounded analytic matrix function and a continuous matrix function), and shortly after proved for the class of admissible matrix functions in [5]. While it is possible to compute the superoptimal singular values of a given matrix function in concrete examples, it is not known how to verify if a matrix function that is not continuous is admissible or not. Thus a complete characterization of the smallest non-zero superoptimal singular value of a given matrix function is an important problem for superoptimal approximation. This remains an open problem. We refer the reader to Chapter 14 of [2] which contains proofs to all of the previously mentioned results and many other interesting results concerning superoptimal approximation. 1.2. An extremal problem Throughout this note, m denotes normalized Lebesgue measure on T so that m(T) = 1. For 1 p ∞, Lp (Mm,n ) denotes the space of m × n matrix-valued functions on T whose entries belong to Lp . We equip Lp (Mm,n ) with the norm · Lp (Mm,n ) , where

662

A.A. Condori / Journal of Functional Analysis 257 (2009) 659–682 p F Lp (Mm,n )

=

F (ζ )p

Mm,n

dm(ζ )

for 1 p < ∞,

and

T

F

L∞ (Mm,n )

= ess supF (ζ )M

m,n

ζ ∈T

.

p

H p (Mm,n ) and H0 (Mm,n ) consist of matrix-valued functions in Lp (Mm,n ) whose entries bep p long to the Hardy space H p and H0 , respectively. (Recall that H p and H0 denote the spaces of p L functions on T whose Fourier coefficients of negative and non-positive index vanish, respectively.) Definition 1.2. Let m, n > 1 and 1 k min{m, n}. For Φ ∈ L∞ (Mm,n ), we define σk (Φ) by def (1.1) σk (Φ) = sup trace Φ(ζ )Ψ (ζ ) dm(ζ ), Ψ ∈An,m k

T

where = Ψ ∈ H01 (Mn,m ): Ψ L1 (Mn,m ) 1 and rank Ψ (ζ ) k a.e. ζ ∈ T . An,m k def

Whenever n = m, we use the notation Ank = An,m k . We are interested in the following extremal problem: Extremal problem 1.1. For a matrix function Φ ∈ L∞ (Mm,n ), when is there a matrix function such that Ψ ∈ An,m k

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = σk (Φ)?

T

The importance of this problem arose from the following observation due to Peller [3]. Theorem 1.3. Let 1 k min{m, n}. If Φ ∈ L∞ (Mm,n ) is admissible, then σk (Φ) t0 (Φ) + · · · + tk−1 (Φ).

(1.2)

Proof. Let Ψ ∈ An,m k . We may assume, without loss of generality, that Φ is very badly approximable. Indeed, trace Φ(ζ )Ψ (ζ ) dm(ζ ) = trace (Φ − Q)(ζ )Ψ (ζ ) dm(ζ ) T

T

holds for any Q ∈ H ∞ (Mm,n ), and so we may replace Φ with Φ − Q if necessary, where Q is the superoptimal approximation to Φ in H ∞ (Mm,n ). m Let S m 1 denote the collection of m × m matrices equipped with the trace norm AS 1 = ∗ 1/2 trace(A A) = j 0 sj (A).


663

It follows from the well-known inequality |trace(A)| AS m1 that the inequalities trace Φ(ζ )Ψ (ζ ) Φ(ζ )Ψ (ζ )

k−1 sj Φ(ζ ) Ψ (ζ )M

Sm 1

n,m

j =0

hold for a.e. ζ ∈ T. Thus, k−1 trace Φ(ζ )Ψ (ζ ) dm(ζ ) sj Φ(ζ ) Ψ (ζ )M dm(ζ ) n,m T

j =0

T

k−1 j =0

T

k−1

tj (Φ) Ψ (ζ )M

n,m

dm(ζ )

tj (Φ) Ψ L1 (Mn,m )

j =0

k−1

(1.3)

tj (Φ),

j =0

because the singular values of Φ satisfy sj (Φ(ζ )) = tj (Φ) for a.e. ζ ∈ T since Φ is very badly approximable. 2 Before proceeding, let us observe that equality holds in (1.2) for some simple cases. Let r be a positive integer and t0 , t1 , . . . , tr−1 be positive numbers satisfying t0 t1 · · · tr−1 . Suppose Φ is an n × n matrix function of the form ⎛

u0 ⎜O def ⎜ . Φ=⎜ ⎜ .. ⎝O O

O t1 u1 .. .

... ... .. .

O O .. .

O O .. .

O O

... ...

tr−1 ur−1 O

⎞

⎟ ⎟ ⎟, ⎟ O⎠ Φ#

(1.4)

¯ h with θj an inner where Φ# L∞ tr−1 and uj is a unimodular function of the form uj = z¯ θ¯j h/ function for 0 j r − 1 and h an outer function in H 2 . Without loss of generality, we may assume that hL2 = 1. It can be seen that if ⎛

zθ0 h2 ⎜ O def ⎜ . Ψ =⎜ ⎜ .. ⎝ O O

O zθ1 h2 .. . O O

... ... .. . ... ...

O O .. . zθr−1 O

h2

⎞ O O⎟ .. ⎟ ⎟ . ⎟, O⎠ O

(1.5)

664


then Ψ ∈ H01 (Mn ), rank Ψ (ζ ) = r a.e. on T, Ψ L1 (Mn ) = 1, and

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 + · · · + tr−1 .

T

Thus we obtain that σr (Φ) = t0 (Φ) + · · · + tr−1 (Φ). On the other hand, one cannot expect the inequality (1.2) to hold with equality in general. After all, by the Hahn–Banach theorem, distL∞ (S n1 ) Φ, H ∞ (Mn ) = σn (Φ),

(1.6)

and there are admissible very badly approximable 2 × 2 matrix functions Φ for which the strict inequality distL∞ (S 2 ) Φ, H ∞ (M2 ) < t0 (Φ) + t1 (Φ) 1

holds. For instance, consider the matrix function 1 z¯ O 1 1 z¯ z¯ z¯ 2 Φ= . =√ √ O z¯ 2 −z 1 2 −1 z¯ Clearly, Φ has superoptimal singular values t0 (Φ) = t1 (Φ) = 1. Let 1 O O . F=√ 2 −1 O It is not difficult to verify that √ 1 s0 (Φ − F )(ζ ) = 3+ 5 2

√ 1 and s1 (Φ − F )(ζ ) = 3− 5 2

for all ζ ∈ T. Therefore distL∞ (S 2 ) Φ, H ∞ (M2 ) Φ − F L∞ (S 2 ) < 2 = t0 (Φ) + t1 (Φ). 1

1

(1.7)

1.3. What is done in this paper? In virtue of Theorem 1.3 and the remarks following it, one may ask whether it is possible to characterize the matrix functions Φ for which (1.2) becomes an equality. So let Φ be an admissible n × n matrix function with a superoptimal approximant Q in H ∞ (Mn ) for which equality in Theorem 1.3 holds with k = n. In this case, it must be that n−1 n−1 distL∞ (S n1 ) Φ, H ∞ (Mn ) = tj (Φ) = sj (Φ − Q)(ζ ) = Φ − QL∞ (S n1 ) j =0

j =0


665

by (1.6) and thus the superoptimal approximant Q must be a best approximant to Φ under the L∞ (S n1 ) norm as well. Hence, we are led to investigate the following problems: 1. For which matrix functions Φ does Extremal problem 1.1 have a solution? 2. If Q$ is a best approximant to Φ under the L∞ (S n1 )-norm, when does it follow that Q$ is the superoptimal approximant to Φ in L∞ (Mn )? 3. Can we find necessary and sufficient conditions on Φ to obtain equality in (1.2) of Theorem 1.3? Before addressing these problems, we recall certain standard principles of functional analysis in Section 2 that are used throughout the paper. In particular, we give their explicit formulation for the spaces Lp (S m,n q ). {k}

In Section 3, we introduce the Hankel-type operators HΦ on spaces of matrix functions and {k} k-extremal functions, and prove that the number σk (Φ) equals the operator norm of HΦ . We {k} also show that Extremal problem 1.1 has a solution if and only if the Hankel-type operator HΦ has a maximizing vector, and thus answer question 1 in terms Hankel-type operators. In Section 4, we establish the main results of this paper concerning best approximation under the L∞ (S m,n 1 ) norm (Theorem 4.7) and the sum of superoptimal singular values (Theorem 4.13). The latter result characterizes the smallest number k for which trace Φ(ζ )Ψ (ζ ) dm(ζ ) T

equals the sum of all non-zero superoptimal singular values for some function Ψ ∈ An,m k . These results serve as partial solutions to problems 2 and 3. Lastly, in Section 5, we restrict our attention to unitary-valued very badly approximable matrix functions. For any such matrix function U , we provide a representation of any function Ψ for which the formula trace U (ζ )Ψ (ζ ) dm(ζ ) = n T

holds. 2. Best approximation and dual extremal problems We now provide explicit formulation of some basic results concerning best approximation q m,n in H q (S m,n p ) for functions in L (S p ) and the corresponding dual extremal problem. We first consider the general setting. 2.1. Best approximation Definition 2.1. Let X be a normed space, M be a closed subspace of X, and x0 ∈ X. We say that m0 is a best approximant to x0 in M if m0 ∈ M and def x0 − m0 X = dist(x0 , M) = inf x0 − mX : m ∈ M .

666


It is known that if X is a reflexive Banach space and M is a closed subspace of X, then each x0 ∈ X \ M has a best approximant m0 in M. Two standard principles from functional analysis are used throughout this note. Namely, if X is a normed space with a linear subspace M, then for any Λ0 ∈ X ∗ and x0 ∈ X sup

Λ0 (m) = min Λ0 − Λ: Λ ∈ M ⊥

max

Λ(x0 ) = dist(x0 , M)

m∈M, m1

Λ∈M ⊥ , Λ1

and

whenever M is closed.

We now discuss these results in the case of the spaces Lq (S m,n p ). 2.2. The spaces Lq (S m,n p ) Let 1 q < ∞ and 1 p ∞. Let p denote the conjugate exponent to p, i.e. p = p/(p − 1). Let S m,n p denote the space of m × n matrices equipped with the Schatten–von Neumann norm , i.e. for A ∈ Mm,n · S m,n p def

AS m,n = AMm,n ∞

def

and AS m,n = p

1/p

p sj (A)

for 1 p < ∞.

j 0

def

We also use the notation S np = S n,n p . If X is a normed space of functions on T with norm · X , then X(S m,n p ) denotes the space ), we define of m × n matrix functions whose entries belong to X. For Φ ∈ X(S m,n p def

= ρX , ΦX(S m,n p )

def where ρ(ζ ) = Φ(ζ )S m,n for ζ ∈ T. p

n,m q It is known that the dual space of Lq (S m,n p ) is isometrically isomorphic to L (S p ) via the

mapping Φ → ΛΦ , where Φ ∈ Lq (S n,m p ) and ΛΦ (Ψ ) =

trace Φ(ζ )Ψ (ζ ) dm(ζ )

. for Ψ ∈ Lq S m,n p

T q

n,m q m,n In particular, it follows that the annihilator of H q (S m,n p ) in L (S p ) is given by H0 (S p ), and so q m,n trace Φ(ζ )Ψ (ζ ) dm(ζ ), Φ, H S = max distLq (S m,n p p ) Ψ

q n,m 1 H0 (S ) p

T

by our remarks in Section 2.1. Moreover, if 1 < q < ∞, then Φ ∈ Lq (S m,n p ) has a best approxiq m,n mant Q in H q (S m,n p ) (as L (S p ) is reflexive); that is, Φ, H q S m,n . = distLq (S m,n Φ − QLq (S m,n p p ) p )


667

∞ m,n The situation is similar in the case of L∞ (S m,n p ). Indeed, L (S p ) is a dual space, and so m,n there is a Q ∈ H ∞ (S p ) such that

Φ, H ∞ S m,n . Φ − QL∞ (S m,n = distL∞ (S m,n p p ) p ) Again, it also follows from our remarks in Section 2.1 that distL∞ (S m,n Φ, H ∞ S m,n = p p )

trace Φ(ζ )Ψ (ζ ) dm(ζ ).

sup

Ψ H 1 (S n,m ) 1 0

p

T

However, an extremal function may fail to exist in this case even if Φ is a scalar-valued function. An example can be deduced from Section 1 of Chapter 1 in [2]. 3. σk (Φ) as the norm of a Hankel-type operator and k-extremal functions {k}

We now introduce the Hankel-type operators HΦ which act on spaces of matrix functions. {k} {k} We prove that the number σk (Φ) equals the operator norm of HΦ and characterize when HΦ has a maximizing vector. Recall that for an operator T : X → Y between normed spaces X and Y , a vector x ∈ X is called a maximizing vector of T if x is non-zero and T xY = T · xX . We begin by establishing the following lemma. Lemma 3.1. Let 1 k min{m, n}. If Ψ ∈ H 1 (Mn,m ) is such that rank Ψ (ζ ) = k for a.e. ζ ∈ T, then there are functions R ∈ H 2 (Mn,k ) and Q ∈ H 2 (Mk,m ) such that R(ζ ) has rank equal to k for almost every ζ ∈ T, Ψ = RQ

2 and R(ζ )M

n,k

2 = Q(ζ )M

k,m

= Ψ (ζ )M

n,m

for a.e. ζ ∈ T.

Proof. Consider the set A = closL1 (Cn ) f ∈ H 1 Cn : f (ζ ) ∈ Range Ψ (ζ ) a.e. on T . Since A is a non-trivial invariant subspace of H 1 (Cn ) under multiplication by z, there is an n × r inner function Θ such that A = ΘH 1 (Cr ). We first show that r = k. Let {ej }rj =1 be an orthonormal basis for Cr . Then for almost every ζ ∈ T, we have that {Θ(ζ )ej }rj =1 is a linearly independent set, since Θ is inner. Moreover, {Θ(ζ )ej }rj =1 is a basis for Range Θ(ζ ) = Range Ψ (ζ ) for a.e. ζ ∈ T. Since dim Range Ψ (ζ ) = k a.e. on T, it follows that r = dim Range Θ(ζ ) = dim Range Ψ (ζ ) = k. In particular, we obtain that A = ΘH 1 Ck . Therefore, Ψ = ΘF for some k × m matrix function F ∈ H 1 (Mk,m ), because the columns of Ψ belong to A .

668

A.A. Condori / Journal of Functional Analysis 257 (2009) 659–682 1/2

Let h be an outer function in H 2 such that |h(ζ )| = Ψ (ζ )Mn,m for a.e. ζ ∈ T. (The existence

of h is a consequence of the fact that log Ψ (ζ )Mn,m ∈ L1 as Ψ ∈ H 1 (Mn,m ).) Thus, the matrix functions and Q = h−1 F

R = hΘ 2

have the desired properties.

2 m,k 2 m,k Definition 3.2. Let Φ ∈ L∞ (Mm,n ), 1 k min{m, n}, and ρ :L2 (S m,k 1 ) →L (S 1 )/H (S 1 ) {k} denote the natural quotient map. We define the Hankel-type operator HΦ : H 2 (Mn,k ) → 2 m,k L2 (S m,k 1 )/H (S 1 ) by setting {k}

def

HΦ F = ρ(ΦF )

for F ∈ H 2 (Mn,k ).

2 m,k The norm in the quotient space L2 (S m,k 1 )/H (S 1 ) is the natural one; that is, the norm of a m,k coset equals the infimum of the L2 (S 1 )-norms of its elements.

Theorem 3.3. Let 1 k min{m, n}. If Φ ∈ L∞ (Mm,n ), then {k} σk (Φ) = HΦ H 2 (M

2 m,k 2 m,k n,k )→L (S 1 )/H (S 1 )

.

Proof. Consider the collection Bkn,m = RQ: RH 2 (Mn,k ) 1, QH 2 (Mk,m ) 1 . 0

Bkn,m

An,m k .

We claim that = Indeed if Ψ ∈ Ak satisfies rank Ψ (ζ ) = j for ζ ∈ T, where 1 j k, then by Lemma 3.1 there are functions R ∈ H 2 (Mn,j ) and Q ∈ H02 (Mj,m ) such that R(ζ ) has rank equal to j for almost every ζ ∈ T, Ψ = RQ

2 and R(ζ )M

n,j

2 = Q(ζ )M

j,m

= Ψ (ζ )M

n,m

for a.e. ζ ∈ T.

We may now add zeros, if necessary, to obtain n × k and k × m matrix functions Q R# = ( R O ) and Q# = , O respectively, from which it follows that Ψ = R# Q# ∈ Bkn,m . Therefore An,m ⊂ Bkn,m . The reverse k inclusion is trivial and so these sets are equal. Hence σk (Φ) = sup sup trace Φ(ζ )R(ζ )Q(ζ ) dm(ζ ) RH 2 (M

=

n,k )

sup

RH 2 (M

n,k )

1 QH 2 (M 0

1

{k} = HΦ H 2 (M

k,m )

1

T

distL2 (S m,k ) ΦR, H 2 (Mm,k ) 1

2 m,k 2 m,k n,k )→L (S 1 )/H (S 1 )

.

2


669

Definition 3.4. Let Φ ∈ L∞ (Mm,n ) and 1 k min{m, n}. We say that Ψ is a k-extremal and function for Φ if Ψ ∈ An,m k σk (Φ) =

trace Φ(ζ )Ψ (ζ ) dm(ζ ).

T

Thus a matrix function Φ has a k-extremal function if and only if Extremal problem 1.1 has a positive solution. We can now describe matrix functions that have a k-extremal function in terms of Hankel-type operators. Theorem 3.5. Let Φ ∈ L∞ (Mm,n ). The matrix function Φ has a k-extremal function if and only {k} if the Hankel-type operator HΦ has a maximizing vector. Proof. To simplify notation, let {k} def {k} H = H Φ

Φ

2 m,k H 2 (Mn,k )→L2 (S m,k 1 )/H (S 1 )

.

Suppose Ψ is a k-extremal function for Φ. Let j ∈ N be such that j k and rank Ψ (ζ ) = j

for a.e. ζ ∈ T.

By Lemma 3.1, there is an R ∈ H 2 (Mn,j ) and a Q ∈ H02 (Mj,m ) such that Ψ = RQ

2 and R(ζ )M

n,j

2 = Q(ζ )M

j,m

= Ψ (ζ )M

n,m

for a.e. ζ ∈ T.

As before, adding zeros if necessary, we obtain n × k and k × m matrix functions Q R# = ( R O ) and Q# = , O respectively, so that Ψ = R# Q# and Q# (ζ )2

Mk,m

2 = Q(ζ )M

j,m

= Ψ (ζ )M

n,m

for a.e. ζ ∈ T.

{k}

Let us show that R# is a maximizing vector for HΦ . Since Q# belongs to H02 (Mk,m ), we have that for any F ∈ H 2 (S m,k 1 ) σk (Φ) = T

= T

and so

trace Φ(ζ )Ψ (ζ ) dm(ζ ) =

trace Φ(ζ )R# (ζ )Q# (ζ ) dm(ζ )

T

trace (ΦR# − F )(ζ )Q# (ζ ) dm(ζ ),

670


σk (Φ) = trace (ΦR# − F )(ζ )Q# (ζ ) dm(ζ ) T

trace (ΦR# − F )(ζ )Q# (ζ ) dm(ζ )

T

(ΦR# − F )(ζ )Q# (ζ )

Sm 1

dm(ζ )

T

(ΦR# − F )(ζ )

S m,k 1

Q# (ζ )

Mk,m

dm(ζ )

T

ΦR# − F L2 (S m,k ) Q# L2 (Mk,m ) 1

= ΦR# − F L2 (S m,k ) Ψ L1 (Mn,m ) 1

ΦR# − F L2 (S m,k ) . 1

By Theorem 3.3, we obtain that {k} {k} σk (Φ) HΦ R# L2 (S m,k )/H 2 (S m,k ) HΦ = σk (Φ), 1

1

and therefore {k} {k} H = H R # Φ

Φ

2 m,k L2 (S m,k 1 )/H (S 1 )

.

Thus, R# is a maximizing vector of HΦ . {k} Conversely, suppose the Hankel-type operator HΦ has a maximizing vector R ∈ H 2 (Mn,k ). Without loss of generality, we may assume that RL2 (Mn,k ) = 1. Then {k} = HΦ . distL2 (S m,k ) ΦR, H 2 S m,k 1 1

By the remarks in Section 2.2, there is a function G ∈ H02 (Mk,m ) such that GL2 (Mk,m ) 1 and . trace (ΦR)(ζ )G(ζ ) dm(ζ ) = distL2 (S m,k ) ΦR, H 2 S m,k 1 1

T {k}

On the other hand, since R is a maximizing vector of HΦ , it follows from Theorem 3.3 that

{k} trace Φ(ζ )(RG)(ζ ) dm(ζ ) = HΦ = σk (Φ).

T def

Hence Ψ = RG is a k-extremal function for Φ.

2


671

Before stating the next result, let us recall that the Hankel operator HΦ : H 2 (Cn ) → H−2 (Cm ) is defined by HΦ f = P− Φf for f ∈ H 2 (Cn ). The following is an immediate consequence of the previous theorem when k = 1. Corollary 3.6. Let Φ ∈ L∞ (Mm,n ). The Hankel operator HΦ has a maximizing vector if and only if Φ has a 1-extremal function. Proof. By Theorem 3.5, Φ has a 1-extremal function if and only if the Hankel-type opera{1} tor HΦ : H 2 (Cn ) → L2 (Cm )/H 2 (Cm ) has a maximizing vector. The conclusion now follows by considering the “natural” isometric isomorphism between the spaces H−2 (Cm ) = L2 (Cm ) H 2 (Cm ) and L2 (Cm )/H 2 (Cm ). 2 Remark 3.7. It is worth mentioning that if a matrix function Φ is such that the Hankel operator HΦ has a maximizing vector (e.g. Φ ∈ (H ∞ + C)(Mn )), then any 1-extremal function Ψ of Φ satisfies trace Φ(ζ )Ψ (ζ ) dm(ζ ) = HΦ = t0 (Φ). T

This is a consequence of Corollary 3.6 and Theorem 3.3. Remark 3.8. There are other characterizations of the class of bounded matrix functions Φ such that the Hankel operator HΦ has a maximizing vector. These involve “dual” extremal functions and “thematic” factorizations. We refer the interested reader to [4] for details. {k}

Corollary 3.9. Let 1 k n and Φ ∈ L∞ (Mn ). Suppose that σk (Φ) = σ (Φ). If HΦ has { } a maximizing vector, then HΦ also has a maximizing vector. Proof. This is an immediate consequence of Theorem 3.5.

2

4. How about the sum of superoptimal singular values? In this section, we prove in Theorem 4.7 that equality is obtained in (1.2) under some natural conditions. For the rest of this note, we assume that m = n. Consider the non-decreasing sequence σ1 (Φ), . . . , σn (Φ). Recall that σn (Φ) = distL∞ (S n1 ) Φ, H ∞ (Mn ) and the distance on the right-hand side is in fact always attained, i.e. a best approximant Q to Φ under the L∞ (S n1 ) norm always exists as explained in Section 2.2. Theorem 4.1. Let Φ ∈ L∞ (Mn ) and 1 k n. Suppose Q is a best approximant to Φ in {k} H ∞ (Mn ) under the L∞ (S n1 )-norm. If the Hankel-type operator HΦ has a maximizing vector F 2 in H (Mn,k ) and σk (Φ) = σn (Φ), then

672


1. QF is a best approximant to ΦF in H 2 under the L2 (S n,k 1 )-norm, 2. for each j 0, sj (Φ − Q)(ζ )F (ζ ) = sj (Φ − Q)(ζ ) F (ζ )M

for a.e. ζ ∈ T,

n,k

k−1 3. j =0 sj ((Φ − Q)(ζ )) = σk (Φ) holds for a.e. ζ ∈ T, and 4. sj ((Φ − Q)(ζ )) = 0 holds for a.e. ζ ∈ T whenever j k. Proof. By our assumptions, {k} 2 H F 2 2 Φ

L (Mn,k )

2 {k} 2 = HΦ F L2 (S n,k )/H 2 (S n,k ) = ρ(ΦF ) 1

1

2 = ρ (Φ − Q)F

2 (Φ − Q)F L2 (S n,k ) =

(Φ − Q)(ζ )F (ζ )2 n,k dm(ζ ) S1

1

T

(Φ − Q)(ζ )2 n F (ζ )2

Mn,k

S1

dm(ζ )

T

Φ − Q2L∞ (S n ) F 2L2 (M

n,k )

1

= σk (Φ)2 F 2L2 (M

n,k )

.

It follows from Theorem 3.3 that all inequalities are equalities. In particular, we obtain that QF is a best approximant to ΦQ under the L2 (S n,k 1 )-norm since the first inequality is actually an equality. For almost every ζ ∈ T, (Φ − Q)(ζ )F (ζ ) n = (Φ − Q)(ζ ) n F (ζ ) S1 S1 Mn,k (Φ − Q)(ζ ) n = Φ − QL∞ (S n ) = σk (Φ), S 1

and (4.1)

1

because the second and third inequalities are equalities as well. It follows from (4.1) that for each j 0, sj (Φ − Q)(ζ )F (ζ ) = sj (Φ − Q)(ζ ) F (ζ )M

n,k

for a.e. ζ ∈ T.

We claim that if j k, then sj ((Φ − Q)(ζ )) = 0 for a.e. ζ ∈ T. By Theorem 3.5, we can choose a k-extremal function, say Ψ , for Φ. Since Ψ belongs to H01 (Mn ), σk (Φ) =

trace Φ(ζ )Ψ (ζ ) dm(ζ ) =

T

T

trace (Φ − Q)(ζ )Ψ (ζ ) dm(ζ )

T

(Φ − Q)(ζ )Ψ (ζ )

dm(ζ ) Sn 1

(Φ − Q)(ζ ) n Ψ (ζ ) dm(ζ ) S M 1

T

Φ − QL∞ (S n1 ) Ψ L1 (Mn ) Φ − QL∞ (S n1 ) = σk (Φ),

n


673

and so all inequalities are equalities. It follows that trace (Φ − Q)(ζ )Ψ (ζ ) = (Φ − Q)(ζ ) n Ψ (ζ ) S M

for a.e. ζ ∈ T.

n

1

(4.2)

In order to complete the proof, we need the following lemma. Lemma 4.2. Let A ∈ Mn and B ∈ Mn . Suppose that A and B satisfy trace(AB) = AM BS n . n 1 If rank A k, then rank B k as well. We first finish the proof of Theorem 4.1 before proving Lemma 4.2. It follows from (4.2) and Lemma 4.2 that rank (Φ − Q)(ζ ) k

for a.e. ζ ∈ T.

In particular, if j k, then sj (Φ − Q)(ζ ) = 0

for a.e. ζ ∈ T,

and so k−1 sj (Φ − Q)(ζ ) = (Φ − Q)(ζ )S n = σk (Φ) 1

j =0

This completes the proof.

for a.e. ζ ∈ T.

2

Remark 4.3. Lemma 4.2 is a slight modification of Lemma 4.6 in [1]. Although the proof of Lemma 4.2 given below is almost the same as that given in [1] for Lemma 4.6, we include it for the convenience of the reader. Proof of Lemma 4.2. Let B have polar decomposition B = U P and set C = AU , where P = (B ∗ B)1/2 . Let e1 , . . . , en be an orthonormal basis of eigenvectors for P and P ej = λj ej . It is easy to see that the following inequalities hold: n n ∗ ∗ trace(AB) = trace(CP ) = P ej , C ej = λj ej , C ej j =1

j =1

n n n = λj (Cej , ej ) λj (Cej , ej ) λj Cej j =1

CMn

j =1

n j =1

λj .

j =1

674


On the other hand, AMn B

S n1

= CMm P

S n1

= CMn

n

λj

j =1

and so, by the assumption |trace(AB)| = AMn BS n1 , it follows that n j =1

λj Cej = CMn

n

λj .

j =1

Therefore λj Cej = CMn λj for each j . However, if rank A k, then rank C k. Thus there are at most k vectors ej such that Cej = CMn . In particular, there are at least n − k vectors ej such that Cej < CMn . Thus, λj = 0 for those n − k vectors ej , rank P k, and so rank B k. 2 Remark 4.4. Note that the distance function dΦ defined on T by def dΦ (ζ ) = (Φ − Q)(ζ )S n 1

equals σk (Φ) for almost every ζ ∈ T and is therefore independent of the choice of the best approximant Q. This is an immediate consequence of Theorem 4.1. A similar phenomenon occurs in the case of matrix functions Φ ∈ Lp (Mn ) for 2 < p < ∞. We refer the reader to [1] for details. Corollary 4.5. Let Φ ∈ L∞ (Mn ) be an admissible matrix function and 1 k n. If the Hankel{k} type operator HΦ has a maximizing vector and σk (Φ) = σn (Φ), then k−1 k−1 sj (Φ − Q)(ζ ) tj (Φ) j =0

j =0

for any best approximation Q of Φ in H ∞ (Mn ) under the L∞ (S n1 )-norm. Proof. This is an immediate consequence of Theorems 1.3 and 4.1.

2

Definition 4.6. A matrix function Φ ∈ L∞ (Mn ) is said to have order if is the smallest number { } such that HΦ has a maximizing vector and σ (Φ) = distL∞ (S n1 ) Φ, H ∞ (Mn ) . If no such number exists, we say that Φ is inaccessible. The interested reader should compare this definition of “order” with the one made in [1] for matrix functions in Lp (Mn ) for 2 < p < ∞. Also, due to Corollary 3.9, it is clear that if {k} Φ ∈ L∞ (Mn ) has order , then the Hankel-type operator HΦ has a maximizing vector and


675

σk (Φ) = distL∞ (S n1 ) Φ, H ∞ (Mn ) holds for each k . Theorem 4.7. Let Φ ∈ L∞ (Mn ) be an admissible matrix function of order k. The following statements are equivalent. (1) Q ∈ H ∞ is a best approximant to Φ under the L∞ (S n1 )-norm and the functions ζ → sj (Φ − Q)(ζ ) ,

0 j k − 1,

are constant almost everywhere on T. (2) Q is the superoptimal approximant to Φ, tj (Φ) = 0 for j k, and σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ). Proof. We first prove that (1) implies (2). By Corollary 4.5, we have that, for almost every ζ ∈ T, k−1 k−1 k−1 k−1 sj (Φ − Q)(ζ ) tj (Φ) ess sup sj (Φ − Q)(ζ ) = sj (Φ − Q)(ζ ) . j =0

j =0

j =0

ζ ∈T

j =0

This implies that tj (Φ) = ess sup sj (Φ − Q)(ζ ) = sj (Φ − Q)(ζ ) ζ ∈T

for 0 j k − 1,

Q ∈ Ωk−1 (Φ), and k−1

tj (Φ) =

j =0

k−1 sj (Φ − Q)(ζ ) = σk (Φ). j =0

Moreover, Theorem 4.1 gives that sj ((Φ − Q)(ζ )) = 0 a.e. on T for j k, and so tj (Φ) = 0 for j k, as Q ∈ Ωk−1 (Φ). Hence, Q is the superoptimal approximant to Φ. Let us show that (2) implies (1). Clearly, it suffices to show that if (2) holds, then Q is a best approximant to Φ under the L∞ (S n1 )-norm. Suppose (2) holds. In this case, we must have that σk (Φ) =

k−1

tj (Φ) =

j =0

k−1 sj (Φ − Q)(ζ ) = Φ − QL∞ (S n1 ) . j =0

Since Φ has order k, it follows that σn (Φ) = Φ − QL∞ (S n1 ) and so the proof is complete.

2

676


For the rest of this section, we restrict ourselves to admissible matrix functions Φ which are also very badly approximable. Recall that, in this case, the function ζ → sj (Φ(ζ )) equals tj (Φ) a.e. on T for 0 j n − 1, as mentioned in Section 1.1. The next result follows at once from Theorem 4.7. Corollary 4.8. Let Φ be an admissible very badly approximable n × n matrix function of order k. The zero matrix function is a best approximant to Φ under the L∞ (S n1 )-norm if and only if tj (Φ) = 0 for j k and σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ). It is natural to question at this point whether or not the collection of admissible very badly approximable matrix functions of order k is non-empty. It turns out that one can easily construct examples of admissible very badly approximable matrix functions of order k (see Examples 4.14 and 4.15). Theorem 4.10 below gives a simple sufficient condition for determining when a very badly approximable matrix function has order k. We first need the following lemma. Lemma 4.9. Let Φ ∈ L∞ (Mn ). Suppose there is Ψ ∈ Ank such that

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = ΦL∞ (S n1 ) .

T

Then Ψ is a k-extremal function for Φ, σk (Φ) = σn (Φ), and the zero matrix function is a best approximant to Φ under the L∞ (S n1 )-norm. Proof. By the assumptions on Ψ , we have ΦL∞ (S n1 ) =

trace Φ(ζ )Ψ (ζ ) dm(ζ ) σk (Φ).

T

On the other hand, σk (Φ) distL∞ (S n1 ) Φ, H ∞ ΦL∞ (S n1 ) always holds. Since all the previously mentioned inequalities are equalities, the conclusion follows. 2 Theorem 4.10. Let Φ ∈ L∞ (Mn ) be an admissible very badly approximable matrix function. Suppose there is Ψ ∈ Ank such that

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 (Φ) + · · · + tn−1 (Φ).

T

If tk−1 (Φ) > 0, then Φ has order k and the zero matrix function is a best approximant to Φ under the L∞ (S n1 )-norm.


677

Proof. By the remarks preceding Corollary 4.8, it is easy to see that ΦL∞ (S n1 ) = t0 (Φ) + · · · + tn (Φ). It follows from Lemma 4.9 that Ψ is a k-extremal function for Φ, σk (Φ) = σn (Φ), and the zero matrix function is a best approximant to Φ under the L∞ (S n1 )-norm. Thus ΦL∞ (S n1 ) = σk (Φ). Moreover, by Theorem 1.3, σk−1 (Φ) t0 (Φ) + · · · + tk−2 (Φ) < t0 (Φ) + · · · + tk−1 (Φ) ΦL∞ (S n1 ) . Therefore σk−1 (Φ) < σk (Φ).

2

Remark 4.11. Notice that under the hypotheses of Theorem 4.10, one also obtains that tk−1 (Φ) is the smallest non-zero superoptimal singular value of Φ. This is an immediate consequence of Corollary 4.8. We now formulate the corresponding result for admissible very badly approximable unitaryvalued matrix functions. These functions are considered in greater detail in Section 5. Corollary 4.12. Let U ∈ L∞ (Mn ) be an admissible very badly approximable unitary-valued matrix function. If there is Ψ ∈ Ann such that

trace U (ζ )Ψ (ζ ) dm(ζ ) = n,

T

then U has order n and the zero matrix function is a best approximant to U under the L∞ (S n1 )norm. Proof. This is a trivial consequence of Theorem 4.10 and the fact that tj (U ) = 1 for 0 j n − 1.

2

We are now ready to state the main result of this section. Theorem 4.13. Let Φ be an admissible very badly approximable n × n matrix function. The following statements are equivalent: (1) k is the smallest number for which there exists Ψ ∈ Ank such that

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 (Φ) + · · · + tn−1 (Φ);

T

(2) Φ has order k, tj (Φ) = 0 for j k and σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ).

678


Proof. Let κ(Φ) = inf j 0: there exists a Ψ ∈ Anj such that def

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 (Φ) + · · · + tn−1 (Φ) .

T

Clearly, κ(Φ) may be infinite for arbitrary Φ. Suppose κ = κ(Φ) is finite. Then Lemma 4.9 implies that Φ has a κ-extremal function, σκ (Φ) = σn (Φ), and the zero matrix function is a best approximant to Φ under the L∞ (S n1 )norm. In particular, Φ has order k κ(Φ), tj (Φ) = 0 for j k, and σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ), by Corollary 4.8. On the other hand, if Φ has order k, tj (Φ) = 0 for j k, and σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ), then Φ has a k-extremal function Ψ ∈ Ank such that

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = σk (Φ) = t0 (Φ) + · · · + tk−1 (Φ).

T

Since tj (Φ) = 0 for j k, it follows that

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 (Φ) + · · · + tn−1 (Φ).

T

Thus κ(Φ) k. Hence, if either κ(Φ) is finite or Φ satisfies (2), then k = κ(Φ).

2

We end this section by illustrating existence of very badly approximable matrix functions of order k by giving two simple examples; a 2 × 2 matrix function of order 2 and a 3 × 3 matrix function of order 2. Example 4.14. Let 1 Φ=√ 2

1 −1 1 1

z¯ 2 O

O . z¯

It is easy to see that Φ is a continuous (and hence admissible) unitary-valued very badly approximable matrix function with superoptimal singular values t0 (Φ) = t1 (Φ) = 1. We claim that Φ has order 2. Indeed, the matrix function


Ψ=

z2 O

O z

1 √ 2

1 1 −1 1

679

satisfies

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = 2,

T

and so Φ has order 2 by Corollary 4.12. Example 4.15. Let t0 and t1 be two positive numbers satisfying t0 t1 . Let

Φ=

t0 z¯ a O O

O t1 z¯ b O

O O , O

where a and b are positive integers. It is easy to see that Φ is a continuous (and hence admissible) very badly approximable matrix function with superoptimal singular values t0 (Φ) = t0 , t1 (Φ) = t1 , and t2 (Φ) = 0. Again, we have that Φ has order 2. After all, the matrix function

Ψ=

za O O

O zb O

O O O

satisfies

trace Φ(ζ )Ψ (ζ ) dm(ζ ) = t0 + t1 = t0 (Φ) + t1 (Φ) + t2 (Φ),

T

and so Φ has order 2 by Theorem 4.10, since t1 (Φ) = t1 > 0. 5. Unitary-valued very badly approximable matrix functions We lastly consider the class Un of admissible very badly approximable unitary-valued matrix functions of size n × n and provide a representation of any n-extremal function Ψ for a function U ∈ Un such that

trace U (ζ )Ψ (ζ ) dm(ζ ) = t0 (U ) + · · · + tn−1 (U )

T

holds. Note that for any such U we have that tj (U ) = 1 for 0 j n − 1. For a matrix function Φ ∈ L∞ (Mm,n ), we define the Toeplitz operator TΦ by TΦ f = P+ Φf

for f ∈ H 2 Cn ,

where P+ denotes the orthogonal projection from L2 (Cn ) onto H 2 (Cm ).

(5.1)

680


It is well known that, for any function U ∈ Un , the Toeplitz operator TU is Fredholm and ind TU > 0. (As usual, for a Fredholm operator T , its index, ind T , is defined by dim ker T − dim ker T ∗ .) In particular, ind Tdet U = ind TU . We refer the reader to Chapter 14 in [2] for more information concerning functions in Un . Theorem 5.1. Suppose U ∈ Un has an n-extremal function Ψ such that (5.1) holds. Then Ψ admits a representation of the form Ψ = zh2 Θ, where h ∈ H 2 is an outer function such that hL2 = 1 and Θ is a finite Blaschke–Potapov product. Moreover, the scalar functions det(U Θ) and trace(U Θ) are admissible badly approximable functions that admit the factorizations det(U Θ) = z¯ n

h¯ n hn

h¯ trace(U Θ) = n¯z . h

and

We refer the reader to Section 5 of Chapter 2 in [2] for the definition and other results concerning Blaschke–Potapov products. Proof. It follows from (5.1) that all inequalities in (1.3) are equalities and so trace U (ζ )Ψ (ζ ) = U (ζ )Ψ (ζ )S n = nΨ (ζ )M

n

1

(5.2)

holds for a.e. ζ ∈ T. Since U is unitary-valued, then U (ζ )Ψ (ζ )

S n1

= Ψ (ζ )S n , 1

and so Ψ (ζ )

S n1

= nΨ (ζ )M

n

must hold for a.e. ζ ∈ T. Therefore we must have Ψ (ζ ) = Ψ (ζ )M V (ζ ) n

for a.e. ζ ∈ T,

for some unitary-valued matrix function V , because sj Ψ (ζ ) = Ψ (ζ )M

n

for a.e. ζ ∈ T, 0 j n − 1.

Let h ∈ H 2 be an outer function such that h(ζ ) = Ψ (ζ )1/2 Mn

on T.

(5.3)


681

def

Consider also the matrix function Ξ = h−2 Ψ . It follows from (5.3) that ∗ Ξ Ξ (ζ ) =

1 ∗ Ψ Ψ (ζ ) = In |h(ζ )|4

for a.e. ζ ∈ T,

and so Ξ is an inner function. Thus Ψ admits the factorization Ψ = zh2 Θ for some n × n unitary-valued inner function Θ and an outer function h ∈ H 2 such that hL2 = 1. def

Note that the first equality in (5.2) indicates that the scalar function ϕ = trace(U Θ) satisfies zh2 ϕ = n|h|2

on T,

or equivalently h¯ ϕ = n¯z . h Moreover, HU Θ e HU e < 1, hence Hϕ e < n = Hϕ implying that ϕ is an admissible badly approximable scalar function on T. We conclude that the Toeplitz operator Tϕ is Fredholm and ind Tϕ > 0 (cf. Theorem 7.5.5 in [2]). Returning to (5.2), it also follows that each eigenvalue of U (ζ )Ψ (ζ ) equals Ψ (ζ )Mn = |h(ζ )|2 for a.e. ζ ∈ T. In particular, h(ζ )2n = det U (ζ )Ψ (ζ ) = zn h2n (ζ ) · det U (ζ ) · det Θ(ζ ) holds a.e. ζ ∈ T. By setting def

θ = det Θ

def

and u = det U,

we have that u admits the factorization u = θ¯ z¯ n

h¯ n = θ¯ ωn , hn

¯ h = ϕ/n. Since the Toeplitz operator Tω is Fredholm with positive index, Tuω¯ n is where ω = z¯ h/ Fredholm as well. We conclude now that def

dim H 2 θ H 2 = dim ker Tθ∗ = dim ker Tθ¯ = ind Tθ¯ < ∞ because ker Tθ = {O} and uω¯ n = θ¯ . Therefore Θ is a Blaschke–Potapov product, because Θ is a unitary-valued inner function and det Θ is a finite Blaschke product. 2

682


Acknowledgments This article is based in part on the author’s PhD dissertation at Michigan State University. I would like to thank Professor Vladimir V. Peller for communicating Theorem 1.3 and for suggesting corrections on earlier versions of this paper. I also thank the reviewer for remarks that led to improvements in the paper’s presentation. References [1] L. Baratchart, F.L. Nazarov, V.V. Peller, Analytic approximation of matrix functions in Lp , J. Approx. Theory, in press. [2] V.V. Peller, Hankel Operators and Their Applications, Springer Monogr. Math., Springer, New York, 2003. [3] V.V. Peller, personal communication. [4] V.V. Peller, Analytic approximation of matrix functions and dual extremal functions, Proc. Amer. Math. Soc., in press. [5] V.V. Peller, S.R. Treil, Approximation by analytic matrix functions. The four block problem, J. Funct. Anal. 148 (1997) 191–228. [6] V.V. Peller, N.J. Young, Superoptimal analytic approximation of matrix functions, J. Funct. Anal. 120 (1994) 300– 343.


Relationships between combinatorial measurements and Orlicz norms Ron Blei ∗ , Lin Ge Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States Received 17 November 2008; accepted 23 February 2009 Available online 10 March 2009 Communicated by K. Ball

Abstract We establish in a setting of harmonic analysis precise relationships between combinatorial measurements and Orlicz norms. These relationships further extend and sharpen prior results concerning extensions of the Littlewood 2n/(n + 1)-inequalities, the n-dimensional Khintchin inequalities, and the Kahane–Khintchin inequality. © 2009 Elsevier Inc. All rights reserved. Keywords: Rademacher system; Littlewood 2n/(n + 1)-inequalities; n-Dimensional Khintchin inequalities; Kahane–Khintchin inequality; Combinatorial measurement; α-Orlicz function; Orlicz norm; Quasi-asymptotic

1. Introduction In this paper we focus on connections between measurements reflecting purely combinatorial data and measurements that are based on harmonic-analytic and probabilistic properties. Given an infinite set Y and F ⊂ Y n (n 1), we consider a function associated with F , ΨF : N → N such that for s ∈ N, ΨF (s) = max F ∩ (A1 × · · · × An ): Aj ⊂ Y, |Aj | s, j = 1, . . . , n . Define * Corresponding author.

E-mail addresses: [email protected] (R. Blei), [email protected] (L. Ge). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.020

(1.1)

684

R. Blei, L. Ge / Journal of Functional Analysis 257 (2009) 683–720

dim F = lim log ΨF (s)/ log s;

(1.2)

dF (a) = sup ΨF (s)/s a : s ∈ N ,

(1.3)

s→∞

equivalently, for a > 0 define

and observe that if |F | = ∞, then dim F = inf a: dF (a) < ∞ = sup a: dF (a) = ∞ .

(1.4)

The function ΨF is viewed as a gauge of the combinatorial complexity in F : ΨF (s) is the smallest integer k such that for all s-sets A1 ⊂ Y, . . . , An ⊂ Y , the number of samplings a1 ∈ A1 , . . . , an ∈ An with (a1 , . . . , an ) ∈ F is no greater than k. The index dim F is viewed as the combinatorial dimension of F , conveying that ΨF (s) “grows like” s dim F , in the sense that ΨF (s) 0 if β > dim F, lim = ∞ if β < dim F. s→∞ s β

(1.5)

We distinguish between two cases: (i) if lims→∞ ΨF (s)/s dim F < ∞ (dF (dim F ) < ∞), then dim F is exact; (ii) if lims→∞ ΨF (s)/s dim F = ∞ (dF (dim F ) = ∞), then dim F is asymptotic. (E.g., see [2], XIII.4.) In this paper we further analyze the asymptotic case, and establish a precise resolution of it. We take Y to be N (without loss of generality), and identify it with the Rademacher system {rj }j ∈N := R, a set of projections from {−1, 1}N : = Ω onto {−1, 1}: rj (ω) = ω(j ),

j ∈ N, ω = ω(j ) j ∈N ∈ Ω.

(1.6)

Here we view Ω as a compact Abelian group (endowed with the product topology, coordinatewise multiplication, and the normalized Haar measure P), and view R as an independent set of characters on Ω. (E.g., see [2], II.1, VII.2.) For F ⊂ R n (n 1), let CF (Ω n ) and L2F (Ω n ) be, respectively, the spaces of continuous functions and Pn -square integrable functions on Ω n , whose Fourier–Walsh transforms are supported in F . For t > 0, let · t be the t norm, and for f ∈ C(Ω n ), let fˆ be the Fourier–Walsh transform of f . For F ⊂ R n and t > 0, let ζF (t) = sup fˆt : f ∈ BCF (Ω n ) ,

(1.7)

where BCF (Ω n ) denotes the closed unit ball in CF (Ω n ), and define σF = inf t: ζF (t) < ∞ = sup t: ζF (t) = ∞ ; if ζF (σF ) < ∞, then σF is exact, and if ζF (σF ) = ∞, then σF is asymptotic.

(1.8)


685

For F ⊂ R n and t > 0, let ηF (t) = sup f Lp /p t : p > 2, f ∈ BL2 (Ω n ) , F

(1.9)

where BL2 (Ω n ) is the closed unit ball in L2F (Ω n ), and define F

δF = inf t: ηF (t) < ∞ = sup t: ηF (t) = ∞ ;

(1.10)

again, if ηF (δF ) < ∞, then δF is exact, and if ηF (δF ) = ∞, then δF is asymptotic. The main results in [1] were: dF (t) < ∞

⇐⇒

ζF 2t/(t + 1) < ∞

⇐⇒

ηF (t/2) < ∞.

(1.11)

In particular, σF =

2 dim F dim F + 1

(1.12)

dim F , 2

(1.13)

and δF =

where σF and δF are exact if and only if dimF is exact. These results in effect were extensions of the classical Littlewood 2n/(n + 1)-inequalities [6,9], and the n-dimensional Khintchin inequalities [5,7]. In this paper, we use Orlicz functions and their associated Orlicz norms to precisely resolve the case dF (dim F ) = ∞. Our work is divided into four parts. In the first part we focus on the combinatorial gauge ΨF , F ⊂ R n (n 1). Given functions Ψ : N → N and Φ : R → R, we say that Ψ is quasi-asymptotic to Φ, and write Ψ ∼q Φ, if 0 < lim

Ψ (s)

s→∞ Φ(s)

0, then there exists an α-Orlicz function (Definition 2.1) Φ such that ΨF ∼q Φ. Conversely, we show (Theorem 2.3) that for every α-Orlicz function Φ (α 1) there exists F ⊂ R n such that ΨF ∼q Φ. These results extend prior constructions in [3] and [4]. In the next three parts we derive precise relations between ΨF (F ⊂ R n ) and corresponding Orlicz norms associated with ΨF in CF (Ω n ) and L2F (Ω n , Pn ) (Theorem 3.4, Corollary 4.3 and Theorem 5.4). These results naturally extend prior results stated in (1.11), (1.12) and (1.13) above, concerning relations between combinatorial dimension and Littlewood-type inequalities and Khintchin-type inequalities. The results in this paper form a portion of the second author’s PhD dissertation written at the University of Connecticut under the guidance of the first author.

686


2. Combinatorial structures and α-Orlicz functions An R-valued function Φ on [0, ∞) is an Orlicz function if Φ is continuous, non-decreasing, convex, Φ(0) = 0, and limx→∞ Φ(x) = ∞. (E.g., see [8], 4.a.) For F ⊂ Nn , and Orlicz function Φ, define (extending the definition in (1.3)) dF (Φ) = sup ΨF (s)/Φ(s): s ∈ N .

(2.1)

If Φ(x) = x a for some a 1, then we write dF (a) for dF (Φ). Note that dim F = α is exact (α 1) and lims→∞ ΨF (s)/s α > 0 if and only if ΨF is quasiasymptotic to Φ(x) = x α , x 0. If dim F = α is asymptotic, then we focus on φ(s) = ΨF (s)/s α , where (necessarily) lims→∞ φ(s) = ∞, and φ(s) is o(s ) for all > 0. To this end, for technical reason that will later become apparent, we introduce the notion of an α-Orlicz function: Definition 2.1. For α 1, an Orlicz function Φ is said to be an α-Orlicz function if φ ∈ C 2 [0, ∞) and Φ(x) = x α φ(x) for x 0, where either φ ≡ 1, or φ satisfies the following properties: (i) φ is concave and strictly increasing to ∞; (ii) xφ(x) is convex for x 0; (iii) φ(x) = o(x ) for all > 0, and for each > 0 there exists K > 0, such that φ(x)/x is decreasing with increasing x for x K. An example. Suppose we want to construct an α-Orlicz function whose graph contains (s, s α (log s)β ) for s large, for some α 1 and β > 0. Note that (log x)β is not concave for x < eβ−1 , x(log x)β is not convex for x < e1−β , and the y-intercept of the tangent line to the graph of (log x)β at x for x < eβ is less than 0. Let x0 = max{e1−β , eβ } + 1, and let be the linear function whose graph is the tangent line to the graph of (log x)β at x0 ; that is (x) = (log x0 )β + βx0−1 (log x0 )β−1 (x − x0 ),

−∞ < x < ∞.

(2.2)

Let (log x)β ˜ φ(x) = (x)

if x x0 , if 0 x < x0 .

(2.3)

Smooth φ˜ at x0 so that the smoothed function φ is in C 2 [0, ∞), φ is concave, and xφ(x) is convex. Then the function Φ(x) = x α φ(x) for x 0 is the desired α-Orlicz function. Theorem 2.2. Let n ∈ N. If F ⊂ Nn is infinite with dim F = α, and lims→∞ ΨF (s)/s α > 0, then there exists an α-Orlicz function Φ such that ΨF ∼q Φ. Proof. Because F ⊂ Nn is infinite, we have α 1. If dF (α) < ∞, then Φ(x) = x α for x 0 is an α-Orlicz function such that ΨF ∼q Φ. Suppose dF (α) = ∞. First we choose a sequence {sj }, sj ↑ ∞. For any positive integers s and s , let s,s be the linear function whose graph is the line passing through (s, ΨF (s)/s α ) and (s , ΨF (s )/(s )α ). (Let 0,1 be the linear function whose graph is the line passing through (0, 0) and (1, 1).) Let s1 = 0, and s2 = 1. To choose sj for j > 2, we proceed by (double) induction.


687 (1)

(j )

Suppose we have chosen sj for j 2. To choose sj +1 , we consider the j points sj , . . . , sj such that ΨF (sj ) ΨF (s) (1) sj = min s > sj : < < (s) , (2.4) sj −1 ,sj sjα sα and for 1 sj : < < sj −1 ,sj (s) . sjα sα

(2.5)

(j )

The existence of sj , . . . , sj for any j is guaranteed because dF (α) = ∞, and because ΨF (s)/s α = o(s ) for all > 0 (because dim F = α). Denote the slope of s,s by ms,s for any s and s . Let (j )

sj +1 = max s ∈ sj(1) , . . . , sj : msj ,s ms

(i) j ,sj

for all i = 1, . . . , j .

(2.6)

Continuing this process, we obtain a sequence sj ↑ ∞ that satisfies (1) ΨF (sj )/sjα is strictly increasing to ∞ with increasing j ; (2) msj −1 ,sj > msj ,sj +1 > 0 for all j > 1; (j )

(3) for each j , and sj s sj , either ΨF (s) sj −1 ,sj (s), sα

(2.7)

ΨF (s) sj ,sj +1 (s). sα

(2.8)

or

Claim 1. For each j , there are only finitely many s ∈ N such that ΨF (s) sj ,sj +1 (s). sα

(2.9)

Proof of Claim 1. Suppose the claim is false. Then there exist j , and a sequence sk ↑ ∞ such that ΨF (sk ) sj ,sj +1 (sk ). (sk )α

(2.10)

sj ,sj +1 (x) = msj ,sj +1 x + bj ,

(2.11)

For x 0, write

where msj ,sj +1 > 0, and bj > 0. By (2.10) and (2.11),

688


ΨF (sk ) msj ,sj +1 (sk )α+1 + bj (sk )α , which contradicts dim F = α, and the claim follows.

(2.12)

2

Let be the piecewise-linear function defined by (x) = sj ,sj +1 (x),

sj x sj +1 , j 1.

(2.13)

Claim 2. sup ΨF (s)/ s α (s) : s ∈ N < ∞.

(2.14)

Proof of Claim 2. Suppose the claim is false. Then there exists a sequence {si } such that ΨF (si )/(si )α > (si ), and limi→∞ ΨF (si )/((si )α (si )) = ∞. By Claim 1 and because (j ) (j ) [sj , sj + j ] ⊂ [sj , sj ], there exist j sufficiently large, and si ∈ [sj , sj ] such that sj ,sj +1 (si ) < α ΨF (si )/(si ) < sj −1 ,sj (si ), which contradicts (2.7) and (2.8), and the claim follows. 2 Next we construct a spline function as follows. Note that for b > 0, (log x)b is concave for b < log x + 1, and x(log x)b is convex for x > e. We start from s4 (because s4 > e). For s4 x s5 , let P4 (x) = a4 (log x)b4 + c4 x + d4 ,

(2.15)

where a4 > 0, 0 < b4 < log s4 + 1, c4 0, and d4 are chosen such that P4 (s4 ) = (s4 ),

P4 (s5 ) = (s5 ),

(P4 )+ (s4 ) =

ms3 ,s4 + ms4 ,s5 , 2

(2.16)

where (P4 )+ (x) denotes the right derivative of P4 at x. (Similarly (P4 )− (x) denotes the left derivative of P4 at x.) For s5 x s6 , let P5 (x) = a5 (log x)b5 + c5 x + d5 ,

(2.17)

where a5 > 0, 0 < b5 < log s5 + 1, c5 0, and d5 are chosen such that: (4) if (P4 )− (s5 ) > ms5 ,s6 , then P5 (s5 ) = (s5 ),

P5 (s6 ) = (s6 ),

and (P5 )+ (s5 ) = (P4 )− (s5 );

(2.18)

(5) if (P4 )− (s5 ) ms5 ,s6 , then P5 (s5 ) = (s5 ),

P5 (s6 ) = (s6 ),

and (P5 )− (s6 ) =

ms5 ,s6 + ms6 ,s7 . 2

(2.19)


689

We proceed as follows. For j 6, and sj x sj +1 , let Pj (x) = aj (log x)bj + cj x + dj ,

(2.20)

where aj > 0, 0 < bj < log sj + 1, cj 0, and dj are chosen such that: (6) if (Pj −1 )− (sj ) > msj ,sj +1 , then Pj (sj ) = (sj ),

Pj (sj +1 ) = (sj +1 ),

and (Pj )+ (sj ) = (Pj −1 )− (sj );

(2.21)

(7) if (Pj −1 )− (sj ) msj ,sj +1 , then Pj (sj ) = (sj ),

Pj (sj +1 ) = (sj +1 ),

(2.22)

and (Pj )− (sj +1 ) =

msj ,sj +1 + msj +1 ,sj +2 2

.

(2.23)

For any j 5 such that (7) holds, (Pj −1 )− (sj ) msj ,sj +1 < (Pj )+ (sj ). By the mean value theorem, there exist xj −1 ∈ (sj −1 , sj ), and xj ∈ (sj , sj +1 ) such that Pj −1 (xj −1 ) = msj −1 ,sj , and Pj (xj ) = msj ,sj +1 . Because Pj −1 and Pj are concave, and because msj −1 ,sj > msj ,sj +1 , there are tj −1 ∈ (xj −1 , sj ), and tj ∈ (sj , xj ) such that Pj −1 (tj −1 ) = Pj (tj ).

(2.24)

Tj (x) = Pj −1 (tj −1 ) + Pj −1 (tj −1 )(x − tj −1 ),

(2.25)

For x 0, let

that is, Tj is the linear function whose graph is both the tangent line to the graph of Pj −1 at tj −1 , and the tangent line to the graph of Pj at tj . Let φ˜ be the spline function such that ˜ is the linear function whose graph is the tangent line to the graph of P4 (8) for 0 x s4 , φ(x) at s4 ; (9) for any x s4 , let p(x) = Pj (x), sj −1 x sj for j 5 and let ˜ φ(x) =

Tj (x) p(x)

if tj −1 x tj , [tj −1 , tj ] ⊂ [xj −1 , xj ], j 5, otherwise,

(2.26)

where tj −1 and tj , j 5, are indicated in (2.24). ˜ ˜ Then φ˜ is concave, and x φ(x) is convex. Let Φ˜ = x α φ(x). By Claim 2 and because φ˜ , ˜ sup ΨF (s)/Φ(s): s ∈ N sup ΨF (s)/ s α (s) : s ∈ N < ∞.

(2.27)

690


Claim 3. There are infinitely many j such that ˜ j ) = (sj ). φ(s

(2.28)

Proof of Claim 3. For each j 5, by the definition of φ˜ in (2.26), and the construction of Pj ˜ j ) = Pj (sj ) = (sj ), or φ(s ˜ j +1 ) = Pj (sj +1 ) = (sj +1 ). Hence in (6) and (7), we have either φ(s the claim follows. ˜ Smooth By (2.27) and (2.28), and because (sj ) = ΨF (sj )/sjα for all j , we obtain ΨF ∼q Φ. 2 φ˜ at the points sj and tj for all j so that the smoothed function φ is in C [0, ∞), φ is concave, and xφ(x) is convex. Let Φ = x α φ(x). Then Φ is an α-Orlicz function such that ΨF ∼q Φ. 2 Next we establish the converse to Theorem 2.2. Theorem 2.3. (Cf. [2], Theorem XIII.19.) For n 2, and 1 α < n, if Φ is an α-Orlicz function, then there exist F ⊂ Nn such that ΨF ∼q Φ. Lemma 2.4. (Cf. [2], Lemma XIII.17.) Let n 2 be an integer, and 1 γ < n. Let Φ be an Orlicz function such that x Φ(x) x γ for all x ∈ [1, ∞) and Φ(x)/x γ is decreasing with increasing x. Then for every k ∈ N, there exist F ⊂ [k]n ([k] = {1, . . . , k}) such that ΨF (s) CΦ(s),

s ∈ [k],

(2.29)

and 1 |F | = ΨF (k) Φ(k), 2

(2.30)

where C > 0 depends only on n and γ . (k)

Proof. For k ∈ N, let {Xi : i ∈ [k]n } be the Bernoulli system of statistically independent {0, 1}valued variable on (Ω, P) such that (k) Φ(k) P Xi = 1 = n . k

(2.31)

(k)

Consider the random set F = {i: Xi = 1}. We use the following elementary fact about binomial probabilities: for p ∈ (0, 1), and integers m > 0 and i 2mp, m i m p (1 − p)m−i , p i+1 (1 − p)m−i−1 i i+1

2

(2.32)

which implies m

m j m i m−i 2 p (1 − p)m−j , p (1 − p) j i i=j

j 2mp.

(2.33)


691

Fix s ∈ [k], and let A be a s-hypercube in [k]n (A = A1 × · · · × An , where |A1 | = · · · = |An | = s). Denote n+2 n + 1 . (2.34) C = max 2e , n−γ Let j (s) = [CΦ(s)] (= smallest integer CΦ(s)). Then Φ(k) Φ(s) k n k n s n Φ(k) Φ(k) 2s n n because Φ(s)/s γ is decreasing . k

j 2Φ(s) = 2s n

(2.35)

By (2.33) and (2.35), n s n n s Φ(k) s −i Φ(k) i (k) 1 − Xi j = P i kn kn i=j

i∈A

n n Φ(k) s −j Φ(k) j s 1− n 2 j kn k j nj Φ(k) s 2 j! kn

2s nj (Φ(k))j . (CΦ(s))j e−j k nj

(2.36)

Then, (k) P Xi CΦ(s) for some s-hypercube A i∈A

n 2s nj (Φ(k))j k s (CΦ(s))j e−j k nj k ns

2s nj (Φ(k))j

s ns e−ns

C j (Φ(s))j e−j k nj

nj −ns s Φ(k) j ej +ns by (2.34) n+2 j Φ(s) (2e ) k nj −ns γj 1 s k n(j −s)+j because Φ(s)/s γ is decreasing k s e (n−γ )j −ns s because n(j − s) + j j s e−s k s s e−s because (n − γ )j (n − γ )CΦ(s) (n + 1)s . k

2

(2.37)

692


Hence, s k s (k) Xi CΦ(s) for some s-hypercube A, s ∈ [k] e−s . P k

(2.38)

s=1

i∈A

Therefore, (k) lim P Xi CΦ(s) = 0.

k→∞

(2.39)

i∈A

By Chebyshev’s inequality, Φ(k) (k) P Xi − Φ(k) > 2 n i∈[k]

(k) Var( i∈[k]n Xi − Φ(k)) 2 ( Φ(k) 2 )

(k) 4k n Var(Xi ) 4k n ( Φ(k) k n )(1 − = = Φ(k)2 Φ(k)2

Φ(k) kn )

4 . Φ(k)

(2.40)

Hence Φ(k) (k) = 0. lim P Xi k→∞ 2 n

(2.41)

i∈[k]

By (2.39) and (2.41), lim P F satisfies (2.29) and (2.30) = 1.

k→∞

2

(2.42)

Let π1 , . . . , πn be the canonical projections from Nn onto N. We say F ⊂ Nn and G ⊂ Nn are n-disjoint if π (F ) ∩ π (G) = ∅ for all = 1, . . . , n. Lemma 2.5. (Cf. [2], Lemma XIII.18.) Suppose Fj , j ∈ N, is a sequence of pairwise n-disjoint subsets of Nn , and let F = j Fj . For an Orlicz function Φ, and for every m ∈ N, sup ΨF (s)/Φ(s): s ∈ [m] n sup ΨFj (s)/Φ(s): s ∈ [m], j ∈ N .

(2.43)

Proof. Let m ∈ N and let s ∈ [m]. For A1 × · · · × An ⊂ Nn such that |A1 | s, . . . , |An | s, let si,j = πi (Fj ) ∩ Ai ,

i ∈ [n], j ∈ N,

(2.44)

and sj = max si,j : i ∈ [n] ,

j ∈ N.

(2.45)


693

Then ∞

si,j |Ai | s,

i ∈ [n].

(2.46)

j =1

Let L = sup ΨFj (s)/Φ(s): s ∈ [m], j ∈ N .

(2.47)

By (2.44), (2.45) and (2.47), for any j ∈ N, Fj ∩ (A1 × · · · × An ) = Fj ∩ π1 (Fj ) ∩ A1 × · · · × πn (Fj ) ∩ An LΦ(sj ).

(2.48)

Then |F ∩ (A1 × · · · × An )| = Φ(s)

∞

j =1 |Fj

L

∩ (A1 × · · · × An )| Φ(s)

∞

j =1 Φ(sj )

Φ(s)

(2.49)

.

Because Φ is increasing, ∞

Φ(sj )

j =1

n ∞

i=1 j =1

n

Φ

i=1

nΦ(s)

by (2.45)

Φ(si,j ) ∞

j =1

si,j

(because Φ is convex) by (2.46) .

(2.50)

By (2.49) and (2.50), |F ∩ (A1 × · · · × An )| nL. Φ(s)

2

(2.51)

Proof of Theorem 2.3. Let α < γ < n, and let Φ be an α-Orlicz function. Then x Φ(x) x γ for large x, and Φ(x)/x γ is eventually decreasing. By Lemma 2.4, we produce a collection {Fj } of pairwise n-disjoint subsets of Nn such that sup{ΨFj (s)/Φ(s): s ∈ N} < C, and for each j ∈ N, we have |π (Fj )| = j for ∈ [n], |Fj | Φ(j )/2. Let F = j Fj , and apply Lemma 2.5. 2

694


3. A relation between combinatorial and functional analytic structures 3.1. An Orlicz norm associated with an α-Orlicz function Suppose Φ(x) = x α φ(x) for x 0 is an α-Orlicz function. Because φ is concave, increasing, and φ(0) 0, we have φ (x) φ(x)/x for all x 0. Hence 0

φ (x) x 1, φ(x)

x 0.

(3.1)

Let Θ(x) = x

α+1 2

1 φ(x) 2 ,

x 0,

(3.2)

x 0.

(3.3)

and θ (x) =

1 φ(Θ −1 (1/x))

,

Note that φ(x)θ 1/Θ(x) = 1,

x 0.

(3.4)

For x 0, define 1 2α MΦ (x) = x α+1 θ (x) α+1 .

(3.5)

Then 1 2α θ (x) 1 x α+1 −1 θ (x) α+1 2α + x , α+1 θ (x)

(3.6)

1 2α(α − 1) 2α 1 −2 α+1 α+1 x + D(x) , θ (x) α+1 α+1

(3.7)

(x) = MΦ

and MΦ (x) =

where 4α θ (x) α θ (x) 2 θ (x) 2 D(x) = x− x + x . α + 1 θ (x) α + 1 θ (x) θ (x)

(3.8)

We now establish that MΦ is an Orlicz function. (In Section 3.2 we will use the Orlicz norm associated with MΦ .) Lemma 3.1. MΦ (defined in (3.5)) is an Orlicz function. Moreover, except for the case Φ(x) = x (x) > 0 and M (x) > 0 for x > 0. for x 0, we have MΦ Φ


695

(x) > 0 for x > 0. Now we consider M . Taking derivatives on Proof. It is obvious that MΦ Φ both sides of (3.4), we have

φ (x)θ 1/Θ(x) + φ(x)θ 1/Θ(x) 1/Θ(x) = 0.

(3.9)

θ (1/Θ(x)) φ (x) x+ 1/Θ(x) x = 0. φ(x) θ (1/Θ(x))

(3.10)

α + 1 1/Θ(x) x = − 1 + E(x) 1/Θ(x) , 2

(3.11)

Hence

By (3.2),

where E(x) =

1 φ (x) x. α + 1 φ(x)

(3.12)

Note α 1. By (3.10) and (3.11), and by substituting 1/Θ(x) = y, we have 2 φ (x) φ (x) θ (y) y= x x 1 by (3.1) . θ (y) (α + 1)(1 + E(x)) φ(x) φ(x)

(3.13)

Taking derivatives on both sides of (3.9), we have φ (x)θ 1/Θ(x) + 2φ (x)θ 1/Θ(x) 1/Θ(x) 2 + φ(x)θ 1/Θ(x) 1/Θ(x) + φ(x)θ 1/Θ(x) 1/Θ(x) = 0.

(3.14)

Hence 2 φ (x) 2 φ (x) θ (1/Θ(x)) θ (1/Θ(x)) x +2 x 1/Θ(x) x + 1/Θ(x) x 2 φ(x) φ(x) θ (1/Θ(x)) θ (1/Θ(x)) θ (1/Θ(x)) 1/Θ(x) x 2 = 0. + θ (1/Θ(x))

(3.15)

By (3.2), (α + 1)(α + 3) 1 + F (x) 1/Θ(x) , 1/Θ(x) x 2 = 4

(3.16)

α + 1 φ (x) 3 φ (x) 2 1 φ (x) 2 4 x+ x − x . F (x) = (α + 1)(α + 3) 2 φ(x) 4 φ(x) 2 φ(x)

(3.17)

where

Bringing (3.11) and (3.16) into (3.15), and substituting 1/Θ(x) = y, we have

696


φ (x) θ (y) φ (x) 2 x − (α + 1) 1 + E(x) x y φ(x) φ(x) θ (y) θ (y) 2 θ (y) 2 (α + 1)(α + 3) α + 1 2 y + 1 + F (x) y = 0. 1 + E(x) + 2 θ (y) 4 θ (y)

(3.18)

By (3.12) and (3.17), 2 1 + F (x) = 1 + E(x) −

φ (x) 2 2 x − G(x), (α + 1)(α + 3) φ(x)

(3.19)

where G(x) =

φ (x) α φ (x) 4 x 1− x . (α + 1)(α + 3) φ(x) 2(α + 1) φ(x)

(3.20)

Then by (3.1), G(x) 0 for all x 0. Applying (3.13) and (3.19) to (3.18), we have φ (x) 2 (α + 1)2 (1 + E(x))2 θ (y) 2 x − y φ(x) 2 θ (y) +

2 θ (y) (α + 1)2 (1 + E(x))2 θ (y) 2 (α + 1)(α + 3) 1 + E(x) y + y 4 θ (y) 4 θ (y)

−

(α + 1)(α + 3) θ (y) 1 φ (x) 2 θ (y) x y− G(x) y = 0. 2 φ(x) θ (y) 4 θ (y)

(3.21)

Then (α + 1)2 (1 + E(x))2 θ (y) 2 θ (y) 2 α + 3 θ (y) −2 y + y + y 4 θ (y) θ (y) α + 1 θ (y) 1 θ (y) (α + 1)(α + 3) θ (y) φ (x) 2 x 1− y + G(x) y0 =− φ(x) 2 θ (y) 4 θ (y) φ (x) 2 θ (y) because − x 0, 0 y 1, and G(x) 0 . φ(x) θ (y)

(3.22)

Hence for all y 0,

θ (y) y −2 θ (y)

2 +

θ (y) 2 α + 3 θ (y) y + y 0. θ (y) α + 1 θ (y)

(3.23)

Then, by (3.8), for all x 0, θ (x) 2 θ (x) 2 α + 3 θ (x) x + x + x D(x) = −2 θ (x) θ (x) α + 1 θ (x) α + 2 θ (x) 2 3α − 3 θ (x) + x + x 0. α + 1 θ (x) α + 1 θ (x)

(3.24)


697

0. If α > 1, then 2α(α − 1)/(α + 1) > 0, and hence By (3.7) and (3.24), we have MΦ (y) y > 0 in (3.13), and hence MΦ (x) > 0 for x > 0. If α = 1, and φ is strictly increasing, then θθ(y) D(x) > 0 for x > 0. Then MΦ (x) > 0 for x > 0. Because Φ is an α-Orlicz function, either (x) > 0 φ ≡ 1, or φ is strictly increasing. Therefore, except for the case Φ(x) = x, we have MΦ and MΦ (x) > 0 for x > 0. 2

The following property will be needed in Section 3.2. (x) − xM (x) > 0 for all x > 0. Lemma 3.2. For MΦ defined in (3.5), MΦ Φ (y) − yM (y) > 0 for all y > 0, where y = 1/Θ(x). For Proof. It suffices to show that MΦ Φ simplicity, we denote MΦ by M. By (3.6) and (3.7),

M (y) − yM (y) =

2α 1 1 y α+1 −1 θ (y) α+1 H (y), α+1

(3.25)

where H (y) = 2α +

2α(α − 1) θ (y) y− − D(y), θ (y) α+1

(3.26)

where D(y) is defined in (3.8). By (3.24) and (3.22), we have D(y) =

4 φ (x) 2 1 θ (y) (α + 1)(α + 3) θ (y) − 1 − x y + G(x) y φ(x) 2 θ (y) 4 θ (y) (α + 1)2 (1 + E(x))2 2 α + 2 θ (y) 3α − 3 θ (y) + y + y. (3.27) α + 1 θ (y) α + 1 θ (y)

Because xφ(x) is convex for x 0, we have xφ(x) = 2φ (x) + xφ (x) 0,

x 0.

(3.28)

Hence −

φ (x) 2 φ (x) x 2 x. φ(x) φ(x)

By (3.26), (3.27), (3.29) and (3.20), H (y)

4α φ (x) θ (y) 4 1 θ (y) 2 + y− x 1 − y α+1 θ (y) φ(x) 2 θ (y) (α + 1)2 (1 + E(x))2 α φ (x) θ (y) φ (x) x 1− x y + φ(x) 2(α + 1) φ(x) θ (y) α + 2 θ (y) 2 3α − 3 θ (y) y − y − α + 1 θ (y) α + 1 θ (y)

(3.29)

698


θ (y) 4 θ (y) α θ (y) 3 4α + y− y+ y = α+1 θ (y) (α + 1)(1 + E(x)) θ (y) 2(α + 1) θ (y) α + 2 θ (y) 2 3α − 3 θ (y) y − y by (3.13) . − α + 1 θ (y) α + 1 θ (y)

(3.30)

By (3.12), θ (y) 4 4 y= (α + 1)(1 + E(x)) θ (y) α+1+

θ (y) y φ (x) θ (y) x φ(x)

φ (x) x (x) α + 1 + φφ(x) x φ(x)

4 α+2

4

by (3.13)

by (3.1) .

(3.31)

By (3.30) and (3.31), 4α 4 3α − 3 θ (y) α + 2 θ (y) 2 − + 1− y− y α+1 α+2 α + 1 θ (y) α + 1 θ (y) 3α − 2 −2α + 4 θ (y) α + 2 θ (y) 2 α2 + + y− y = (α + 1)(α + 2) α+1 α + 1 θ (y) α + 1 θ (y) 3α − 2 −2α + 4 θ (y) α + 2 θ (y) 2 α2 + + y− y (α + 1)(α + 2) α+1 α+1 θ (y) α + 1 θ (y) θ (y) because 0 y1 θ (y) α + 2 θ (y) α + 2 θ (y) 2 α2 + y− y > 0, = (α + 1)(α + 2) α + 1 θ (y) α + 1 θ (y)

H (y)

as desired.

(3.32)

2

3.2. Precise relations between combinatorial measurements and Orlicz norms We recall the following definitions of Orlicz norms ([8], 4.a and 4.b). For an Orlicz function M and a sequence of scalars a = (a1 , a2 , . . .), define aM = inf ρ > 0:

∞

M |an |/ρ 1 ,

(3.33)

n=1

M ∗ (u) = max ux − M(x): x > 0 ,

(3.34)


699

and |||a|||M = sup

∞

an bn :

n=1

∞

M |bn | 1 . ∗

(3.35)

n=1

The two Orlicz norms · M and ||| · |||M are equivalent and aM |||a|||M 2aM .

(3.36)

Definition 3.3. For F ⊂ R n and α-Orlicz function Φ, let ζF (Φ) = sup fˆMΦ : f ∈ BCF (Ω n ) ,

(3.37)

where MΦ is given in (3.5). This definition naturally extends the definition in (1.7). If Φ(x) = x α , x 0, for some α 1, then ζF (Φ) and ζF (2α/(α + 1)) have the same meaning. Let n ∈ N. For F ⊂ R n and α-Orlicz function Φ, let δ(α) =

1 2α

α 2(α 2 +α+1)

if dF (Φ) 1, if dF (Φ) > 1.

(3.38)

Theorem 3.4. (Cf. [2], Theorem XIII.20.) For n ∈ N, there exist Cn > 0 and Dn > 0 such that for all F ⊂ R n and α-Orlicz functions Φ, δ(α) 1 Cn dF (Φ) ζF (Φ) Dn max dF (Φ) 2α , 1 .

(3.39)

Proof. Let Φ be an α-Orlicz function, and let F ⊂ R n such that dF (Φ) < ∞. First we assume limx→∞ Φ(x)/x = ∞. (That is, we exclude the case Φ(x) = x for x 0.) Then, by Lemma 3.1, (x) > 0 and M (x) > 0 for x > 0. (The case Φ(x) = x, MΦ (defined in (3.5)) satisfies MΦ Φ x ∈ [0, ∞), will be discussed later.) For simplicity, we denote MΦ by M. In (3.34), for each u > 0, the maximum of ux − M(x) occurs at the unique point x satisfying M (x) = u. Hence we can treat x as a function of u, and write M ∗ as a function satisfying the two equations M ∗ (u) = ux − M(x),

where x is such that M (x) = u.

(3.40)

We define M2 on [0, ∞) in a similar way by M2 (w) =

√ wx − M(x),

where x is such that M (x) =

√ w.

(3.41)

Then for x satisfying (3.41), dx √ x 1 = , M2 (w) = √ x + w − M (x) dw 2M (x) 2 w

(3.42)

700


and M2 (w) =

M (x) − xM (x) dx . dw 2(M (x))2

(3.43)

By (3.42), (3.43) and Lemma 3.2, M2 is an Orlicz function such that M2 (w) > 0, M2 (w) > 0 for w > 0. By (3.34), M2∗ (y) = max yw − M2 (w): w > 0 .

(3.44)

For each y > 0, the maximum of yw − M2 (w) occurs at the unique point w satisfying y = M2 (w). Hence we can treat w as a function of y. But x is a function of w in (3.41). Therefore by (3.41) and (3.44), x M2∗ (y) = − M (x) + M(x), 2

where x is such that

x = y. 2M (x)

(3.45)

Our aim is to apply (3.45) and the duality expressed in (3.35) to prove (3.39). To this end, let s ∈ N, and consider a s-hypercube A1 × · · · × An ⊂ R n such that |F ∩ (A1 × · · · × An )| = ΨF (s). By (3.33), 1F ∩(A1 ×···×An ) M2∗

= inf ρ > 0:

M2∗

1F (w)/ρ 1

w∈A1 ×···×An

= inf ρ > 0: M2∗ (1/ρ)ΨF (s) 1 .

(3.46)

Let ρs > 0 be such that M2∗ (1/ρs )ΨF (s) = 1.

(3.47)

Then ρs = 1F ∩(A1 ×···×An ) M2∗ . Replacing y by 1/ρs in (3.45), and then combining (3.45) with (3.47), we have the system of equations x 1 = M(x) − M (x) ΨF (s) 2 ρs =

and

2M (x) . x

(3.48) (3.49)

We want to estimate ρs using Eqs. (3.48) and (3.49). To this end, we first estimate x as a solution to Eq. (3.48). By (3.48), 1 x 1 by (2.1) M(x) M(x) − M (x) = α 2 ΨF (s) dF (Φ)s φ(s) 1 1 = φ(s)θ 1/Θ(s) α+1 by (3.4) α dF (Φ)s φ(s) 1 1 2α 1 − α+1 s 2 φ(s)− 2 α+1 θ 1/Θ(s) α+1 = dF (Φ)


2α 1 1 1/Θ(s) α+1 θ 1/Θ(s) α+1 dF (Φ) 1 = M 1/Θ(s) by (3.5) . dF (Φ)

=

701

by ( 3.2) (3.50)

If dF (Φ) 1, then α+1 2α α+1 1 α+1 M dF (Φ) 2α x = dF (Φ) 2α x α+1 θ dF (Φ) 2α x α+1 by (3.5) 1 2α dF (Φ)x α+1 θ (x) α+1 (because θ is increasing) = dF (Φ)M(x) by (3.5) M 1/Θ(s) by (3.50) .

(3.51)

Because M is increasing, the comparison of both sides of (3.51) implies − α+1 x dF (Φ) 2α /Θ(s).

(3.52)

M(x) M 1/Θ(s) .

(3.53)

x 1/Θ(s).

(3.54)

If dF (Φ) < 1, then by (3.50),

Hence

For simplicity, let d˜F (Φ) = max{dF (Φ), 1}. By (3.52), (3.54), − α+1 x d˜F (Φ) 2α /Θ(s), which is the estimate that we need. Now we estimate ρs . By (3.13), we have 0 (3.49), ρs =

θ (x) θ(x) x

1 for all x 0. Then by (3.6) and

1 1 2α θ (x) 2 x α+1 −2 θ (x) α+1 2α + x 4 x −2 θ (x) α+1 . α+1 θ (x)

(3.55)

(3.56)

1

(x) x 1 for all x 0, (x −2 θ (x)) α+1 is decreasing with increasing x. Then by Because 0 θθ(x) (3.55) and (3.56),

− α+1 − 2 − α+1 1 ρs 4 d˜F (Φ) 2α /Θ(s) α+1 θ d˜F (Φ) 2α /Θ(s) α+1 1 1 1 by (3.2) and because d˜F (Φ) 1 4 d˜F (Φ) α s φ(s) α+1 θ 1/Θ(s) α+1 1 = 4 d˜F (Φ) α s by (3.4) , (3.57) which is the estimate we need.

702


Let h be a function with support in F such that

M ∗ h(w) 1.

(3.58)

w∈A1 ×···×An

By (3.40) and (3.41), for all w ∈ A1 × · · · × An , 2 M ∗ h(w) = M2 h(w) .

(3.59)

Hence

2 M2 h(w) 1.

(3.60)

w∈A1 ×···×An

Then,

h(w)2 =

w∈A1 ×···×An

w∈A1 ×···×An

h(w)2 1F |||1F ∩(A ×···×A ) |||M ∗ n 1 2

by (3.60) and the duality in (3.35) 21F ∩(A1 ×···×An ) M2∗ by (3.36) 1 8 d˜F (Φ) α s

by ( 3.57) .

(3.61)

Let πi be the canonical projection from R n onto R. By [2], Lemma XIII.21, there exists a cover {G1 , . . . , Gn } of A1 × · · · × An such that for every i = 1, . . . , n, max r∈Ai

1 h(w)2 1G (w) 8 d˜F (Φ) α . i

(3.62)

w∈πi−1 [r]

Suppose f is an R n -polynomial in C(Ω n ) with spectrum in A1 × · · · × An . (We identify (rj1 , . . . , rjn ) ∈ R n with the character w = rj1 ⊗ · · · ⊗ rjn on Ω n .) By the Cauchy–Schwarz inequality, (3.62) and [2], Lemma XIII.22, we obtain for i ∈ [n],

w∈A1 ×···×An

ˆ f (w)h(w)1Gi (w)

r∈Ai w∈π −1 [r] i

r∈Ai

max r∈Ai

ˆ f (w)h(w)1Gi (w)

2 fˆ(w)2

1

2 h(w)2 1G (w) · i w∈πi−1 [r]

1

w∈πi−1 [r]

1

2 h(w)2 1G (w) · i w∈πi−1 [r]

√ 1 n−1 2 2 d˜F (Φ) 2α ζR (1)2 2 f ∞ ,

r∈Ai

2 fˆ(w)2 1

w∈πi−1 [r]

(3.63)


703

where ζR (1) = sup{fˆ1 : f ∈ BCR (Ω) } 2. Therefore,

w∈A1 ×···×An

n ˆ f (w)h(w)

i=1 w∈A1 ×···×An

ˆ f (w)h(w)1Gi (w)

√ 1 n−1 2 2n d˜F (Φ) 2α ζR (1)2 2 f ∞ .

(3.64)

By (3.58), (3.64) and the duality in (3.35), |||fˆ|||M = sup

fˆ(w)h(w) :

w∈A1 ×···×An

M ∗ h(w) 1

w∈A1 ×···×An

√ 1 n−1 2 2n d˜F (Φ) 2α ζR (1)2 2 f ∞ .

(3.65)

Then by (3.36), √ 1 n−1 fˆM 2 2n d˜F (Φ) 2α ζR (1)2 2 f ∞ ,

(3.66)

√ n−1 which implies (3.39) with Dn = 2 2nζR (1)2 2 . Next suppose that Φ(x) = x for all x 0. (Recall we excluded this case in the beginning of our proof.) Then MΦ (x) = M(x) = x, x 0, and · M = · 1 (R n ) . Let h be in the unit ball of ∞ (R n ) with support in F . Then

h(w)2 F ∩ (A1 × · · · × An ) dF (Φ)s,

(3.67)

w∈A1 ×···×An

which corresponds to (3.61). Following the steps from (3.62) to (3.66), we have 1 n−1 fˆM n dF (Φ) 2 ζR (1)2 2 f ∞ ,

(3.68)

which implies (3.39) in this case. Now we prove the left side inequality of (3.39). For s ∈ N, let A1 × · · · × An be a s-hypercube in R n such that |F ∩ (A1 × · · · × An )| = ΨF (s). Identify (rj1 , . . . , rjn ) ∈ R n with the character w = rj1 ⊗ · · · ⊗ rjn on Ω n . By the Kahane–Salem–Zygmund probabilistic estimates ([2], Theorem X.8), there exists a {−1, +1}-valued n-array {w : w = ±1, w ∈ F ∩ (A1 × · · · × An )} such that if fs =

1 1 2

s (ΨF (s))

1 2

w w,

(3.69)

w∈F ∩(A1 ×···×An )

then 1 1 1 1 1 1 fs ∞ Cfs 2 log 2ns 2 = Cs − 2 (ns) 2 (log 2) 2 = Cn 2 (log 2) 2 , where C > 0 is a constant. By (3.33),

(3.70)

704


fˆs MΦ = inf ρ > 0: = inf ρ > 0:

MΦ fˆs (w)/ρ 1

w∈F ∩(A1 ×···×An )

1 − 1 MΦ s − 2 ΨF (s) 2 /ρ 1

w∈F ∩(A1 ×···×An )

1 − 1 = inf ρ > 0: MΦ s − 2 ΨF (s) 2 ρ −1 ΨF (s) 1 .

(3.71)

For each s ∈ N, let ρs > 0 be such that 1 − 1 MΦ s − 2 ΨF (s) 2 ρs−1 ΨF (s) = 1.

(3.72)

Then ρs = fˆs MΦ . By the definition of dF (Φ) in (2.1) and because Φ is an α-Orlicz function, we have, ΨF (s) dF (Φ)Φ(s) = dF (Φ)s α φ(s).

(3.73)

By the definition of MΦ in (3.5) and by (3.72), 1 2α 1 1 − 1 − 1 1 = s − 2 ΨF (s) 2 ρs−1 α+1 θ s − 2 ΨF (s) 2 ρs−1 α+1 ΨF (s) 2α − α+1

ρs

− 1 1 1 − 1 α α+1 ΨF (s) α+1 s − α+1 θ s − 2 dF (Φ) 2 φ(s) 2 ρs−1 α+1 by (3.73), and because θ is increasing .

(3.74)

By (3.74) and (3.2), − 1 ρs2α ΨF (s)s −α θ dF (Φ) 2 ρs−1 /Θ(s) .

(3.75)

−1/2 −1 ρs . c = dF (Φ)

(3.76)

ρs2α ΨF (s)s −α θ c/Θ(s) .

(3.77)

−1 ΨF (s) . ρs2α ΨF (s)s −α θ 1/Θ(s) = ΨF (s)s −α φ(s) = Φ(s)

(3.78)

Let

By (3.75),

If c > 1, by (3.77) and (3.4),

Then (by taking supremum) 1 sup fˆs MΦ : s ∈ N = sup{ρs : s ∈ N} sup ΨF (s)/Φ(s): s ∈ N = dF (Φ) 2α .

(3.79)


705

If c 1, by (3.2) and because φ is increasing, c/Θ(s) = cs −

α+1 2

− 1 − α+1 − 2 − 1 2 2 2 2 = 1/Θ c − α+1 s . φ(s) 2 c− α+1 s φ c α+1 s

(3.80)

Then −1 2 2 θ c/Θ(s) θ 1/Θ c− α+1 s = φ c− α+1 s by (3.4) .

(3.81)

By (3.77) and (3.81), −1 2 ρs2α ΨF (s)s −α φ c− α+1 s .

(3.82)

Because φ is concave and c < 1, 2

φ(c− α+1 s) − φ(0) c

2 − α+1

s

φ(s) − φ(0) . s

(3.83)

Because φ(0) 0, 2

φ(c− α+1 s) c

2 − α+1

φ(s) − φ(0) +

φ(0) 2

c− α+1

φ(s).

(3.84)

By (3.82), (3.84) and (3.76), −1 2 − 1 − 2 ΨF (s) dF (Φ) α+1 ρs α+1 . ρs2α ΨF (s)s −α φ(s) c α+1 = Φ(s)

(3.85)

Hence 2(α 2 +α+1) α+1

ρs

− 1 ΨF (s) dF (Φ) α+1 . Φ(s)

(3.86)

Then (by taking supremum) 1 (1− α+1 )( 2α+1 ) 2(α +α+1) sup fˆs MΦ : s ∈ N = sup{ρs : s ∈ N} dF (Φ) α = dF (Φ) 2(α2 +α+1) .

(3.87)

By (3.70), (3.79) and (3.87), we obtain the left side inequality of (3.39) with Cn = 1 1 (Cn 2 (log 2) 2 )−1 . 2 Corollary 3.5. For n ∈ N, F ⊂ R n , and α-Orlicz function Φ, lim

s→∞

ΨF (s) 0, k (i) βi log x ,

φ(x) =

x N,

(3.89)

i=1

for k 1 and βi 0 for i = 1, . . . , k. We want to show that the Orlicz function MΦ defined in (3.5) can be approximated in a neighborhood of 0 by Mα,β1 ,...,βk (x) =

α+1 2

β1 α+1

x

2α α+1

− βi k α+1 (i) 1 log , x

(3.90)

i=1

in the sense that limx→0 Mα,β1 ,...,βk (x)/MΦ (x) = 1. By (3.5) and (3.90), Mα,β1 ,...,βk (x) x→0 MΦ (x) lim

β1

= lim

= = =

·

α+1 2 α+1 2 α+1 2 k i=2

k

β

i=1 (log

2α

x→0

2α

α+1 x α+1 ( α+1 2 )

β1 α+1

β1 α+1

1

x α+1 (θ (x)) α+1 βi k − α+1 (i) i=1 (log Θ(y)) lim 1 y→∞ (θ (1/Θ(y))) α+1 lim φ(y)

1 α+1

y→∞

β1 α+1

by substituting x = 1/Θ(y)

βi k 1 − α+1 (i) α+1 2 log y 2 φ(y)

by (3.2) and (3.4)

i=1

− 1 k α+1 (i) βi α + 1 log y α+1 lim log y y→∞ 2 β

i=1

log

i (i) 1 − α+1 x)

(i−1)

k − βi α+1 βi α+1 1 (i) log y + log log x 2 2

k

(by 3.89 )

i=1

k

(i)

βi

y) α+1 = lim βi y→∞ k (i−1) α+1 ( 2 log y + 12 log( ki=1 (log(i) x)βi ))} α+1 i=2 i=2 {log = 1 (by L’Hopital’s rule ), as desired.

i=2 (log

(3.91)


707

4. Relations between combinatorial structures and Lp norms Definition 4.1. (Cf. [2], VII.9, remark (ii).) For n ∈ N, F ⊂ R n , and α-Orlicz function Φ, let ηF (Φ) = sup f Lp /Φ(p): p > 2, f ∈ BL2 (Ω n ) . F

(4.1)

This definition extends the definition in (1.9). Our aim is to establish a link between ηF (Φ) and dF (Φ), where F ⊂ R n . To this end, we first analyze analogous measurements in the context of (TN )n , where T = {e2πit : t ∈ [0, 1]}. We let S = {βj : j ∈ N} be the set of the canonical projections from TN onto T: βj (t) = t(j ),

t = t(j ) : j ∈ N ∈ TN .

(4.2)

We refer to S as the Steinhaus system, and view it as an independent set of characters on the compact Abelian group TN with the normalized Haar measure P . For F ⊂ S n and α-Orlicz function Φ, the definition of ηF (Φ) is the same as in (4.1). (Replace n Ω by (TN )n , and replace P by P.) Lemma 4.2. (Cf. [2], Theorem XIII.27.) For n ∈ N, F ⊂ S n , and α-Orlicz function Φ, 1 1 1 16−n dF (Φ) 2 ηF Φ 2 dF (Φ) 2 .

(4.3)

Proof. By [2], Lemma XII.6 and Lemma XIII.26, for all f ∈ L2F ((TN )n ), 1 f L2s ΨF (s) 2 f L2 ,

s ∈ N.

(4.4)

Because ΨF (s) dF (Φ)Φ(s) for all s ∈ N, 1 1 f L2s dF (Φ) 2 Φ(s) 2 f L2 , Let λ =

2s 2 +2s−ps . p

s ∈ N.

(4.5)

By Hölder’s inequality, for 2s 2s s + 1 and Φ is increasing, λ 1−λ Φ(s) Φ(s + 1) Φ(p).

(4.8)

708


Therefore, 1 1 f Lp dF (Φ) 2 Φ(p) 2 f L2 .

(4.9)

To verify the left side inequality of (4.3), let s ∈ N and let A1 × · · · × An be a s-hypercube in S n such that |F ∩ (A1 × · · · × An )| = ΨF (s). Consider the Riesz product β + β¯ β + β¯ 1+ 1+ ⊗ ··· ⊗ . Hs = 2 2 β∈A1

(4.10)

β∈An

Then Hs L1 = 1 and Hs L2 = 2ns/2 . Hence for 1 p 2, 1− 2

2

ns

Hs Lp Hs L1 q Hs Lq 2 = 2 q ,

1/p + 1/q = 1.

(4.11)

Let

hs =

β1 ⊗ · · · ⊗ βn .

(4.12)

(β1 ,...,βn )∈F ∩(A1 ×···×An )

Let E (expectation) denote integration with respect to Haar measure, either on Ω or on TN . Let En denote the n-fold iteration of E. By Hölder’s inequality and (4.11) with q = s, n E Hs hs Hs Lp hs Lq 2 nss hs Ls 2n ηF (Φ)Φ(s)hs 2 . L

(4.13)

Because n E Hs hs = 2−n ΨF (s),

(4.14)

1 hs L2 = ΨF (s) 2 ,

(4.15)

1 4−n ΨF (s) 2 ηF (Φ)Φ(s),

(4.16)

and

we obtain

which implies the left side of (4.3).

2

Corollary 4.3. (Cf. [2], Corollary XIII.28.) For n ∈ N, F ⊂ R n , and α-Orlicz function Φ, 1 1 1 16−n dF (Φ) 2 ηF Φ 2 4n dF (Φ) 2 .

(4.17)


709

Proof. For each j ∈ N, let rj be the Rademacher function in R such that rj (ω) = ω(j ),

ω ∈ Ω = {−1, 1}N ,

(4.18)

and let βj be the Steinhaus function in S such that βj (t) = t(j ),

t ∈ TN .

(4.19)

Let f be an F -polynomial (i.e., spect f = support fˆ ⊂ F , and spect f is finite). Define for t = (t1 , . . . , tn ) ∈ (TN )n , ft =

fˆ(rj1 ⊗ · · · ⊗ rjn )βj1 (t1 ) · · · βjn (tn )rj1 ⊗ · · · ⊗ rjn .

(4.20)

(rj1 ,...,rjn )∈F

For t ∈ (TN )n , there exists θt ∈ L1 (Ω n ) such that θˆt (rj1 ⊗ · · · ⊗ rjn ) = βj1 (t1 ) · · · βjn (tn ),

rj1 ⊗ · · · ⊗ rjn ∈ spect f,

(4.21)

and θt L1 4n .

(4.22)

(See [2], VII.8 and 12). Then q

q

q

f Lq = ft ∗ θt Lq 4nq ft Lq ,

(4.23)

where ∗ denotes convolution. Integrating both sides of (4.23) with respect to the Haar measure on (TN )n , applying Fubini’s Theorem, and then the right side of (4.3), we obtain 1 1 f Lq 4n dF (Φ) 2 Φ(q) 2 f L2 ,

(4.24)

which implies the right side of (4.17). The proof of the left side of (4.17) is a transcription of the proof of the left side of (4.3).

2

5. Extensions of the Kahane–Khintchin inequality Suppose (A , P) is a probability space. For any Orlicz function ψ , consider the Orlicz norm corresponding to ψ, Xψ = inf ρ > 0: Eψ |X|/ρ 1 ,

X ∈ L0 (A , P).

(5.1)

The classical Kahane–Khintchin inequality states that: if ψ(x) = exp(x 2 ) − 1 for x 0, then there exists K > 0 such that, Xψ KXL2 (Ω,P) ,

X ∈ L2R (Ω, P).

(5.2)

710


(E.g., see [10].) We will extend the inequality in (5.2) to F ⊂ R n . Let Φ = x α φ(x) be an α-Orlicz function (as per Definition 2.1). Define 1 f (x) = x α φ x 2 2 ,

x 0,

(5.3)

and let g = f −1 .

(5.4)

Lemma 5.1. (Cf. [2], Lemma X.18.) Suppose (A , P) is a probability space, and X ∈ BL2 (A ,P) . Then the following are equivalent: (i) there exists 0 < A < ∞ such that 2 lim exp A g(x) P |X| > x < ∞;

x→∞

(5.5)

(ii) there exists 0 2 B;

(5.6)

(iii) there exists 0 < C < ∞ such that lim E exp tg |X| − Ct 2 < ∞;

t→∞

(5.7)

(iv) there exist 0 < D < ∞ such that 2 < ∞. E exp D g |X|

(5.8)

Proof. (i) ⇒ (ii). Suppose limx→∞ exp(A(g(x))2 )P(|X| > x) := B1 < ∞. For p > 2 sufficiently large, ∞ E|X| = p

P |X|p > x dx

0

p

αp 2

p φ(p) 2 + B1 αp p 2

∞

1 2 dx. exp −A g x p

(5.9)

p (φ(p)) 2

1

Let y = g(x p ). Then p p p x = g −1 (y) = f (y) = y αp φ y 2 2 . Hence

(5.10)


p 1 φ (y 2 ) 2 y dx/dy = αpy αp−1 φ y 2 2 1 + α φ(y 2 ) p 2αpy αp−1 φ y 2 2 by (3.1) . When x = p

αp 2

p

(φ(p)) 2 , we have y = αp 2

E|X| p p

711

(5.11)

√ p. Hence by (5.9) and (5.11),

p φ(p) 2 + B1

∞

√

p 2αpy αp−1 φ y 2 2 exp −Ay 2 dy.

(5.12)

p

By the Cauchy–Schwarz inequality, E|X| p p

αp 2

1 ∞ 2 p 2(αp−1) 2 2 φ(p) + 2B1 αp π/A A/π y exp −Ay dy

∞ · √

0

p A/π φ y 2 exp −Ay 2 dy

1 2

(5.13)

.

p

The first integral on the right side of (5.13) is the 2(αp − 1) moment of a Gaussian random variable with mean 0 and variance 1/2A. Hence there exists B2 > 0 such that ∞

p A/πy 2(αp−1) exp −Ay 2 dy B2 p αp−1 .

(5.14)

0

Next we estimate the second integral on the right side of (5.13). By property (iii) in Definition 2.1, √ φ(y 2 )/y is eventually decreasing. Because p is sufficiently large, for all y p, √ φ y 2 /y φ(p)/ p. (5.15) Then ∞ 2 p 1 2 dy A/π φ y exp −Ay (φ(p))p √ p

∞ =

p A/π φ y 2 /φ(p) exp −Ay 2 dy

√ p

∞

√

√ A/π (y/ p )p exp −Ay 2 dy

by (5.15)

p

1 p

p2

∞ p A/π y p exp −Ay 2 dy B3 , 0

(5.16)

712


for some B3 > 0 (by estimating p-th moments of Gaussian random variables). Then ∞ √

p p p A/π φ y 2 exp −Ay 2 dy B3 φ(p) .

(5.17)

p

By (5.13), (5.14) and (5.17), there exists B > 0 such that 1 α XLp Bp 2 φ(p) 2 .

(5.18)

(ii) ⇒ (iii). We assume B 1. For t > 0, ∞ k t k E exp tg |X| = E g |X| . k!

(5.19)

k=0

For each k 1, let 2 k fk (x) = x α φ x k 2 ,

x ∈ [0, ∞),

(5.20)

and let gk = fk−1 .

(5.21)

Then f1 = f and g1 = g. We will show that gk is increasing for k 1, and is concave for k 2. To this end, it suffices to show that fk is increasing for k 1, and is convex for k 2. By (5.20), fk (x) = x α−1

k φ (y) 2 α+ y 0, φ(y) φ(y)

(5.22)

where y = x 2/k . Hence gk = fk−1 is increasing for all k 1. By (5.22), k 2 kα(α − 1) k φ (y) fk (x) = x α−2 φ(y) 2 + kα − + 1 y k 2 2 φ(y) φ (y) 2 φ (y) 2 k + . y + −1 y φ(y) 2 φ(y)

(5.23)

Because Φ is an Orlicz function, for all x 0, φ (x) 2 φ (x) Φ (x) = x α φ(x) = x α−2 φ(x) α(α − 1) + 2α x+ x 0. φ(x) φ(x) For k 2, the expression inside the brackets of (5.23) is

(5.24)


713

k φ (y) φ (y) 2 φ (y) 2 kα(α − 1) k + kα − + 1 y+ y + −1 y 2 2 φ(y) φ(y) 2 φ(y) α(α − 1) + 2α 0

φ (y) 2 φ (y) y+ y φ(y) φ(y)

by (5.24) .

(5.25)

Hence fk 0 for k 2. Therefore gk = fk−1 is concave for k 2, as desired. By (5.3), (5.4), (5.20) and (5.21), k k g(x) = f −1 (x) = fk−1 x k = gk x k .

(5.26)

Then, by Jensen’s inequality, for k 2, k E g |X| = E gk |X|k gk E|X|k .

(5.27)

By assumption (ii) and because XL2 1, for k 2, E|X|k B k k

αk 2

k φ(k) 2 .

(5.28)

Because B 1, we have φ(B 2/α k) φ(k). Hence 2 k 2 αk 2 k E|X|k B α k 2 φ B α k 2 = fk B α k 2 .

(5.29)

By (5.27) and (5.29), for k 2, k 2 k 2 k k k E g(|X|) (gk ◦ fk ) B α k 2 = B α k 2 = B α k 2 .

(5.30)

Next we estimate E(g(|X|)). By (5.3), 0 f (x) (φ(1))1/2 for 0 x 1. Then 0 g(x) 1 for 0 x (φ(1))1/2 (because g = f −1 ). Also by (5.3), for x (φ(1))1/2 , we have − 1 − 1 1 1 g(x) x α φ(1) 2α = x φ(1) 2 α − 1 − 1 2 because x φ(1) 2 1 x φ(1) 2 −1 = x 2 φ(1) .

(5.31)

Let K = max{2(φ(1))−1 , 2}. Then E g |X| = E g |X| 1

1

{|X|(φ(1)) 2 }

+ E g |X| 1

−1 1 + φ(1) E|X|2 K

1

{|X|>(φ(1)) 2 }

because E|X|2 1 .

Applying (5.30) and (5.32) to (5.19), we obtain for t sufficiently large,

(5.32)

714

R. Blei, L. Ge / Journal of Functional Analysis 257 (2009) 683–720 ∞ k

k k t Bαk2 E exp tg |X| 1 + Kt + k! k=2

k because k 2 /k! < 2k /(k/2)! exp Ct 2

(5.33)

for some C > 0. (iii) ⇒ (i). Because g is increasing (g = g1 ), for x > 0 and t > 0, P |X| > x P g(|X|) > g(x) E exp(tg(|X|)) exp(tg(x))

(by Chebyshev’s inequality).

(5.34)

Then by assumption (iii), for t > 0 sufficiently large, exp(Ct 2 ) P |X| > x . exp(tg(x))

(5.35)

Put t = g(x)/2C in (5.34), and obtain (5.5) with A = 1/4C. (i) ⇒ (iv). Suppose limx→∞ exp(A(g(x))2 )P(|X| > x) := M1 < ∞, and let M2 > 0 be sufficiently large so that P(|X| > x) M1 exp(−A(g(x))2 ) for x M2 . Choose 0 < D < A. Then 2 E exp D g |X| =

∞

2 > x dx P exp D g |X|

0

∞ M2 +

1 1 P |X| > g −1 (log x) 2 /D 2 dx

(because g is increasing)

M2

∞ M2 + M1

1 1 2 dx exp −A g g −1 (log x) 2 /D 2

(by assumption (i))

M2

∞ = M2 + M 1

A

x − D dx M2 + M1 .

(5.36)

M2

(iv) ⇒ (i). Because g is increasing, for x > 0 sufficiently large, 2 2 P |X| > x P D g |X| > D g(x) which implies (5.5).

2

E exp(D(g(|X|))2 ) exp(D(g(x))2 )

(by Chebyshev’s inequality),

(5.37)


715

Lemma 5.2. Let f and g be the functions defined in (5.3) and (5.4). Let 2 h(x) = exp g(x) − 1,

x 0.

(5.38)

Then there exists N > 0 such that h (x) 0 for all x N . Proof. 2 h (x) = 2 exp g(x) g(x)g (x),

(5.39)

2 h (x) = 2 exp g(x) I (x),

(5.40)

2 2 2 I (x) = 2 g(x) g (x) + g (x) + g(x)g (x).

(5.41)

and

where

Because g = f −1 , we have g f (x) f (x) = 1,

x 0.

(5.42)

Hence 2 g f (x) f (x) + g f (x) f (x) = 0.

(5.43)

f (x) . g f (x) = − (f (x))3

(5.44)

By (5.42) and (5.43),

By (5.41) and (5.44), 2 2 2 I f (x) = 2 g f (x) g f (x) + g f (x) + g f (x) g f (x) 2 2 because g = f −1 = 2x 2 g f (x) + g f (x) + xg f (x) f (x) (f (x)3 ) 1 f (x) 2 2x + 1 − = x . f (x) (f (x))2 = 2x 2

1

(f (x))2

+

1

(f (x))2

−x

by ( 5.42) and (5.44) (5.45)

By (5.22) with k = 1, 1 1 φ (x 2 ) 2 x f (x) = x α−1 φ x 2 2 α + αx α−1 φ x 2 2 . 2 φ(x )

(5.46)

716


By (5.23) with k = 1, and because 0

φ (x 2 ) 2 x φ(x 2 )

1 and φ 0,

2 1 φ (x 2 ) 2 φ (x 2 ) 4 φ (x ) 2 2 f (x) = x α−2 φ x 2 2 α(α − 1) + (2α + 1) x x x + 2 − φ(x 2 ) φ(x 2 ) φ(x 2 ) 1 (5.47) x α−2 φ x 2 2 α(α − 1) + 2α + 1 . Hence 1 f (x) x α + 1 + α + 2. f (x) α

(5.48)

By (5.45) and (5.48), I f (x)

1 (f (x))2

2x 2 − α − 1 .

(5.49)

Replacing x by g(x) in (5.49), we have I (x)

1 (f (g(x)))2

2 2 g(x) − α − 1 .

Then for x g −1 ((α + 1)/2)1/2 , we have I (x) 0 which implies h (x) 0.

(5.50) 2

Let ψΦ be an Orlicz function such that for some N > 0, 2 ψΦ (x) = exp g(x) − 1,

x N,

(5.51)

where g is defined in (5.4). Lemma 5.3. (Cf. [2], Remark X.9.i.) Suppose (A , P) is a probability space, and X ∈ BL2 (A ,P) . Then the following are equivalent: F

(i) there exists 0 < D < ∞ such that 2 < ∞; E exp D g |X|

(5.52)

XψΦ < ∞.

(5.53)

(ii)


717

Proof. (i) ⇒ (ii). Suppose 2 M, E exp D g |X|

(5.54)

for some M 1. Let β > 0 be such that β max{4M, D} and ψΦ (N/β) 12 . Then 1 EψΦ |X|/β 1{|X| 0, φ(cx) − φ(0) φ(x) − φ(0) . cx x

(5.56)

φ(cx) φ(x) φ(0) φ(0) φ(x) − + . cx x x cx x

(5.57)

Then, because φ(0) 0,

Let − α+1 2 x , L(x) = g βD −1

x 0.

(5.58)

Then α+1 − α+1 2 x x = βD −1 2 βD −1 α+1 = βD −1 2 g −1 L(x) α+1 α 2 1 2 by (5.3) and (5.4) = βD −1 2 L(x) φ L(x) α α 2 1 2 = βD −1 2 L(x) βD −1 φ L(x) 1 2 1 α 2 by (5.57) and because βD −1 1 βD −1 2 L(x) φ βD −1 L(x) 1 by (5.3) and (5.4) . (5.59) = g −1 βD −1 2 L(x) By (5.58) and (5.59), 1 1 − α+1 2 x . g(x) βD −1 2 L(x) = βD −1 2 g βD −1

(5.60)

Because β 2M, we have − α+1 1 1 2 x . (2M)− 2 D 2 g(x) g βD −1 Then

(5.61)

718

R. Blei, L. Ge / Journal of Functional Analysis 257 (2009) 683–720 ∞

− α+1 2 k 1 2 |X| 2 1 + E (2M)−1 D g |X| E exp g βD −1 k! k=1

1+

∞ 1 1 2 k D g |X| E 2M k!

(because M 1)

k=1

1+

3 2

2 1 E exp D g |X| 2M by (5.54) .

(5.62)

By the definition of ψΦ in (5.51), we have − α+1 −1 − α+1 2 |X| 1 2 |X| 2 1 EψΦ βD −1 {|X|N } = E exp g βD {|X|N } − 1 Let K = max{β, (βD −1 )

α+1 2

1 2

by (5.62) .

(5.63)

}. By (5.55) and (5.63), we have EψΦ |X|/K 1.

(5.64)

XψΦ K.

(5.65)

Therefore

(ii) ⇒ (i). If XψΦ K for some K > 0, then EψΦ |X|/K 1.

(5.66)

Hence by the definition of ψΦ in (5.51), 2 − 1 1{|X|N } 1. E exp g |X|/K

(5.67)

2 . M = max 4, 2E exp g(N/K)

(5.68)

2 M + 2 M. E exp g |X|/K 2

(5.69)

Let

By (5.67) and (5.68),

We may assume K 1. By (5.3) and (5.4), for x 0, α 2 1 2. x = f g(x) = g(x) φ g(x) Then

(5.70)


719

α 2 1 2 1 f g(x)/K α = g(x)/K 1/α φ g(x) /K α 2 2 1 α 2 /K g(x) φ g(x) = x/K by ( 5.70) = f g(x/K) .

(5.71)

Hence 1

g(x)/K α g(x/K).

(5.72)

Let D = 1/K 2/α . By (5.72) and (5.69), we obtain 2 2 E exp D g |X| E exp g |X|/K M,

(5.73)

2

as desired.

The following is a link between the combinatorial structure of F ⊂ R n and tail probability estimates involving random variables in L2F (Ω n , Pn ). Theorem 5.4. For n ∈ N, F ⊂ R n , and α-Orlicz function Φ, dF (Φ) < ∞

⇐⇒

sup XψΦ : X ∈ BL2 (Ω n ) < ∞. F

(5.74)

Proof. Observe that statement (iv) in Lemma 5.1 is the same as statement (i) in Lemma 5.3. Then by Lemma 5.1 and Lemma 5.3, 1 α sup XLp /p 2 φ(p) 2 : p > 2, X ∈ BL2 (Ω n ) < ∞ F ⇐⇒ sup XψΦ : X ∈ BL2 (Ω n ) < ∞. F

1

(5.75)

1

Because Φ 2 (p) = p α/2 (φ(p)) 2 , 1 1 α ηF Φ 2 = sup XLp /p 2 φ(p) 2 : p > 2, X ∈ BL2 (Ω n ) F

(Definition 4.1).

(5.76)

Hence 1 ηF Φ 2 < ∞

⇐⇒

which, by Corollary 4.3, implies (5.74).

sup XψΦ : X ∈ BL2 (Ω n ) < ∞, F

2

(5.77)

720


References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

R. Blei, Combinatorial dimension and certain norms in harmonic analysis, Amer. J. Math. 106 (1984) 847–887. R. Blei, Analysis in Integer and Fractional Dimensions, Cambridge University Press, Cambridge, 2001. R. Blei, T.W. Körner, Combinatorial dimension and random sets, Israel J. Math. 47 (1984) 65–74. R. Blei, Y. Peres, J.H. Schmerl, Fractional products of sets, Random Structures Algorithms 6 (1995) 113–119. A. Bonami, Étude des coefficients de Fourier des fonctions de Lp (G), Ann. Inst. Fourier (Grenoble) 20 (1970) 335–402. G. Johnson, G. Woodward, On p-Sidon sets, Indiana Univ. Math. J. 24 (1974) 161–167. A. Khintchin, Über dyadische Brüche, Math. Z. 18 (1923) 109–116. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, Berlin, Heidelberg, New York, 1977. J.E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford 1 (1930) 164–174. G. Pe˘skir, Best constants in Kahane–Khintchine inequalities for complex Steinhaus function, Proc. Amer. Math. Soc. 123 (10) (October 1995) 3101–3111.


Local “superlinearity” and “sublinearity” for the p-Laplacian Djairo G. de Figueiredo a , Jean-Pierre Gossez b,∗ , Pedro Ubilla c a IMECC-UNICAMP, Caixa Postal 6065, 13081-970 Campinas, SP, Brazil b Département de Mathématique, C.P. 214, Université Libre de Bruxelles, 1050 Bruxelles, Belgium c Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Received 23 November 2008; accepted 1 April 2009 Available online 29 April 2009 Communicated by J. Coron

Abstract We study the existence, nonexistence and multiplicity of positive solutions for a family of problems 1,p −p u = fλ (x, u), u ∈ W0 (Ω), where Ω is a bounded domain in RN , N > p, and λ > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti–Brezis–Cerami type in a more general form, namely λa(x)uq + b(x)ur , where 0 q < p − 1 < r p ∗ − 1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis–Nirenberg 1,p result on local minimization in W0 and C01 , a C 1,α estimate for equations of the form −p u = h(x, u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper–lower solutions for the p-Laplacian. © 2009 Elsevier Inc. All rights reserved. 1,p

Keywords: p-Laplacian; Concave-convex nonlinearities; Critical exponent; C01 versus W0 comparison principle; C 1,α estimate; Upper–lower solutions

local minimization; Strong


E-mail addresses: [email protected] (D.G. de Figueiredo), [email protected] (J.-P. Gossez), [email protected] (P. Ubilla). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.001

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D.G. de Figueiredo et al. / Journal of Functional Analysis 257 (2009) 721–752

1. Introduction This paper is concerned with the existence, nonexistence and multiplicity of solutions for the family of problems ⎧ ⎨ −p u = fλ (x, u) u>0 ⎩ u=0

in Ω, in Ω, on ∂Ω

(1.1)

where p u := div(|∇u|p−2 ∇u) is the usual p-Laplacian, Ω is a bounded domain in RN , and λ > 0 is a real parameter. A basic feature of the family considered here is its monotone dependence on λ, i.e. fλ (x, s) fλ (x, s) if λ < λ . In the context of (1.1) the conditions of local “sublinearity” at 0 and of local “superlinearity” at ∞ mean, roughly speaking, that for x in a subdomain Ω1 of Ω, one has lim fλ (x, s)/s p−1 = +∞,

s→0 s>0

while for x in another subdomain Ω2 of Ω, one has lim fλ (x, s)/s p−1 = +∞

s→+∞

(see (HΩ1 ) and (HΩ2 ) in Section 2 for the precise statements). There are several motivations to our study of (1.1). The main one comes from the following example: ⎧ q r ⎨ −p u = λa(x)u + b(x)u u>0 ⎩ u=0


(1.2)

where 0 < q < p − 1 < r. Problem (1.2) was originally considered in [2] when p = 2 and a(x) ≡ 1, b(x) ≡ 1. It was in particular shown there that if r 2∗ − 1, then there exists 0 < Λ < ∞ such that (1.2) has at least two solutions for λ < Λ, at least one solution for λ = Λ, and no solution for λ > Λ. This result of “global multiplicity” was extended in [15] to the case p = 2 and variable coefficients a(x), b(x). It was also extended in [19] to the case p = 2 and a(x) ≡ 1, b(x) ≡ 1, although here under some restrictions on the exponents p, q in the critical case r = p ∗ − 1. In this paper we consider the general case: p = 2 and variable coefficients a(x), b(x). As in [15] a(x) is restricted to be 0 and b(x) is allowed to change sign, even in the critical case r = p ∗ − 1. For the existence of a second solution, we will however need here a stronger restriction on a(x), namely a(x) > 0. This difference with respect to the semilinear case is connected with the use of a strong comparison principle for the p-Laplacian (see the comments after hypothesis (M) in Section 2). In the critical case r = p ∗ − 1 we will meet similar restrictions as in [19] on the exponents p, q (see Remark 2.7). As observed on p. 454 of [7], critical problems become more delicate in the presence of variable coefficients. In this respect our basic assumption on b(x) in (1.2) for r = p ∗ − 1 is of the same nature as that introduced in [15] when p = 2: b(x) should be sufficiently close to b L∞ (Ω) on a small ball (cf. condition (b) in Theorems 2.5 and 2.6).


723

Our general results relative to (1.1) can also be applied to various situations rather different from (1.2). For instance the p version of example (1.3) from [15] can be handled, as we shall see in Section 6. There are several preliminary results in our study which have an independent interest. We mention in particular the extension to the p-Laplacian context of the well-known result of Brezis 1,p and Nirenberg [8] on local minimization in C01 and W0 (cf. Proposition 3.9). Several works have been devoted to this problem, e.g. [19,23,21,16]. Our approach differs from that in these papers and is more in the line of that introduced recently in [9] in the subcritical case. It avoids in particular the consideration of equations involving two p-Laplacians. An important step in the proof of this minimization result is Proposition 3.7 which provides a C 1,α estimate in the critical case. Another result of independent interest concerns the strong comparison principle for the p-Laplacian. This is known to be a delicate question, which has not received yet a complete answer (see [29] for a recent survey). The version we present here (cf. Proposition 3.4) is based on ideas from [4] and [17]. There is finally a variational approach to the method of upper–lower solutions for the p-Laplacian (cf. Proposition 3.1). It is adapted from [30] and could also prove useful in other situations. Our method to obtain multiple solutions to (1.1) follows the classical way of obtaining a first solution via upper–lower solutions and a second solution via the mountain pass theorem. To handle the (PS) condition in the critical case we use some of the techniques initially developed for p = 2 in [7,2] and later extended for p = 2 in [18]. Our results relative to (1.1) are stated in detail in Section 2 and their proofs given in Sections 4 and 5. Section 3 is devoted to various preliminaries, including those mentioned above. Section 6 is devoted to some applications of the results of Section 2, in particular to (1.2). 2. Statement of results In this section we state our results relative to (1.1). We successively consider fλ (x, u) of arbitrary growth, of subcritical growth, and finally of critical growth. Let Ω be a smooth bounded domain in R N . We assume 1 < p < N . Our results, however, can be easily adapted to the case p N , by replacing the subcritical or critical growth by an arbitrary power growth. Our general assumptions on the family fλ (x, s) are the following: (H ) For each λ > 0, fλ : Ω × [0, ∞[ → R is a Carathéodory function with the property that for any s0 > 0, there exists a constant A (depending on λ and s0 ), such that |fλ (x, s)| A for a.e. x ∈ Ω and all s ∈ [0, s0 ]. Moreover if λ < λ , then fλ (x, s) fλ (x, s) for a.e. x ∈ R and all s 0. (H0 ) For each λ > 0 and any s0 > 0, there exists B 0 (depending on λ and s0 ), such that fλ (x, s) −Bs p−1 for a.e. x ∈ Ω and all s ∈ [0, s0 ].

724


Assumption (H0 ) concerns the behavior of fλ (x, s) near s = 0 and is used to apply the maximum principle; it implies fλ (x, 0) 0. From now on we will always understand that fλ (x, s) has been extended for s < 0 by putting fλ (x, s) = fλ (x, 0). 1,p At this stage, if u ∈ W0 (Ω) ∩ L∞ (Ω) satisfies −p u = fλ (x, u) in the weak sense, then (H ) and the regularity theory for the p-Laplacian (cf. e.g. [26]) imply u ∈ C 1,α (Ω) for some α = α(N, p) ∈ ]0, 1[. Moreover u 0 (take −u− as testing function and use fλ (x, 0) 0). In addition, by the strong maximum principle of [31], if u ≡ 0, then u > 0 in Ω and ∂u/∂ν < 0 on ∂Ω, where ν denotes the exterior normal. We thus have a solution of (1.1). Observe also that the associated functional 1 Iλ (u) := p

|∇u| − p

Ω

Fλ (x, u)

(2.1)

Ω

s 1,p where Fλ (x, s) := 0 fλ (x, t) dt, is well defined for u ∈ W0 (Ω) ∩ L∞ (Ω). The following two assumptions will also be used throughout the paper: (He ) There exist λ > 0 and a nondecreasing function g with inf{g(s)/s p−1 : s > 0} < p−1 1/ e ∞ such that fλ (x, s) g(s) 1,p

for a.e. x ∈ Ω and all s 0; here e ∈ W0 (Ω) ∩ C 1,α (Ω) is the solution of −p e = 1 and ∞ denotes the L∞ (Ω) norm. (HΩ1 ) For any λ > 0 there exists a smooth subdomain Ω1 , s1 > 0 and θ1 > λ1 (Ω1 ) such that fλ (x, s) θ1 s p−1 for a.e. x ∈ Ω1 and all s ∈ [0, s1 ]; here λ1 (Ω) denotes the principal eigenvalue of −p 1,p on W0 (Ω). Assumption (He ) is used to guarantee the existence of an upper solution for that specific value of λ. More comments about (He ) can be found for instance in [14]. Assumption (HΩ1 ) is a local (i.e. on Ω1 ) “sublinearity” condition at 0, which is satisfied for instance if lim fλ (x, s)/s p−1 = +∞

s→0 s>0

uniformly for x ∈ Ω1 . (HΩ1 ) is used to construct a lower solution. Theorem 2.1 (Existence of one solution without growth condition). Under the assumptions (H ), (H0 ), (He ) and (HΩ1 ), there exists 0 < Λ +∞ such that problem (1.1) has at least one 1,p solution u ∈ W0 (Ω) ∩ L∞ (Ω) (with Iλ (u) < 0) for 0 < λ < Λ and no solution for λ > Λ. As observed in [15, p. 272], in the present generality, Λ can be +∞.


725

Theorem 2.2 (Nonexistence for λ large). In addition to the hypotheses of Theorem 2.1 assume: ˜ m ˜ with m ˜ m) (HΩ˜ ) There exist λ > 0, a smooth subdomain Ω, ˜ ∈ L∞ (Ω) ˜ 0, ≡ 0, μ > λ1 (Ω, ˜ such that p−1 fλ (x, s) μm(x)s ˜

˜ m) for a.e. x ∈ Ω˜ and all s 0; here λ1 (Ω, ˜ denotes the principal eigenvalue of −p 1,p ˜ on W0 (Ω) for the weight m. ˜ Then problem (1.1) for the value of λ provided by (HΩ˜ ) has no solution (and consequently Λ < +∞). Assumption (HΩ˜ ) can be seen as a localized version of the trivial sufficient condition of nonexistence for the semilinear problem −u = l(u) in Ω, u > 0 in Ω and u = 0 on ∂Ω, namely inf{l(s)/s: s > 0} > λ1 (Ω), where λ1 (Ω) denotes here the first eigenvalue of − on H01 (Ω). p−1 for some function h with Assumption (HΩ˜ ) is satisfied in particular if fλ (x, s) h(λ)m(x)s ˜ h(λ) → +∞ as λ → ∞. Due to the absence of growth condition, we have up to now included in the definition of a solution u the requirement that u belongs to L∞ (Ω). We will now assume the following growth condition on fλ (x, u), where p ∗ := Np/(N − p): (G) For any λ > 0, there exist d1 , d2 and σ p ∗ − 1 such that fλ (x, s) d1 + d2 s σ for a.e. x ∈ Ω and all s 0. 1,p

Condition (G) implies that any u ∈ W0 (Ω) which solves −p u = fλ (x, u) in the weak sense belongs to L∞ (Ω) (cf. e.g. [3] in the subcritical case, [22] in the critical case), and consequently, as before, belongs to C 1,α (Ω) for some α = α(p, N ). Moreover, when σ < p ∗ − 1, the norm of u 1,p in C 1,α (Ω) can be estimated in terms of the constants from (G) and the norm of u in W0 (Ω) (by using successively [3] and [26]). Such an estimate does not hold anymore when σ = p ∗ − 1, as can be seen by considering for p = 2 the family of instantons uε (x) =

ε 2 ε + |x|2

N−2 2

−

ε 2 ε +1

N−2 2

on Ω = B(0, 1). Note that Proposition 3.7 from Section 3 will provide a substitute to this estimate in the critical case. Condition (G) also implies that the functional Iλ (u) is now well defined for 1,p all u in W0 (Ω). The following Ambrosetti–Rabinowitz type condition, introduced in [14] to handle indefinite nonlinearities, will play a role in our subsequent results:

726


(AR)d For any λ > 0, there exist θ > p, ρ < p, s0 0 and d 0 such that θ Fλ (x, s) sfλ (x, s) + ds ρ for a.e. x ∈ Ω and all s s0 . Theorem 2.3 (Existence of one solution for λ = Λ in the subcritical case). In addition to the hypotheses of Theorem 2.2 assume the continuity of fλ (x, s) with respect to λ ( for a.e. x and uniformly for s bounded ). Assume also that (G) with σ 0} (i.e. the various constants appearing in (G) and (AR)d ) can be chosen independently of λ for λ ∈ [r, R]). Then problem (1.1) has at least one solution u ∈ 1,p W0 (Ω) ∩ L∞ (Ω) (with Iλ (u) 0) for λ = Λ. Next we state our results on the existence of at least two solutions for 0 < λ < Λ. We will first deal with the subcritical case σ 0 and any s0 > 0, there exists B 0 such that for a.e. x ∈ Ω, the function s → fλ (x, s) + Bs p−1 is nondecreasing on [0, s0 ]; moreover fλ (x, 0) 0 for all λ 0 and a.e. x ∈ Ω. The monotonicity of the family fλ is also strengthened in the following way, where we write h(x) ≺ l(x) to mean that for any compact K ⊂ Ω there exists ε > 0 such that h(x) + ε l(x) for a.e. x ∈ K: (M) For any λ < λ and any u ∈ C01 (Ω) with u > 0 in Ω, one has

fλ x, u(x) ≺ fλ x, u(x) . Note that (M) is significantly stronger than the corresponding requirement in the semilinear case: fλ (z, u(x)) ≡ fλ (x, u(x)) (cf. [15, p. 273]). This difference is related to the use of a strong comparison principle for the p-Laplacian (cf. Proposition 3.4 below). Our last additional assumption is: (HΩ2 ) For any λ > 0, these exist a subdomain Ω2 , s2 > 0 and θ2 > 0 such that Fλ (x, s) θ2 s p for a.e. x ∈ Ω2 and all s s2 . Assumption (HΩ2 ) is a local (i.e. on Ω2 ) “superlinearity” condition at ∞, which is satisfied for instance if lim fλ (x, s)/s p−1 = +∞

s→+∞


727

uniformly for x ∈ Ω2 . It is used in conjunction with (AR)d to derive the geometry of the mountain pass. Theorem 2.4 (Second solution in the subcritical case). In addition to the hypotheses of Theorem 2.1, assume (G) with σ < p ∗ − 1 as well as (AR)d , (H0 ) , (M) and (HΩ2 ). Then problem (1.1) has at least two solutions u, v for 0 < λ < Λ, with u ≡ v in Ω and Iλ (u) < 0. We finally consider multiplicity in the case where fλ has critical growth. Here we write fλ as fλ (x, s) = hλ (x, s) + b(x)s p

∗ −1

(2.2)

,

where hλ : Ω × [0, ∞[ → R is a Carathéodory function and b ∈ L∞ (Ω). We will distinguish two cases depending on the growth of the subcritical part hλ (x, s) : either (i) hλ satisfies (G) with σ < p − 1, b(x) may change sign, or (ii) hλ satisfies (G) with σ N(N − p)/(p 2 + N − p), one has 0 b ∞ − b(x) M|x − x0 |γb for a.e. x ∈ B1 . (Recall that ∞ denotes the L∞ (Ω) norm.) Moreover, when p 3, the following condition on hλ will also be assumed: 2 0, there exist c0 > 0, δ > 0, and q with p ∗ − p−1 such that

Hλ (x, s + u) − Hλ (x, s) hλ (x, s)u + c0 uq+1 for all u 0, s ∈ [0, s0 ] and a.e. x ∈ B(x0 , δ), where Hλ (x, s) := the point involved in assumption (b) above.

s 0

hλ (x, t) dt, and x0 is

Assumption (b) implies some control on the negative part of b: b− ∞ b+ ∞ , with in addition some limitation on the way b(x) approaches b ∞ . It trivially holds if b(x) = b ∞ on a small ball. Assumption (Hh ) provides some control on the way Hλ (x, s) is increasing. It is used to handle the case p 3 in Lemma 5.3. A simple calculation based on Lemma A4, part (4), from [18] shows that (Hh ) holds for instance if hλ (x, s) = λa(x)(s q + g(s)) with g nondecreas2 < q + 1 0 near x0 . Note that the condition p ∗ − p−1 rather strong restriction on the dimension: N > p(1 + and 6.4. We first deal with the critical case (i).

p(p−1) ). 2

See in this respect Remarks 2.7

Theorem 2.5 (Second solution in the critical case with σ < p − 1 for hλ ). In addition to the assumptions of Theorem 2.1, assume fλ (x, s) satisfies (H0 ) and (M). Suppose also that fλ (x, s) can be written as in (2.2) with hλ (x, s) nondecreasing with respect to s and satisfying (G) with

728


σ < p − 1. Suppose also that b(x) in (2.2) is ≡ 0, and satisfies (b). Then the conclusion of Theorem 2.4 holds provided that in addition either (i) 2N/(N + 1) < p < 3, or (ii) p 3 and (Hh ) is satisfied. We now deal with the critical case (ii). Theorem 2.6 (Second solution in the critical case with σ < p ∗ − 1 for hλ ). In addition to the assumptions of Theorem 2.1, assume fλ (x, s) satisfies (H0 ) and (M). Suppose also that fλ (x, s) can be written as in (2.2) with hλ (x, s) nondecreasing with respect to s and satisfying (G) with σ < p ∗ − 1, and hλ (x, s) satisfying (AR)d . Suppose that b in (2.2) is ≡ 0, 0 in Ω and satisfies (b). Then the conclusion of Theorem 2.4 holds provided that in addition either (i) 2N/(N + 1) p < 3, or (ii) p 3 and (Hh ) is satisfied. In Theorem 2.6, hλ (x, s) is allowed any subcritical growth, at the expense of assuming (AR)d for hλ (x, s) and b(x) 0. Remark 2.7. The condition p > N2N +1 in Theorems 2.5 and 2.6 is slightly more restrictive than 2 considered in [18,19]. The condition p ∗ − p−1 < q + 1 N2N +2 already appears in [18,19]. 3. Some preliminaries 3.1. Upper–lower solutions We start by recalling the version of the method of upper–lower solutions which we will use repeatedly. Let g(x, s) be a Carathéodory function on Ω × R with the property that for any s0 > 0, there exists a constant A such that |g(x, s)| A for a.e. x ∈ Ω and all s ∈ [−s0 , s0 ]. A function u ∈ W 1,p (Ω) ∩ L∞ (Ω) is called a (weak) lower solution of the problem

−p u = g(x, u) u=0

in Ω, on ∂Ω,

(3.1)

if u 0 on ∂Ω and

|∇u|p−2 ∇u∇ϕ

Ω

g(x, u)ϕ Ω

for all ϕ ∈ Cc∞ (Ω), ϕ 0. An upper solution is defined by reversing the inequality signs. Proposition 3.1. Assume that u and u are respectively lower and upper solutions for (3.1), with u u a.e. in Ω. Consider the associated functional Φ(u) :=

1 p

|∇u|p − Ω

G(x, u) Ω


where G(x, s) :=

s 0

729

g(x, t) dt, and the interval

1,p M := u ∈ W0 (Ω): u u u a.e. in Ω .

Then the infimum of Φ on M is achieved at some u, and such a u is a solution of (3.1). Proof. It is adapted from [30] which deals with the semilinear case. By coercivity and weak lower semicontinuity, one easily sees that the infimum of Φ on M is achieved at some u. Let ϕ ∈ Cc∞ (Ω), ε > 0, and define

vε := min u, max{u, u + εϕ} = u + εϕ − ϕ ε + ϕε where ϕ ε := max{0, u + εϕ − u} and ϕε := − min{0, u + εϕ − u}. Since u minimizes Φ on M, one has Φ (u), vε − u 0, which gives Φ (u), ϕ Φ (u), ϕ ε − Φ (u), ϕε /ε.

(3.2)

Since u is an upper solution and −p is monotone, one also has Φ (u), ϕ ε Φ (u) − Φ (u), ϕ ε

p−2 p−2 ε |∇u| ∇u − |∇u| ∇u ∇ϕ − ε g(x, u) − g(x, u)|ϕ|

Ωε

Ωε

where Ωε := {x ∈ Ω: u(x) + εϕ(x) u(x) > u(x)}. As |Ωε | → 0 as ε → 0, this latter relation implies Φ (u), ϕ ε o(ε) as ε → 0. Similarly Φ (u), ϕε o(ε), and so by (3.2), Φ (u), ϕ 0. Replacing ϕ by −ϕ, one concludes that u solves (3.1). 2 Other results asserting the existence of a solution between a lower solution and an upper solution for a p-Laplacian equation can be found e.g. in [12,19]. These results however do not have the level of generality we need (our local “sublinearity” assumption will lead us to deal with weak lower solutions) or do not give enough information on the solution (the minimization property will be crucial for our multiplicity results). 3.2. Integration by parts formula We now turn to an integration by parts formula which will play a role in the use of our local “sublinearity” condition (HΩ1 ). Proposition 3.2. Let u ∈ C 1 (Ω) with p u ∈ L1 (Ω) in the distribution sense. Let ϕ ∈ C ∞ (Ω). Then |∇u|p−2 ∇u∇ϕ = |∇u|p−2 ∇uνϕ − (p u)ϕ (3.3) Ω

∂Ω

where ν denotes the exterior normal vector.

Ω

730


Formula (3.3) is standard when u ∈ W 2,p (Ω). Its proof in the present situation is based on the following Lemma 3.3. (Cf.[11].) Let a ∈ C(Ω)N be a vector field such that div a ∈ L1 (Ω) in the distribution sense. Then Ω div a = ∂Ω aν. Proof of Proposition 3.2. Take a := |∇u|p−2 ∇uϕ. By the well-known formula for the derivation of the product of a distribution by a C ∞ function, one has div a = (p u)ϕ + |∇u|p−2 ∇u∇ϕ. Lemma 3.3 thus applies and yields (3.3).

2

3.3. Strong comparison principle The following strong comparison principle, which is obtained by combining arguments from [4] and [17], will be of importance in our study of multiplicity. We recall that the notation f ≺ g was introduced before the statement of assumption (M) in Section 2; moreover, for ∂v C01 (Ω) functions u and v, we will write u v to mean u(x) < v(x) in Ω and ∂u ∂ν (x) > ∂ν (x) on ∂Ω. Proposition 3.4. Let f, g ∈ L∞ (Ω), and let u, v be solutions of −p u = μ|u|p−2 u + f

in Ω,

u = 0 on ∂Ω,

(3.4)

−p v = μ|v|p−2 v + g

in Ω,

v = 0 on ∂Ω

(3.5)

where μ 0. If 0 f ≺ g, then u v. Several works have been devoted in the last years to the strong comparison principle for the pLaplacian. In particular the strict inequality u < v in Proposition 3.4 was derived recently in [4]. Note that this conclusion u < v does not hold if the hypothesis 0 f ≺ g in Proposition 3.4 is weakened into 0 f g, f ≡ g (cf. [11]). The stronger conclusion u v however holds under this weakened hypothesis on f, g if 0 μ < λ1 (Ω) and ∂Ω is connected (cf. [10]). This conclusion u v also holds if μ = 0 and 0 f g with f ≡ g on any open subset of Ω (cf. [22]). Note that the assumption 0 f ≺ g was also considered recently in a slightly different setting in [21]. Proof of Proposition 3.4. The assumptions imply g 0, ≡ 0, and so, by standard arguments, v 0 (take −v − as testing function in (3.5) to get v 0 and apply the strong maximum principle of [31]). Once this is observed, the proof of Proposition 2.6 in [4] can be followed without any change to reach u(x) < v(x) in Ω. Now to derive the strict inequality of the normal derivatives on ∂Ω, it suffices to apply near any point of ∂Ω a local strong comparison result from [17], which we recall below. 2 Lemma 3.5. (Cf. [17].) Let f, g ∈ L∞ (Ω) and let u, v be solutions of (3.4), (3.5) where μ < λ1 (Ω). Assume 0 f g and call Ωδ := {x ∈ Ω: dist(x, ∂Ω) < δ}. Then for every δ > 0


731

sufficiently small and for every component Σ of Ωδ , one has either (i) u ≡ v in Σ , or (ii) u < v in Σ and ∂u/∂ν > ∂v/∂ν on ∂Ω ∩ Σ . 3.4. Brezis–Lieb lemma The following version of the Brezis–Lieb lemma (cf. [6]) for vector-valued functions will also be needed. Lemma 3.6. Let fk be a bounded sequence in Lp (Ω, Rn ), where 1 p < ∞. Assume fk → f p p p a.e. in Ω. Then fk Lp − f − fk Lp → f Lp , where g Lp denotes ( Ω |g(x)|p )1/p with |g(x)| the Euclidean norm of g(x) ∈ Rn . Proof. It is easily adapted from the proof given for instance in [24] in the scalar case. The only difference is the verification of |a + b|p − |a|p − |b|p ε|a|p + Cε |b|p for a, b ∈ RN . This latter relation follows from the observation that for a, b ∈ RN , (|a + b|p − |a|p − |b|p )/|a|p → 0 as |a| → +∞, uniformly for |b| 1. 2 3.5. C01,α estimates of weak solutions in the critical case 1,p

For the proof in the next subsection on the local minimization in C01 and W0 , we will need the following estimate. 1,p

Proposition 3.7. Let a sequence uk ∈ W0 (Ω) satisfy −p uk = hk (x, uk )

(3.6)

where the Carathéodory functions hk verify the uniform growth condition hk (x, s) C1 + C2 |s|p∗ −1 .

(3.7)

∗ 1,p Assume that uk remains bounded in W0 (Ω). Moreover assume E |uk |p → 0 as |E| → 0, ¯ for some 0 < α < 1. uniformly in k. Then uk remains bounded in C01,α (Ω) As observed in the comments after hypothesis (G) in Section 2, the fact that each uk above ¯ follows from [22]. We are proving here that uk remains bounded in C 1,α (Ω) ¯ belongs to C01,α (Ω) 0 ∗ 1,p provided, in addition to being bounded in W0 (Ω), uk is uniformly equi-integrable in Lp (Ω). The necessity of such an additional requirement is clear from the comments after hypothesis (G). Our proof below combines arguments and results from [18,25,26]. Proof of Proposition 3.7. We break the proof in three steps. (i) There exists q > p ∗ such that the sequence uk remains bounded in Lq (Ω). (ii) The sequence uk remains bounded in L∞ (Ω). ¯ (iii) There exists 0 < α < 1 such that the sequence uk remains bounded in C01,α (Ω).

732


Proof of step (i). This is essentially a consequence of the calculations on pp. 951–952 from [18]. Indeed a careful reading of [18], using our assumption of uniform equi-integrability, shows the existence of R > 0 such that for any nonnegative η ∈ Cc∞ (RN ) with support of diameter R, on has

βp∗ ∗

ηp u+ k

p/p∗

C(η)

Ω

+ βp uk + C|Ω|

(3.8)

Ω

with constants independent of k. Here β is fixed with 1 < β < p ∗ /p. This clearly implies that βp ∗ (Ω). And a similar argument applied to u− yields the conclusion of u+ k remains bounded in L k step (i). Proof of step (ii). Theorem 7.1 from [25] can be applied to (3.6) to derive that uk remains bounded in L∞ (Ω). In fact a particular case of this result from [25] suffices here, which is recalled below as Lemma 3.8. Here are some details on the application of this Lemma 3.8 to (3.6): (3.10) clearly holds with ϕ1 ≡ 0; taking q > p ∗ as given by step (i), one can verify (3.11) with α2 = p ∗ − 1 and ϕ2 a suitable constant by picking r2 sufficiently large. Proof of step (iii). Once the L∞ estimate of step (ii) is obtained, the global regularity result of [26] can be applied to (3.6) and gives that for some 0 < α < 1, uk remains bounded ¯ This concludes the proof of Proposition 3.7. 2 in C01,α (Ω). Lemma 3.8. (Cf. [25].) Let u ∈ W0 (Ω) ∩ Lq (Ω) with q p ∗ satisfy 1,p

a(x, u, ∇u)∇v =

Ω

(3.9)

b(x, u)v Ω

for v of the form (u − c)+ or (u + c)− , c any positive constant. Here the functions a(x, s, η) and b(x, s) are assumed to verify for x ∈ Ω, s ∈ R and η ∈ RN ,

a(x, s, η), η ν|η|p − 1 + |s|α1 ϕ1 (x),

(sign s)b(x, s) 1 + |s|α2 ϕ2 (x),

(3.10) (3.11)

with ν a positive constant, 0 ϕi ∈ Lri (Ω), ri > N/p, 0 α1 < p NN+q − p NN+q

−1− and Ω.

q r2 .

Then

u ∈ L∞ (Ω)

q r1

and 0 α2
0 and all w ∈ C01 (Ω) 1,α ¯

w C 1 ε0 . Then u0 ∈ C0 (Ω) for some 0 < α < 1 and u0 is a local minimizer of Φ for 1,p

0

1,p

1,p

the W0 (Ω) topology, i.e. Φ(u0 ) Φ(u0 + w) for some ε1 > 0 and all w ∈ W0 (Ω) with

w W 1,p ε1 . 0

Proposition 3.9 was proved in [8] when p = 2 and later extended to the case p = 2 with σ < p ∗ − 1 in [19,23] (see also [21,16,9] for recent related works). Its validity in the critical case σ = p ∗ − 1 is also suggested in [19], although not explicitly. For that matter, we feel necessary to give here a complete proof. Our proof below borrows some ideas from [8,19] but follows a different approach, which turns out to be simpler and to yield a slightly stronger result (cf. Remark 3.10). In fact, as in [9], we avoid the consideration of equations involving two p-Laplacians, equations to which the global C 1,α estimates from [26] do not seem to apply. Proof of Proposition 3.9. Since (3.12) with σ < p ∗ − 1 implies a similar condition with σ = 1,p p ∗ − 1, it suffices to consider the latter case. Moreover since u0 ∈ W0 (Ω) satisfies −p u0 = g(x, u0 )

in Ω,

(3.13)

¯ Assume by contradiction it follows from Corollary 1.1 in [22] that u0 belongs to some C01,α (Ω). 1,p that u0 is not a local minimizer of Φ for the W0 (Ω) topology. This means that for any ε > 0 1,p there exists vε ∈ W0 (Ω) with vε − u0 W 1,p ε and Φ(vε ) < Φ(u0 ). For later use of Lagrange 0 multiplier rule, it will be convenient to use only, as in [9], a consequence of that, namely vε − u0 Lp ε and Φ(vε ) < Φ(u0 ). We now consider as in [8] the truncated functional Φj (u) :=

1 p

|∇u|p − Ω

Gj (x, u) Ω

s for j = 1, 2, . . . , where gj (x, s) := g(x, Tj (s)), Gj (x, s) := 0 gj (x, t) dt and Tj (s) = −j if s −j , s if −j s j and +j if s j . Note that (3.12) still holds for each gj , with constants and exponent independent of j . This easily implies, by dominated convergence, that for each 1,p v ∈ W0 (Ω), Φj (v) → Φ(v) as j → ∞. It follows that for each ε > 0, there is some jε such that Φjε (vε ) < Φ(u0 ). On the other hand, since gjε has subcritical growth and since for some constants D1 , D2 independent of ε, 1 Φjε (v) p

|∇v| − p

Ω

∗ D1 + D2 |v|p

(3.14)

Ω

1,p

1,p

for v ∈ W0 (Ω), one deduces that Φjε achieves its infimum on {v ∈ W0 (Ω): v −u0 Lp∗ ε} at some uε ; this easily follows by taking a minimizing sequence and using (3.14). One thus has Φjε (uε ) Φjε (vε ) < Φ(u0 ).

(3.15)

734

D.G. de Figueiredo et al. / Journal of Functional Analysis 257 (2009) 721–752 ∗

By construction uε → u0 in Lp (Ω) as ε → 0, and it follows from (3.14), (3.15) that uε remains 1,p bounded in W0 (Ω). Claim. There exists 0 < α < 1 such that uε remains bounded in C01,α (Ω) as ε → 0. Accepting this claim, one deduces from Ascoli–Arzela theorem that uε → u0 in C01 (Ω). It follows that for ε > 0 sufficiently small, Φ(uε ) = Φjε (uε ) < Φ(u0 ), ¯ topology, and the which contradicts the fact that u0 is a local minimizer of Φ for the C01 (Ω) proof of Proposition 3.9 will be complete. It remains to prove the claim. For this purpose we write the Euler equation satisfied by uε : −p uε = gjε (x, uε ) + με |uε − u0 |p

∗ −2

(uε − u0 )

(3.16)

where με is a Lagrange multiplier associated to the constraint uε − u0 Lp∗ ε. Taking u0 − uε as testing function in (3.16) and using the minimizing property of uε , one gets με 0. We now distinguish two cases according to the behavior of με as ε → 0: either (i) με remains bounded, or (ii) for a subsequence με → −∞. In case (ii) below, for simplicity of notation, we will keep writing ε → 0 instead of considering a subsequence. Case (i). In this case the conclusion of the claim is a direct consequence of Proposition 3.7. Case (ii). In this case there exists ε0 > 0 and a constant M such that for 0 < ε < ε0 , p∗ −2

< 0 for s > M, s − u0 (x) gjε (x, s) + με s − u0 (x) > 0 for s < −M.

(3.17)

∗ Indeed, by (3.12) and the fact that u0 is bounded, |gjε (x, s)| d˜1 + d˜2 |s − u0 (x)|p −1 for some constant d˜1 , d˜2 ; one then picks ε0 such that με −d˜2 − 1 for 0 < ε < ε0 , and observe that the ∗ left-hand side of (3.17) for s > u0 ∞ (resp. s < − u0 ∞ ) and 0 < ε < ε0 is d˜1 − |s − u0 |p −1 ∗ (resp. d˜1 + |s − u0 |p −1 ); consequently (3.17) follows. Taking now (uε − M)+ and (uε − M)− as testing functions in (3.16), one concludes that |uε (x)| M for x ∈ Ω and 0 < ε < ε0 . So uε remains bounded in L∞ (Ω) as ε → 0. We now take |uε − u0 |β−1 (uε − u0 ) with β 1 as testing function in (3.13), (3.16), and use the monotonicity of −p to get

0

|∇uε |p−2 ∇uε − |∇u0 |p−2 ∇u0 ∇ |uε − u0 |β−1 (uε − u0 )

Ω

=

gjε (x, uε ) − g(x, u0 ) |uε − u0 |β−1 (uε − u0 ) + με

Ω

|uε − u0 |p

∗ +β−1

.

Ω

Since uε remains bounded in L∞ (Ω), using Hölder inequality in the integral involving g, we obtain p ∗ −1

−με uε − u0 Lp∗ +β−1 c


735

where the constant c does not depend on β and ε. Letting β → ∞ yields p ∗ −1

−με uε − u0 L∞ c. So the right-hand side of (3.16) remains bounded in L∞ (Ω), and it follows from [26] that for some 0 < α = α(N, p) < 1, uε remains bounded in C01,α (Ω). This concludes the proof of the claim in case (ii). 2 Remark 3.10. The above proof of Proposition 3.9 in the critical case σ = p ∗ − 1 shows that ∗ 1,p u0 is a local minimizer of Φ on W0 (Ω) for the Lp (Ω) topology, i.e. Φ(u0 ) Φ(u0 + w) for 1,p some ε1 > 0 and all w ∈ W0 (Ω) with w Lp∗ ε1 . And in the subcritical case σ < p ∗ − 1, 1,p one would conclude that u0 is a local minimizer of Φ on W0 (Ω) for the Lσ +1 (Ω) topology. Remark 3.11. The above proof of Proposition 3.9 greatly simplifies in the subcritical case σ < ¯ now follows by using successively p ∗ − 1 (cf. [9]): the fact that u0 belongs to some C01,α (Ω) [3] and [26], no truncation of Φ is needed, and Proposition 3.7 can be replaced by another direct application of [3] and [26]. 4. Proofs of Theorems 2.1, 2.2 and 2.3 This section is devoted to the proof of the first theorems stated in Section 2. The general strategy in this section as well as in the following one is rather similar to that in the semilinear case [15] and we will mainly concentrate on the differences with respect to [15]. It will be convenient from now on to denote (1.1) by (1.1)λ . Proof of Theorem 2.1. One starts by proving the existence of an upper solution of (1.1)λ for the value λ provided by (He ). With g and e as in (He ), there exists M > 0 such that

p−1

p−1 1/ e ∞ g M e ∞ / M e ∞ and so one has, for any ϕ ∈ Cc∞ (Ω) with ϕ 0,

∇(Me)p−2 ∇(Me)∇ϕ =

Ω

M p−1 ϕ

Ω

Ω

g(Me)ϕ Ω

g M e ∞ ϕ

fλ (x, Me)ϕ. Ω

This shows that Me is an upper solution. We now construct a lower solution of (1.1)λ by using the local “sublinearity” assumption 1,p (HΩ1 ) at λ. Denote by ϕ1 a positive principal eigenfunction of −p on W0 (Ω1 ); one has ϕ1 ∈ C 1,α (Ω 1 ) and ∂ϕ1 /∂ν < 0 on ∂Ω1 , where ν denotes here the exterior normal on ∂Ω1 . 1,p Extending ϕ1 by 0 on Ω \ Ω1 , the extended function, still denoted by ϕ1 , belongs to W0 (Ω) ∩ ∞ ∞ L (Ω). Call uε := εϕ1 for ε > 0, and let ϕ ∈ Cc (Ω), ϕ 0. One has, for ε sufficiently small so that uε ∞ s1 (where s1 comes from (HΩ1 )),

736


|∇uε |

p−2

∇uε ∇ϕ =

Ω

|∇uε |

p−2

∂Ω1

∇uε νϕ −

(p uε )ϕ

Ω1

λ1 (Ω1 )

|uε |p−2 uε ϕ

Ω1

fλ (x, uε )ϕ

Ω1

fλ (x, uε ) Ω

where we have used successively Proposition 3.2, (HΩ1 ) and fλ (x, 0) 0. This shows that uε is a (weak) lower solution. Moreover, taking ε > 0 smaller if necessary, one has uε Me in Ω. 1,p Proposition 3.1 can thus be applied and yields a solution u ∈ W0 (Ω) ∩ L∞ (Ω) of (1.1)λ for the value of λ provided by (He ). So at this stage, we have proved that

Λ := sup λ > 0: (1.1)λ has a solution > 0. It remains to show that for each 0 < λ < Λ, (1.1)λ has a solution u with Iλ (u) < 0. Let 0 < λ < Λ and take λ with λ < λ < Λ such that (1.1)λ has a solution u. One has, by the monotonicity of the family fλ (cf. (H )), |∇u|p−2 ∇u∇ϕ = fλ (x, u)ϕ fλ (x, u)ϕ Ω

Ω

Ω

for all ϕ ∈ Cc∞ (Ω) with ϕ 0. This shows that u is an upper solution for (1.1)λ . The rest of the argument is then easily adapted from p. 275 in [15]: one first constructs a (weak) lower solution as above in the form εϕ1 by using (HΩ1 ) at λ, and Proposition 3.1 applies again to yield a solution u0 to (1.1)λ ; moreover the minimization property provided by Proposition 3.1 leads to Iλ (u0 ) Iλ (εϕ1 ), and by (HΩ1 ), Iλ (εϕ1 ) < 0 for ε sufficiently small. 2 Proof of Theorem 2.2. Let λ be given by (HΩ˜ ) and suppose by contradiction that (1.1)λ admits 1,p a solution u ∈ W0 (Ω) ∩ L∞ (Ω). Consider the eigenvalue problem with weight

p−2 ˜ v −p v = μm(x)|v|

v=0

˜ in Ω, ˜ on ∂ Ω.

(4.1)

Since by (HΩ˜ ),

|∇u|p−2 ∇u∇ϕ = Ω˜

fλ (x, u)ϕ μ

Ω˜

p−1 m(x)u ˜ ϕ Ω˜

˜ is an upper solution for (4.1). On the other for ϕ ∈ Cc∞ (Ω), ϕ 0, we see that u (restricted to Ω) ˜ m) ˜ and call uε := εϕ1 . Since hand let ϕ1 be a positive eigenfunction associated to λ1 (Ω, p−1 p−1 ˜ m) |∇uε |p−2 ∇uε ∇ϕ = λ1 (Ω, ˜ m(x)u ˜ ϕ < μ m(x)u ˜ ϕ ε ε Ω˜

Ω˜

Ω˜

for ϕ ∈ Cc∞ (Ω), ϕ 0, we see that uε is a lower solution for (4.1). Clearly, for ε > 0 sufficiently ˜ either u(x0 ) > 0 or u(x0 ) = 0 and ∂u/∂ν(x0 ) < 0, where small, uε u on Ω˜ (since for x0 ∈ ∂ Ω,


737

˜ Proposition 3.1 then guarantees the existence of a solution v ν is here the exterior normal on ∂ Ω). ˜ In particular v 0, v ≡ 0, which shows that μ is a principal of (4.1) with uε v u in Ω. ˜ m). ˜ This is a contradiction since μ > λ1 (Ω, ˜ 2 eigenvalue of −p on Ω˜ for the weight m. Remark 4.1. Here is another proof of Theorem 2.2 based on Picone’s identity (cf. Suppose [1]). p > 0. By ˜ with ϕ 0 and ˜ mϕ again that (1.1)λ admits a solution u and let ϕ ∈ Cc∞ (Ω) ˜ Ω Picone’s identity,

|∇ϕ| − p

Ω

∇ ϕ p /up−1 |∇u|p−2 ∇u 0

Ω

and consequently, from the equation satisfied by u and (HΩ˜ ),

|∇ϕ|p

Ω

fλ (x, u)ϕ p /up−1 μ

p m(x)ϕ ˜ . Ω˜

Ω

˜ m) Taking the infimum with respect to ϕ yields λ1 (Ω, ˜ μ, a contradiction. Proof of Theorem 2.3. Let μk → Λ with 0 < μk < Λ, μk increasing, and let uk be a solution of (1.1)μk with Iμk (uk ) < 0. 1,p We first show that uk remains bounded in W0 (Ω). This is carried out as on [15, p. 276]: 1 1,p using Iμk (uk ) < 0 and (AR)d , and denoting by v the W0 (Ω) norm ( Ω |∇v|p ) p , one obtains θ

uk p p

fμk (x, uk )uk + d

Ω

p

uk + c1 Ω

for some constant c1 . One then deduces, using (1.1)μk ,

θ − 1 uk p c2 μk p + c1 p

for some constant c2 , which gives the required bound. Since σ < p ∗ − 1, we have, for a subsequence, uk → u in C 1 (Ω). Clearly u solves −p u = fΛ (x, u) in Ω, u 0 in Ω, u = 0 on ∂Ω, and one has IΛ (u) 0. It remains to see that u ≡ 0. Assume by contradiction u ≡ 0. We will use (HΩ1 ) for λ = μ1 . Let as before Ω1 be the corresponding subdomain and ϕ1 a positive eigenfunction associated to the principal eigenvalue 1,p λ1 (Ω1 ) of −p on W0 (Ω1 ). We have, for ϕ ∈ Cc∞ (Ω), ϕ 0, using the monotonicity of the family fλ ,

|∇uk |

p−2

Ω1

∇uk ∇ϕ Ω1

fμ1 (x, uk )ϕ θ1

p−1

uk

ϕ

Ω1

for k sufficiently large (so that 0 uk (x) s1 on Ω1 , which is possible since uk → 0 uniformly). The above relation shows that uk (restricted to Ω1 ) is an upper solution for the problem

738


−p v = θ1 |v|p−2 v v=0

in Ω1 , on ∂Ω1 .

(4.2)

Since λ1 (Ω1 ) < θ1 , one also has that for ε > 0 sufficiently small, εϕ1 is a lower solution of (4.2) which satisfies εϕ1 uk on Ω1 . Proposition 3.1 then implies the existence of a solution v of (4.2) with εϕ1 v uk . This is a contradiction since θ1 > λ1 (Ω1 ). 2 Remark 4.2. Here is another proof of Theorem 2.3 based on Picone’s identity. One starts by constructing u as above. Assume again by contradiction u ≡ 0. With Ω1 , λ1 (Ω1 ), ϕ1 and k sufficiently large as above, one has λ1 (Ω1 ) Ω1

p

ϕ1 =

|∇ϕ1 |p

Ω1

=

p p−1 |∇uk |p−2 ∇uk ∇ ϕ1 /uk

Ω1 p p−1 fμk (x, uk )ϕ1 /uk

Ω

p p−1 fμk (x, uk )ϕ1 /uk

Ω1

θ1

p

ϕ1 ,

Ω1

which is a contradiction since θ1 > λ1 (Ω1 ). Here one has used successively the definition of λ1 (Ω1 ), Picone’s identity, the equation satisfied by uk , the monotonicity of the family fλ , and (HΩ1 ). 5. Proofs of Theorems 2.4, 2.5 and 2.6 We now turn to the proof of the multiplicity theorems from Section 2. We start with the following result which concerns the first solution. Theorem 5.1. In addition to the assumptions of Theorem 2.1, assume that fλ (x, s) satisfies (H0 ), (M) and the growth condition (G) with σ p ∗ − 1. Let 0 < λ < Λ. Then there exists a solution ¯ of problem (1.1)λ which is a local minimum of Iλ in the W 1,p topology. u0 ∈ C 1,α (Ω) 0 Proof. Pick λ1 , λ2 with 0 < λ1 < λ < λ2 < Λ. We first observe that (1.1)λ1 and (1.1)λ2 have solutions u1 and u2 , respectively, which satisfy u1 u2 . This can be seen as follows. One starts with a solution u2 of (1.1)λ2 and considers (1.1)λ1 . By the monotonicity of the family fλ , u2 is an upper solution of (1.1)λ1 ; moreover, using (HΩ1 ) at λ1 as in the proof of Theorem 2.1, one constructs a lower solution of (1.1)λ1 which is smaller than u2 . Proposition 3.1 thus applies and yields a solution u1 of (1.1)λ1 with u1 u2 in Ω and Iλ1 (u1 ) < 0. We now use u1 and u2 as lower and upper solutions for (1.1)λ and apply as before Proposition 3.1 to obtain a solution u0 of (1.1)λ which minimizes Iλ on {u ∈ W 1,p (Ω): u1 u u2 } and satisfies Iλ (u0 ) < 0 (the latter inequality follows from the minimization property, using Iλ (u1 ) Iλ1 (u1 ) < 0). We claim that

u1 < u0 < u2 ∂u1 /∂ν > ∂u0 /∂ν > ∂u2 /∂ν

in Ω, on ∂Ω.

(5.1)

Let us prove the relations involving u1 and u0 (same argument for u0 and u2 ). We first use (M) and (H0 ) at λ to derive that for a suitable B 0,

D.G. de Figueiredo et al. / Journal of Functional Analysis 257 (2009) 721–752 p−1

−p u1 = fλ1 (x, u1 ) ≺ fλ (x, u1 ) = fλ (x, u1 ) + Bu1 p−1

fλ (x, u0 ) + Bu0

p−1

− Bu1

739

p−1

− Bu1 p−1

= −p u0 + Bu0

p−1

− Bu1

.

So p−1

−p u1 + Bu1

p−1

≺ −p u0 + Bu0

.

Moreover, taking B larger if necessary and applying (H0 ) at λ1 , one also has p−1

−p u1 + Bu1

p−1

= fλ1 (x, u1 ) + Bu1

fλ1 (x, 0) 0.

We are thus in a position to apply the strong comparison principle of Proposition 3.4, which yields u1 u0 , i.e. the assertion (5.1) relative to u1 and u0 . 1,p It follows from (5.1) that {u ∈ W0 (Ω): u1 u u2 } contains a C01 (Ω) neighborhood of u0 and consequently, u0 is a local minimizer of Iλ on C01 (Ω). We now apply Proposition 3.9 to get 1,p that u0 is a local minimizer of Iλ on W0 (Ω). 2 The proofs of Theorems 2.4, 2.5 and 2.6 have a common part, that we will first consider. We keep assuming here the hypothesis of Theorem 5.1. In each theorem the second solution of (1.1)λ will be constructed in the form u0 + w where u0 is the first solution provided by Theorem 5.1 and w is a nonzero solution of

−p (u0 + w) = fλ x, u0 + w + w=0

in Ω, on ∂Ω.

(5.2)

And the construction of a nonzero solution of (5.2) will be carried out using the mountain pass theorem. 1,p Observe that if w ∈ W0 (Ω) solves (5.2), then w 0. Indeed, by (G) and the regularity ∞ theory, w ∈ L (Ω); moreover, using (H0 ) , one has, for a suitable B 0, p−1

p−1

−p u0 = fλ (x, u0 ) + Bu0 − Bu0

p−1 p−1 fλ x, u0 + w + + B u0 + w + − Bu0 p−1

p−1 = −p (u0 + w) + B u0 + w + − Bu0 and consequently, since (B(u0 + w + )p−1 − Bu0

p−1

)w − ≡ 0, we have

p−2 |∇u0 |p−2 ∇u0 − ∇(u0 + w) ∇(u0 + w) ∇w − 0.

Ω

Splitting the preceding integral as an integral on {w > 0} and an integral on {w 0}, one obtains, by strict monotonicity, w − ≡ 0, i.e. w 0.

740


It follows that if w is a nonzero solution of (5.2), then u0 + w will be a second solution of (1.1)λ satisfying all the requirements in Theorem 2.4. To derive the existence of a nonzero solution of (5.2), we write the associated functional 1 Jλ (w) := p

∇(u0 + w)p −

Ω

(5.3)

Gλ (x, w) Ω

where Gλ (x, w) := Fλ (x, u0 + w + ) − Fλ (x, , u0 ) − fλ (x, u0 )w − . We are thus lead to look for a 1,p nonzero critical point of Jλ on W0 (Ω). 1,p We now prove that 0 is a local minimizer of Jλ on W0 (Ω). Indeed, using the fact that u0 is 1,p a local minimizer of Iλ on W0 (Ω), one obtains Jλ (w)

1 p

∇(u0 + w)p − 1 p

Ω

∇ u0 + w + p + 1 p

Ω

|∇u0 |p + Ω

fλ (x, u0 )w −

(5.4)

Ω

for w + sufficiently small. Recall that when p 2,

|ξ2 |p |ξ1 |p + p|ξ1 |p−2 ξ1 , ξ2 − ξ1 + c(p)|ξ2 − ξ1 |p / 2p − 1

(5.5)

for some positive constant c(p) and all ξ1 , ξ2 ∈ R N , and that a similar relation holds when p < 2, with the last term of (5.5) replaced by c(p)|ξ1 − ξ2 |2 /(|ξ2 | + |ξ1 |)2−p (cf. [27,28]). Using (5.5) and the fact that u0 solves (1.1)λ , one derives from (5.4) that when p 2, 1 Jλ (w) p

|∇u0 | + c(p) p

Ω

∇w − p / 2p − 1 Jλ (0),

Ω

i.e. the conclusion that 0 is a local minimizer of Jλ . A similar inequality can be obtained when p < 2. We will now distinguish between the subcritical situation of Theorem 2.4 and the critical situation of Theorems 2.5 and 2.6. Proof of Theorem 2.4. Assumption (G) with σ < p ∗ − 1 and (AR)d imply that Jλ satisfies the 1,p (PS) condition on W0 (Ω). Indeed, if wk is a (PS) sequence, then, for θ as in (AR)d and for some εk → 0 and some constant c, we have θ Jλ (wk ) − Jλ (wk )(u0 + wk ) c + εk u0 + wk . Next, after some computations, using (AR)d , one deduces, for another constant c ,

ρ θ − 1 u0 + wk p + (θ − 1) fλ (x, u0 )wk− c + d u0 + wk+ + εk u0 + wk . p Ω

Ω 1,p

Since ρ < p, this implies that the sequence (wk ) remains bounded in W0 (Ω). Passing to a subsequence, let w0 be the weak limit of (wk ). So it follows that Jλ (wk )(wk − w0 ) → 0. Using


741

the fact that σ < p ∗ − 1 and the (S)+ property of −p , one derives, in a rather standard way, 1,p that for a further subsequence, wk converges in W0 (Ω). 1,p From the above discussion, we know that 0 is a local minimizer of Jλ on W0 (Ω). One now faces the following alternative, as in [15, p. 278]: either Jλ admits near 0 another local minimizer (and we are finished), or for any r > 0 sufficiently small,

1,p Jλ (0) < inf Jλ (w): w ∈ W0 (Ω) and w = r

(5.6)

(by using Theorem 5.10 from [13]). We aim in the latter case at applying the mountain pass 1,p theorem. This will be possible if for some ϕ ∈ W0 (Ω), Jλ (tϕ) → −∞ as t → +∞. Assumption (HΩ2 ) will be used to construct such a ϕ. One first adapts the calculation from p. 462 in [14] to derive from (HΩ2 ) and (AR)d that for some s3 and some c > 0, Fλ (x, s) cs θ

(5.7)

for a.e. x ∈ Ω2 and all s s3 , where θ comes from (AR)d . Take ϕ ∈ Cc∞ (Ω2 ) with ϕ 0, ϕ ≡ 0. Since 1 Jλ (tϕ) = p

∇(u0 + tϕ)p −

Ω

Fλ (x, u0 + tϕ) +

Ω

Fλ (x, u0 ), Ω

one easily derives from (5.7) that Jλ (tϕ) → −∞ as t → ∞. This concludes the proof of Theorem 2.4. 2 Now we study the (PS) condition for the functional Jλ under the hypotheses of either Theorem 2.5 or 2.6. Lemma 5.2. Assume that 0 is the only critical point of Jλ . Then Jλ satisfies the (PS)c condition for all levels c with c < c0 := where S := inf{

Ω

S N/p

u0 p + (N −p)/p p N b ∞

(5.8)

∗ ∗ 1,p |∇u|p /( Ω |u|p )p/p : u ∈ W0 (Ω), u ≡ 0} is the best Sobolev constant.

Proof. Let wk be a (PS)c sequence with c < c0 , i.e. 1

u0 + wk p − p

Gλ (x, wk ) → c,

(5.9)

Ω

∇(u0 + wk )p−2 ∇(u0 + wk )∇ϕ − gλ (x, wk )ϕ εk ϕ , Ω

Ω

1,p

∀ϕ ∈ W0 (Ω)

(5.10)

742


where εk → 0. Recall that in the present situation gλ (x, s) := hλ (x, u0 + s + ) + b(x)(u0 + ∗ ∗ p∗ s + )p −1 and Gλ (x, s) := Hλ (x, u0 + s + ) + b(x)(u0 + s + )p /p ∗ − Hλ (x, u0 ) − b(x)u0 /p ∗ − s p ∗ −1 hλ (x, u0 )s − − b(x)u0 s − , with Hλ (x, s) := 0 hλ (x, t) dt. 1,p We first observe that wk remains bounded in W0 (Ω). This follows by multiplying (5.10) with ϕ = u0 + wk by 1/p ∗ and subtracting from (5.9): the terms of power p ∗ cancel, and the remaining dominating term is wk p , which easily yields the desired bound. In this argument we have used condition (G) with σ < p − 1 in the case of Theorem 2.5, and condition (AR)d as well as b(x) 0 in the case of Theorem 2.6 (the argument in this latter case is a little more involved but can be easily adapted from that on p. 283 in [15]). So, for a subsequence, wk → w0 weakly 1,p in W0 (Ω), strongly in Lr (Ω) for any r < p ∗ , and a.e. in Ω. Writing (5.10) as −p (u0 + wk ) = gλ (x, wk ) + fk

(5.11)

where fk ∈ W −1,p (Ω) with fk −1,p εk , one can apply Theorem 2.1 from [5] to go to the limit in the weak form of (5.11), and so w0 solves (5.2). By the assumption of Lemma 5.2, one then concludes w0 = 0. We now claim that

u0 + wk p /N + u0 p /p ∗ → c.

(5.12)

Indeed, multiplying again (5.10) with ϕ = u0 + wk by 1/p ∗ and subtracting from (5.9), one obtains 1 1

u0 + wk p + ∗ N p

p∗ hλ (x, u0 )u0 + b(x)u0 → c,

Ω

and (5.12) follows by using the equation for u0 . Relation (5.12) and the weak lower semi-continuity of the norm imply c u0 p /p. We distinguish two cases: either (i) c = u0 p /p or (ii) c > u0 p /p. In case (i), one deduces from (5.12) that u0 + wk → u0 , and consequently wk → 0 in 1,p W0 (Ω), which shows that (PS)c holds. We will now prove that case (ii) leads to c c0 , which contradicts assumption (5.8). For that purpose we start from (5.10) with ϕ = u0 + wk and use Eq. (1.1) for u0 to obtain lim u0 + wk = u0 + lim p

p

p∗

p∗ b(x) u0 + wk+ − u0 .

Ω +p

Applying (5.12) to the left-hand side and the mean-value theorem to (u0 + wk+ )p − wk right-hand side, one obtains

N c − u0 p /p = lim

Ω

p∗ b(x) wk+ ,

in the


743

which easily leads to p/p∗

p/p∗ −1

b ∞ /N S lim wk p . c − u0 p /p

(5.13)

On the other hand, by Lemma 3.6, u0 + wk p − wk p → u0 p . Replacing in (5.13) and using again (5.12) together with the fact that we are in case (ii), we come to the inequality c c0 , which is a contradiction. This completes the proof of Lemma 5.2. 2 Proof of Theorems 2.5 and 2.6. Proceeding exactly as before, we know that 0 is a local mini1,p mizer of Jλ on W0 (Ω), and we look for a nonzero critical point of Jλ . Assume by contradiction that 0 is the only critical point of Jλ . Then, for some ball B(0, r) in 1,p W0 (Ω), we have Jλ (0) < Jλ (w)

(5.14)

for all w ∈ B(0, r) with w = 0. Using Lemma 5.2 above and Theorem 5.10 from [13] (which only requires the (PS)c condition to hold at the level of the strict local minimum, here the level Jλ (0) = 0 < c0 ), one deduces from (5.14) that (5.6) holds for all r > 0 sufficiently small, i.e. one has mountain ranges surrounding 0. We aim again at applying the mountain pass theorem. For 1,p this purpose it remains to show the existence of u¯ ∈ W0 (Ω) such that Jλ (u) ¯ < 0 and the inf max value of Jλ over the family of all continuous paths from 0 to u¯ is < c0 . Once this is done, the mountain pass theorem yields the existence of a nonzero critical point of Jλ , and we reach a contradiction with the fact that 0 was supposed to be the only critical point of Jλ . The construction of the desired u¯ is made as follows. One considers as in [2,18] functions of the form uε (x) := ρ(x)Uε (x), where Uε (x) :=

ε (N −p)/p(p−1) (ε p/(p−1) + |x − x0 |p/(p−1) )(N −p)/p

and ρ(x) is a cutoff function near x0 (from assumption (b)). More precisely ρ is smooth, nonnegative, with ρ ≡ 1 near x0 and support in a ball B2 around x0 , where B2 is chosen such that B2 ⊂ B1 ∩ B(x0 , δ) (from assumptions (b) and (Hh )) and b(x) some η > 0 a.e. on B2 . Using Lemma 5.3 below, one easily sees that the function u¯ = tuε satisfies the desired properties if one first selects ε > 0 sufficiently small and then t sufficiently large. 2 Lemma 5.3. (i) For any ε > 0, Jλ (tuε ) → −∞ as t → +∞. (ii) One has sup Jλ (tuε ) < c0 t0

for ε > 0 sufficiently small. Proof. Part (i) easily follows from the “superlinearity” of the problem near x0 . The proof of part (ii) is separated in three cases. We use similar arguments as in [18], in particular their Lemmas A4 and A5.

744


Case 1: 2 p < 3. Let

Jλ (tuε ) := Jλ (tuε ) −

1

u0 p p

where u0 is the first solution from Theorem 5.1. We thus have

Jλ (tuε ) =

1 1 1

u0 + tuε p − u0 p − ∗ p p p −

∗

b(x)(u0 + tuε )p +

1 p∗

Ω

Hλ (x, u0 + tuε ) +

Ω

p∗

b(x)u0 Ω

Hλ (x, u0 ). Ω

Since hλ (x, ·) is nondecreasing, using Lemma A4, parts (1) and (4), with r = p ∗ , as well as the gradient estimate on p. 946 in [18], we obtain

Jλ (tuε )

tp p

|∇uε |p + t Ω

−

Ω

∗ tp

|∇u0 |p−2 ∇u0 ∇uε + Ct γ0 ε β

∗

p ∗ −1

b(x)upε − t

p∗ Ω

− Ct γ

b(x)u0

uε − t p

∗ −1

Ω p ∗ −γ

u0 Ω

b(x)uεp

∗ −1

u0

Ω

uγε − t

hλ (x, u0 )uε Ω

for some β > (N − p)/p and all γ , γ0 with 1 < γ < p ∗ − 1, p − 1 < γ0 < NN(p−1) −1 , where C denotes, here and below, various positive constants possibly depending on γ and γ0 but independent on t and ε. Moreover, as on p. 947 in [18],

|∇uε |p dx = Ω

N−p |∇U1 |p dx + O ε p−1 ,

RN

∗

|uε |p dx =

N ∗ |U1 |p dx + O ε p−1

RN

Ω

when ε → 0. It follows, using Lemma A5, part (1), with α = p ∗ − 1, and the fact that u0 solves (5.2), that

Jλ (tuε )

tp p

∗

N−p

N tp ∗ − ∗ b ∞ |∇U1 |p + O ε p−1 |U1 |p + O ε p−1 p

RN

+ Ct γ0 ε β − Ct p

Ω ∗ −1

ε

N−p p

−

p∗

t p∗

Ω

∗ b(x) − b ∞ upε .


745

Note that

∗ b(x) − b ∞ upε =

Ω

∗ b(x) − b ∞ upε +

Bεδ

∗ b(x) − b ∞ upε

Ω\Bεδ

where Bεδ is the ball centered in x0 and radius ε δ . Using assumption (b), we have that p∗

b ∞ )uε

Bεδ (b(x)−

is of order ε δγb ; we also have, for M large enough,

∗ b(x) − b ∞ upε

M εδ

Ω\Bεδ

N

ε p−1 (ε

p p−1

+r

p p−1

r N −1 dr )N

N(1−δ) = O ε p−1 . Now from the assumption on γb , one can choose δ > 0 such that N − p N (1 − δ) < , p p−1 and N −p < δγb , p and so we have

∗

N−p

b ∞ − b(x) upε o ε p .

Ω

Therefore Jλ (tuε )

tp p

∗

N−p

N−p tp p∗ p−1 p − ∗ b ∞ |U1 | + o ε |∇U1 | + O ε p p

RN

RN

+ Ct γ0 ε β − Ct p

∗ −1

ε

N−p p

.

Denoting Aε =

N−p |∇U1 |p + O ε p−1 ,

Rn

Bε = b ∞ Rn

N−p ∗ |U1 |p + o ε p ,

(5.15)

746


we thus have Jλ (tuε ) f (t), where f is defined by ∗

f (t) :=

N−p tp tp ∗ Aε − ∗ Bε + Ct γ0 ε β − Ct p −1 ε p . p p

It is clear that this function f is bounded from above and reach its maximum at some tε > 0. Note that f (tε ) = 0 if and only if Cγ0 tεγ0 −p ε β + Aε − tεp Since p ∗ > p, −1 γ0 − p
0. And so

tε p f (tε ) = p

p∗

tε

∇U1 − ∗ b ∞ p

N−p

N−p ∗ |U1 |p + o ε p + Cε β − Cε p .

p

RN

(5.17)

RN

But

tp sup t0 p

∗

tp |∇U1 | − ∗ b ∞ p

p

p∗

=

|U1 |

Rn

RN

1 N

1 N−p p

N

Sp.

b ∞

Thus, taking ε > 0 small enough, we have

I (tuε )
∗ implies p > N2N +2 , i.e. p > 2), we obtain

2N N +1


Jλ (tuε )

tp p

−

|∇uε |p + t Ω ∗ tp

p∗

|∇u0 |p−2 ∇u0 ∇uε + t γ0 Ω

∗ b(x)upε

− Ct γ

|∇uε |γ0 Ω

p ∗ −1 b(x)u0 uε

−t

Ω

−t

p ∗ −1

Ω p ∗ −γ γ uε

u0

747

b(x)uεp

∗ −1

Ω

−t

Ω

(5.18)

hλ (x, u0 )uε Ω

for all 1 < γ and β := N −

2N N +1 ,

one can select γ0 such that

N (p−1) N −1

< γ0

N N −p γ0 > . p p

Then, by Lemma A5, part (2), we have |∇uε |γ0 Cε β . Ω

We can now proceed as in Case 1 above to conclude the proof of part (ii) of Lemma 5.3 in Case 2. Case 3: p 3. Since p 3, Lemma A4, part (2), gives 1 1 |∇u0 + t∇uε |p − |∇u0 |p p p

tp |∇uε |p + t|∇u0 |p−2 ∇u0 , ∇uε + Ct 2 |∇uε |2 + Ct p−1 |∇uε |p−1 . p

Thus, using (Hh ) and proceeding as in Case 1 above, using Lemma A4, part (4), we obtain Jλ (tuε )

tp p

|∇uε | + t Ω

|∇u0 |

p

Ω

+ Ct p−1

|∇uε |p−1 −

p∗

t p∗

Ω

− tp

∗ −1

∇u0 ∇uε + Ct

2

p−2

Ω

∗

b(x)uεp

∗ −1

− Ct γ

b(x)u0 Ω

Ω

p ∗ −1

b(x)upε − t Ω

|∇uε |2

p ∗ −γ

u0

uγε − t

Ω

hλ (x, u0 )uε − c0 t q+1

Ω

for all 1 < γ < p ∗ − 1. And so tp Jλ (tuε ) |∇uε |p + Ct 2 |∇uε |2 + Ct p−1 |∇uε |p−1 p −

Ω t∗

Ω ∗

b(x)upε − c0 t q+1

p∗ Ω

Ω

uq+1 ε Ω

uε uq+1 ε Ω

748


=

tp p

∗

|∇uε |p − Ω

tp

b ∞ p∗

|∇uε |p−1 +

∗ tp

p∗

Ω

Since p 3, we have 2

N (p−1) N −p ).

Hence

∗

Jλ (tuε )

2(N−p)

tq tp N − (N−p) p (q+1) =: f (t). Aε − ∗ Bε + C t 2 + t p−1 ε p(p−1) − Ct q+1 ε p p

Using again the hypothesis q + 1 > p ∗ − N−

2 p−1 ,

we have

(N − p) 2(N − p) (q + 1) < . p p(p − 1)

And by the same argument as in Case 1 above, we can finish the proof of Lemma 5.3.

2

6. Applications In this section we will first show how the previous theorems apply to problem (1.2). The verification of the corresponding hypotheses are either easy or can be easily adapted from the arguments in [15]; in Theorem 6.3, the verification of (Hh ) uses Lemma A4, part (4), from [18]. The functional Iλ (u) here reads

q+1

r+1 1 λ 1 |∇u|p − a(x) u+ − b(x) u+ . Iλ (u) := p q +1 r +1 Ω

Ω

As an application of Theorems 2.1, 2.2 and 2.3, we have

Ω


749

Theorem 6.1. Let 0 q 0 on some ball B1 . Then there exists 0 < Λ ∞ such that problem (1.2) has at least one solution u (with Iλ (u) < 0) for 0 < λ < Λ and no solution for λ > Λ. If in addition (iii) b(x) 0 on some ball B2 , with a(x)b(x) ≡ 0 on B2 , then Λ < ∞. Moreover, if r < p ∗ − 1, then problem (1.2) has at least one solution u (with Iλ (u) 0) for λ = Λ. As an application of Theorem 2.4, we have Theorem 6.2. Let 0 q 0 on any compact K ⊂ Ω, (v) b(x) ε > 0 on some ball B2 . Then problem (1.2) has at least two solutions u, v for 0 < λ < Λ, with u ≡ v and Iλ (u) < 0. Note that (iv), (v) imply assumptions (i), (ii) and (iii) of Theorem 6.1. As an application of Theorem 2.5, we finally have Theorem 6.3. Let 0 q < p − 1 and r = p ∗ − 1. Assume that a, b ∈ L∞ (Ω) satisfy respectively condition (iv) of Theorem 6.2 and condition (b) of Theorem 2.5. Assume further that either 2 2N/(N + 1) < p < 3, or p 3 and p ∗ − p−1 < q + 1. Then the conclusion of Theorem 6.2 holds. Note that condition (b) implies assumption (v) of Theorem 6.2. Remark 6.4. Here are some questions which remain unsolved in the context of problem (1.2). (i) In Theorem 6.1, the question of existence of at least one solution for λ = Λ when r = p ∗ − 1. (ii) In Theorems 6.2 and 6.3, the question whether the two solutions u and v satisfy u < v in Ω and ∂u/∂ν > ∂v/∂ν on ∂Ω. (iii) In Theorem 6.3, the question of the necessity of the restrictions on the exponents p, q. Note that these questions are also unsolved in the constant coefficients case. The second application concerns the problem ⎧ r ⎨ −p u = λc(x)(u + 1) u>0 ⎩ u=0


(6.1)

750


where p − 1 < r. This problem was studied in [7,20] when c(x) ≡ 1 and p = 2 and more recently in [15] when p = 2 and c(x) is variable. The functional Iλ (u) here reads

r+1 1 λ |∇u|p − c(x) u+ + 1 . Iλ (u) := p r +1 Ω

Ω

As a consequence of Theorems 2.1, 2.2 and 2.3, we have Theorem 6.5. Assume that c ∈ L∞ (Ω) satisfies (i) c(x) 0 a.e. in Ω, (ii) c(x) ε > 0 on some ball B. Then there exists 0 < Λ < +∞ such that problem (6.1) has at least one solution u (with Iλ (u) < 0) for 0 < λ < Λ and no solution for λ > Λ. Moreover, if r < p ∗ − 1, then problem (6.1) has at least one solution u (with Iλ (u) 0) for λ = Λ. As an application of Theorem 2.4, we have Theorem 6.6. Assume r < p ∗ − 1, and that c ∈ L∞ (Ω) satisfies (iii) c 0. Then problem (6.1) has at least two solutions u, v for 0 < λ < Λ, with u ≡ v and Iλ (u) < 0. Finally, as an application of Theorem 2.6, we have Theorem 6.7. Assume r = p ∗ − 1 and that c(x) satisfies (iii) above as well as condition (b) of Theorem 2.6. Assume further that 2N/(N + 1) < p < 3. Then problem (6.1) has at least two solutions u, v for 0 < λ < Λ, with u ≡ v and Iλ (u) < 0. The critical case r = p ∗ − 1 in Theorem 6.7 requires more care because the right-hand side of (6.1) is not written in the form (2.2). However, u solves (6.1) if and only if v = λg u solves ⎧ p ∗ −1 ⎨ −p v = c(x)(v + μ) ⎩v > 0 v=0


(6.2)

for μ = λg with g = p∗1−p . This implies in particular that (6.2) has at least one solution for μ < Λg and no solutions for μ > Λg . In order to apply Theorem 2.6 to (6.2), we write the nonlinearity in the following way c(x)(v + μ)r = hμ (x, u) + c(x)ur ∗

∗

where hμ (x, s) = c(x)((s + μ)p −1 − s p −1 ). And now is not difficult to verify the hypotheses of Theorem 2.6. Note that the critical case with p 3 in (6.2) remains unsolved; the difficulty lies in the verification of (Hh ).


751

Acknowledgments Most of this work was done with the support of CNPq, FNRS, PRONEX, FAPESP and FONDECYT 1080430 at Unicamp, ULB and USACH. We wish to thank H. Brezis for several comments relative to Proposition 3.7 and the referee for some remarks on condition (Hh ) in Theorems 2.5 and 2.6. References [1] W. Allegretto, Yin Xi Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal. 32 (7) (1998) 819–830. [2] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (2) (1994) 519–543. [3] A. Anane, Etudes des valeurs propres et de la résonance pour l’opérateur p-Laplacien, Thèse de doctorat, Université Libre de Bruxelles, 1988. [4] D. Arcoya, D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations 31 (4–6) (2006) 849–865. [5] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (6) (1992) 581–597. [6] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (3) (1983) 486–490. [7] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437–477. [8] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (5) (1993) 465–472. [9] F. Brock, L. Iturriaga, P. Ubilla, A multiplicity result for the p-Laplacian involving a parameter, Ann. Henri Poincaré 9 (7) (2008) 1371–1386. [10] M. Cuesta, P. Takáˇc, A strong comparison principle for the Dirichlet p-Laplacian, in: Reaction Diffusion Systems, Trieste, 1995, in: Lect. Notes Pure Appl. Math., vol. 194, Dekker, New York, 1998, pp. 79–87. [11] M. Cuesta, P. Takáˇc, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (4–6) (2000) 721–746. [12] C. De Coster, M. Henrard, Existence and localization of solution for second order elliptic BVP in presence of lower and upper solutions without any order, J. Differential Equations 145 (2) (1998) 420–452. [13] D.G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Inst. Fund. Res. Lect. Math. Phys., vol. 81, Springer-Verlag, 1989. [14] D.G. de Figueiredo, J.-P. Gossez, P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2) (2003) 452–467. [15] D.G. de Figueiredo, J.-P. Gossez, P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. 8 (2) (2006) 269–286. [16] X. Fan, On the sub–supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (1) (2007) 665– 682. [17] J. Fleckinger-Pellé, P. Takáˇc, Uniqueness of positive solutions for nonlinear cooperative systems with the pLaplacian, Indiana Univ. Math. J. 43 (4) (1994) 1227–1253. [18] J. García, I. Peral, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 (3) (1994) 941–957. [19] J. García, I. Peral, J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (3) (2000) 385–404. [20] F. Gazzola, A. Malchiodi, Some remarks on the equation −u = (u + 1)p for varying domains, Comm. Partial Differential Equations 27 (2002) 809–845. [21] J. Giacomoni, I. Schindler, P. Takáˇc, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (1) (2007) 117–158. [22] M. Guedda, L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (8) (1989) 879–902. [23] Zongming Guo, Zhitao Zhang, W 1,p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 286 (1) (2003) 32–50.

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Fast rotating Bose–Einstein condensates in an asymmetric trap Amandine Aftalion a,∗ , Xavier Blanc b , Nicolas Lerner c a CMAP, Ecole Polytechnique, CNRS, 91128 Palaiseau cedex, France b Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret,

Paris F-75013, France c Projet analyse fonctionnelle, Institut de Mathématiques de Jussieu, Université Pierre-et-Marie-Curie (Paris 6),

175 rue du Chevaleret, 75013 Paris, France Received 5 December 2008; accepted 5 January 2009 Available online 23 January 2009 Communicated by Paul Malliavin

Abstract We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose– Einstein condensate. This is done in the framework of the 2D Gross–Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena. © 2009 Elsevier Inc. All rights reserved.


E-mail addresses: [email protected] (A. Aftalion), [email protected] (X. Blanc), [email protected] (N. Lerner). URLs: http://www.cmap.polytechnique.fr/~aftalion/ (A. Aftalion), http://www.ann.jussieu.fr/~blanc/ (X. Blanc), http://www.math.jussieu.fr/~lerner/ (N. Lerner). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.002

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A. Aftalion et al. / Journal of Functional Analysis 257 (2009) 753–806

Keywords: Bose–Einstein condensates; Bargmann spaces; Metaplectic transformation; Theta functions; Abrikosov lattice

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The physics problem and its mathematical formulation . . . . . 1.2. The isotropic lowest Landau level . . . . . . . . . . . . . . . . . . . 1.3. Sketch of some preliminary reductions in the anisotropic case 1.4. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. On positive definite quadratic forms on symplectic spaces . . 2.2. Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Effective diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 3. Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The Irving E. Segal formula . . . . . . . . . . . . . . . . . . . . . . . 3.2. The metaplectic group and the generating functions . . . . . . . 3.3. Explicit expression for M . . . . . . . . . . . . . . . . . . . . . . . . 4. The Fock–Bargmann space and the anisotropic LLL . . . . . . . . . . . 4.1. Nonnegative quantization and entire functions . . . . . . . . . . 4.2. The anisotropic LLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The energy in the anisotropic LLL . . . . . . . . . . . . . . . . . . . 4.4. The (final) reduction to a simpler lowest Landau level . . . . . 5. Weak anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Approximation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Energy bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Strong anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Upper bound for the energy . . . . . . . . . . . . . . . . . . . . . . . 6.2. Lower bound for the energy . . . . . . . . . . . . . . . . . . . . . . . 6.3. Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Notations for the calculations of Section 2.3 . . . . . . . . . . . . A.3. Some calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Bose–Einstein condensates (BEC) are a new phase of matter where various aspects of macroscopic quantum physics can be studied. Many experimental and theoretical works have emerged in the past ten years. We refer to the monographs by C.J. Pethick and H. Smith [14], L. Pitaevskii and S. Stringari [15] for more details on the physics and to A. Aftalion [2] for the mathematical aspects. Our work is motivated by experiments in the group of J. Dalibard [11] on rotating condensates: when a condensate is rotated at a sufficiently large velocity, a superfluid behaviour is detected with the observation of quantized vortices. These vortices arrange themselves on a lat-


755

tice, similar to Abrikosov lattices in superconductors [1]. This fast rotation regime is of interest for its analogy with Quantum Hall physics [5,8,18]. In a previous work, A. Aftalion, X. Blanc and F. Nier [3] have addressed the mathematical aspects of fast rotating condensates in harmonic isotropic traps and gave a mathematical description of the observed vortex lattice. This was done through the minimization of the Gross– Pitaevskii energy and the introduction of Bargmann spaces to describe the lowest Landau level sets of states. Nevertheless, the experimental device leading to the realization of a rotating condensate requires an anisotropy of the trap holding the atoms, which was not taken into account in [3]. Several physics papers have addressed the behaviour of anisotropic condensates under rotation and its similarity or differences with isotropic traps. We refer the reader to the paper by A. Fetter [7], and to the related works [13,16,17]. The aim of the present article is to analyze the effect of anisotropy on the energy minimization and the vortex pattern, and in particular to derive a mathematical study of some of Fetter’s computations and conjectures. Two different situations emerge according to the values of the parameters: in one case, the behaviour is similar to the isotropic case with a triangular vortex lattice; in the other case, for very large velocities, we have found a new regime where there are no vortices, and a full mathematical analysis can be performed, reducing the minimization to a 1D problem. The existence of this new regime was apparently not predicted in the physics literature. This feature relies on the analysis of the bottom of the spectrum of a specific operator whose positive lower bound prevents the condensate from shrinking in one direction, contradicting some heuristic explanations present in [7]. Our analysis is based on the symplectic reduction of the quadratic form defining the Hamiltonian (inspired by the computations of Fetter [7]), the characterization of a lowest Landau level adapted to the anisotropy and finally the study of the reduced energy in this space. 1.1. The physics problem and its mathematical formulation Our problem comes from the study of the 3D Gross–Pitaevskii energy functional for a fast rotating Bose–Einstein condensate with N particles of mass m given by EGP (φ) = Hφ, φL2 (R3 ) +

g3d N φ4L4 (R3 ) , 2

(1.1)

where the operator H is H=

m 2 2 1 2 2 h Dx + h2 Dy2 + h2 Dz2 + ωx x + ωy2 y 2 + ωz2 z2 − Ω(xhDy − yhDx ), 2m 2

(1.2)

where h is the Planck constant, Dx = (2iπ)−1 ∂x , ωj is the frequency along the j -axis, Ω is the rotational velocity, and the coupling constant g3d is a positive parameter. In the particular case where ωx = ωy , the fast rotation regime corresponds to the case where Ω tends to ωx and the condensate expands in the transverse direction. It has been proved [4] that the minimizer can be described at leading order by a 2D function ψ(x, y), multiplied by the ground state of the harmonic oscillator in the z-direction (the operator h2 /(2m)Dz2 + mωz2 z2 /2), −1 2 which is equal to (2mωz h−1 )1/4 e−πmωz h z . This property is still true in the anisotropic case if ωy ωz . The reduced 2D energy to study is thus E(ψ) = H0 ψ, ψL2 (R2 ) +

g2d N ψ4L4 (R2 ) , 2

(1.3)

756


where the operator H0 is H0 =

m 2 2 1 2 2 h Dx + h2 Dy2 + ωx x + ωy2 y 2 − Ω(xhDy − yhDx ), 2m 2

(1.4)

and the coupling constant g2d takes into account the integral of the ground state in the z-direction: g2d N =

gh2 , m

where g is dimensionless (and > 0).

(1.5)

Since h has the dimension energy × time, it is consistent to assume that the wave function ψ has the dimension 1/length, with the normalization ψL2 (R2 ) = 1. We define the mean square oscillator frequency ω⊥ by 2 ω⊥ =

1 2 ωx + ωy2 2

and the function u by 1/2 1/2 1/2 ψ(x, y) = h−1/2 m1/2 ω⊥ u h−1/2 m1/2 ω⊥ x, h−1/2 m1/2 ω⊥ y ,

(1.6)

so that uL2 (R2 ) = ψL2 (R2 ) = 1,

g2d N ψ4L4 (R2 ) = ghω⊥ u4L4 (R2 ) .

We also note that the dimension of h−1/2 m1/2 ω⊥ is 1/length, so that 1/2

x1 = h−1/2 m1/2 ω⊥ x, 1/2

x2 = h−1/2 m1/2 ω⊥ y, 1/2

u(x1 , x2 )

are dimensionless.

Assuming ωx2 ωy2 , we use the dimensionless parameter ν to write 2 ωx2 = 1 − ν 2 ω⊥ ,

2 ωy2 = 1 + ν 2 ω⊥ ,

and we get immediately 1 1 1 1 E(ψ) = D1 u2L2 (R2 ) + D2 u2L2 (R2 ) + 1 − ν 2 x1 u2L2 (R2 ) hω⊥ 2 2 2 +

1 Ω g 1 + ν 2 x2 u2L2 (R2 ) − (x1 D2 − x2 D1 )u, u L2 (R2 ) + u4L4 (R2 ) . 2 ω⊥ 2


757

Finally, we have 1 g E(ψ) := EGP (u) = H u, u + u4L4 (R2 ) , hω⊥ 2 2H = D12 + D22 + 1 − ν 2 x12 + 1 + ν 2 x22 − 2ω(x1 D2 − x2 D1 ),

(1.7) ω=

Ω , ω⊥

(1.8)

where ω, ν, u, g are all dimensionless and uL2 (R2 ) = 1. The minimization of this functional is the mathematical problem that we address in this paper. The Euler–Lagrange equation for the minimization of EGP (u), under the constraint uL2 (R2 ) = 1, is H u + g|u|2 u = λu,

(1.9)

where λ is the Lagrange multiplier. We shall always assume that Ω 2 ωx2 , i.e. ω2 + ν 2 1 and define the dimensionless parameter ε by ω2 + ν 2 + ε 2 = 1.

(1.10)

The fast rotation regime occurs when the ratio Ω 2 /ωx2 tends to 1− , i.e. ε tends to 0. Summarizing and reformulating our reduction, we have 1 w g u, u L2 (R2 ) + |u|4 dx, (1.11) EGP (u) = qω,ν,ε 2 2 R2

where qω,ν,ε is the quadratic form qω,ν,ε (x1 , x2 , ξ1 , ξ2 ) = ξ12 + ξ22 + 1 − ν 2 x12 + 1 + ν 2 x22 − 2ω(x1 ξ2 − x2 ξ1 ), (1.12) w which depends on the real parameters ω, ν, ε such that1 (1.10) holds. Here qω,ν,ε is the operator with Weyl symbol qω,ν,ε , that is:

w = D12 + D22 + 1 − ν 2 x12 + 1 + ν 2 x22 − 2ω(x1 D2 − x2 D1 ), qω,ν,ε

(1.13)

where Dj = ∂j /(2iπ). We would like to minimize the energy EGP (u) under the constraint uL2 = 1 and understand what is happening when ε → 0. 1.2. The isotropic lowest Landau level When the harmonic trap is isotropic, i.e. when ν = 0, it turns out that, since ω2 + ε 2 = 1, q = qω,0,ε = (ξ1 + ωx2 )2 + (ξ2 − ωx1 )2 + ε 2 x12 + x22

(1.14)

1 Of course there is no loss of generality assuming that , ν are nonnegative parameters; we may also assume that ω 0, since the change of function u(x1 , x2 ) → u(−x1 , x2 ) preserves the L4 -norm, is unitary in L2 , corresponding to the symplectic transformation (x1 , x2 , ξ1 , ξ2 ) → (−x1 , x2 , −ξ1 , ξ2 ) and leads to the same problem where ω is replaced by −ω.

758


so that 2 1 ω ε2 g u2 + |x|u2 + EGP (u) = (D1 + ωx2 )ψ + i(D2 − ωx1 )u + 2 2π 2 2

|u|4 dx.

We note that, with z = x1 + ix2 , D1 + ωx2 + i(D2 − ωx1 ) =

1 ¯ 1 ¯ ∂ − iωz = (∂ + πωz), iπ iπ

hence the first term of the energy is minimized (and equal to 0) if u ∈ LLLω−1 , where 2 LLLω−1 = u ∈ L2 R2 , u(x) = f (z)e−πω|z| = ker(∂¯ + πωz) ∩ L2 R2 ,

(1.15)

with f holomorphic. We expect the condensate to have a large expansion, hence the term |u|4 to be small. Thus, it is natural to minimize the energy EGP in LLLω−1 . It has been proved in [4] that the restriction to LLLω−1 is a good approximation of the original problem, i.e. the minimization of EGP in L2 (R2 ). We get for u ∈ LLLω−1 , uL2 = 1, 2 g 2 ε2 1 ω + |x|u + EGP (u) = (D1 + ωx2 )u + i(D2 − ωx1 )u + 2 2π 2 2

|u|4 dx,

¯ (iπ)−1 (∂+πωz)u=0

and with u(x) = υ((ωε)1/2 x)(ωε)1/2 (unitary change in L2 (R2 )), EGP (u) =

ε ω + 2π 2ω

2 |y|2 υ(y) dy + ω2 g

υ(y)4 dy .

The minimization problem of EGP (u) in the space LLLω−1 is thus reduced to study 2 ELLL (υ) = |x|υ L2 + ω2 gυ4L4 , −1

υ ∈ LLLε ,

(1.16)

i.e. with z = x1 + ix2 , v(x1 , x2 ) = f (z)e−πε |z| , f entire (and v ∈ L2 (R2 )). This program has been carried out in the paper [3] by A. Aftalion, X. Blanc, F. Nier. In the isotropic case, a key point is the fact that the symplectic diagonalisation of the quadratic Hamiltonian is rather simple: in fact revisiting the formula (1.14), we obtain easily

η12

2

μ21 y12

1−ω 1−ω 2 q= (ξ1 − x2 ) + (ξ2 + x1 )2 2 2 1+ω 1+ω (ξ1 + x2 )2 + (ξ2 − x1 )2 , + 2 2 η22

μ22 y22

(1.17)


759

with

η1 = 2−1/2 (1 − ω)1/2 (ξ1 − x2 ), η2 = 2−1/2 (1 + ω)1/2 (ξ1 + x2 ),

μ1 = 1 − ω, y1 = 2−1/2 (1 − ω)−1/2 (ξ2 + x1 ), μ2 = 1 + ω, y2 = 2−1/2 (1 + ω)−1/2 (x1 − ξ2 ),

(1.18)

so that the linear forms (y1 , y2 , η1 , η2 ) are symplectic coordinates in R4 , i.e. {η1 , y1 } = {η2 , y2 } = 1,

{η1 , η2 } = {η1 , y2 } = {η2 , y1 } = {y1 , y2 } = 0.

In [3], an upper bound for the energy is constructed with a test function which is also an “almost” solution to the Euler–Lagrange equation corresponding to the minimization of (1.16) in LLLε . This almost solution displays a triangular vortex lattice in a central region of the condensate and is constructed using a Jacobi Theta function, which is modulated by an inverted parabola profile and projected onto LLLε . 1.3. Sketch of some preliminary reductions in the anisotropic case The analysis of the reduced energy in the anisotropic case yields two different situations: one is similar to the isotropic case and the other one is quite different, without vortices. To tackle the non-isotropic case where ν > 0 in (1.13), one would like to determine a space playing the role of the LLL and taking into account the anisotropy. Step 1. Symplectic reduction of the quadratic form qω,ν, . Given the quadratic form qω,ν,ε (1.12), identified with a 4 × 4 symmetric matrix, we define its fundamental matrix by the identity F = −σ −1 qω,ν,ε = σ qω,ν,ε where σ=

0 −I2

I2 0

is the symplectic matrix given in 2 × 2 blocks.

The properties of the eigenvalues and eigenvectors of F allow to find a symplectic reduction for qω,ν,ε . Step 2. Determination of the anisotropic LLL. The anisotropic equivalent of the LLL can be determined explicitely, thanks to the results of the first step. We find that it is the subspace of functions u of L2 (R2 ) such that ν2 ν2 πν 2 γ γπ 2 x1 1 − + (β2 x2 )2 1 + exp −i x1 x2 , f (x1 + iβ2 x2 ) exp − 4β2 2α 2α 4α where f is entire. The positive parameters α, γ , β2 are defined in the text and are explicitely known in terms of ω, ν. We also determine an operator M, which can be used to give an explicit expression for the isomorphism between L2 (R) and the anisotropic LLL as well as to express the Gross–Pitaevskii energy in the new symplectic coordinates.

760


Step 3. Rescaling. Introducing a new set of parameters (ω, ν, are positive satisfying (1.10), g > 0 given by (1.5)), κ12

2 2 1+ = 2ν + κ=

κ1 , β2

g0 =

2ν 2 , α − ν 2 + ω2

g1 γ 2 , 4β2

α=

ν 4 + 4ω2 ,

g1 = g

2α 2ωμ2 , , β2 = ω α + 2ω2 + ν 2

γ=

α + 2ω2 + ν 2 , 2α

μ2 = 1 + ω2 + α,

(1.19) (1.20)

we show that, after some rescaling, the minimization of the full energy EGP (u) of (1.11) can be reduced to the minimization of 1 2 2 g0 ε x1 + κ 2 x22 |u|2 + |u|4 (1.21) E(u) = 2 2 R2

on the space 2 Λ0 = u ∈ L2 (R2 ), u(x1 , x2 ) = f (z)e−π|z| /2 , f holomorphic, z = x1 + ix2 .

(1.22)

The point is that, after some scaling, we are able to come back to an isotropic space. The orthogonal projection Π0 of L2 (R2 ) onto Λ0 is explicit and simple: (Π0 u)(x) =

π

e− 2 |x−y|

2 +iπ(x y −y x ) 2 1 2 1

u(y) dy.

(1.23)

R2

We are thus reduced to the following problem: with E(u) given by (1.21), study I (ε, κ) = inf E(u), u ∈ Λ0 , uL2 (R2 ) = 1 .

(1.24)

The minimization of E without the holomorphy constraint yields x12 x22 2 4g0 κ 1/4 4g0 ε 1/4 |u| = 1− 2 − 2 , where R1 = , R2 = . (1.25) πR1 R2 πε 3 πκ 3 R1 R2 2

As ε tends to 0, R1 always tends to infinity (in fact R1 ε −1/2 ), but the behaviour of R2 depends on the respective values of ε and κ, that is of ε and ν. Step 4. Sorting out the various regimes. Recalling that the positive parameter ν stands for the anisotropy, we find two regimes: • ν ε 1/3 (weak anisotropy): R2 → ∞ (in fact, R2 ≈ min(ε −2/3 , ε 1/3 ν −1 )). Numerical simulations (Fig. 1) show a triangular vortex lattice. The behaviour is similar to the isotropic case except that the inverted parabola profile (1.25) takes into account the anisotropy. We will construct an approximate minimizer. 4/3


761

Fig. 1. Plot of the zeroes of the minimizer (left) and the density (right) for ε 2 = 0.002, ν = 0.03. Triangular vortex lattice in an anisotropic trap.

Fig. 2. Plot of the zeroes of the minimizer (left) and the density (right) for ε 2 = 0.002, ν = 0.73. No vortex in the visible region.

• ν ε 1/3 (strong anisotropy): R2 → 0 (in fact R2 ≈ ε 1/3 ν −1 ). Numerical simulations (Fig. 2) show that there are no vortices in the bulk, the behaviour is an inverted parabola in the x1 direction and a fixed Gaussian in the x2 direction. Thus, the size of the condensate does not shrink in the x2 direction and (1.25) is not a good approximation of the minimizer. The shrinking of the condensate in the x2 direction is not allowed in Λ0 (see (1.22)) because the operator x22 is bounded from below in that space by a positive constant and the first eigenfunction is a Gaussian in the x2 direction. We find an asymptotic 1D problem (upper and lower bounds match) which yields a separation of variables. 4/3

762


1.4. Main results 1.4.1. Weakly anisotropic case In a first step,2 we assume that, with κ given by (1.20), ε κ ε 1/3 .

(1.26)

The isotropic case is recovered by assuming κ = ε. This case is similar to the isotropic case and we derive similar results to the paper [3], namely an upper bound given by the Theta function but we lack a good lower bound. We recall that the Jacobi Theta function Θ(z, τ ) associated to a lattice Z ⊕ Zτ is a holomorphic function which vanishes exactly once in any lattice cell and is defined by Θ(z, τ ) =

+∞ 1 2 (−1)n eiπτ (n+1/2) e(2n+1)πiz , i n=−∞

z ∈ C.

(1.27)

This function allows us to construct a periodic function on the same lattice: uτ is defined by π

uτ (x1 , x2 ) = e 2 (z

2 −|z|2 )

√ Θ( τI z, τ ),

z = x1 + ix2 , τ = τR + iτI ,

|uτ | is periodic over the lattice Z ⊕ τ Z, and uτ satisfies Π0 |uτ |2 uτ = λτ uτ ,

(1.28)

(1.29)

with

– |uτ |4 γ (τ ) =√ , λτ =

– |uτ |2 2τI

(1.30)

where – is the mean integral on a cell and

– |uτ |4

. γ (τ ) := ( – |uτ |2 )2

(1.31)

The minimization of γ (τ ) on all possible τ corresponds to the Abrikosov problem. It turns out that the properties of the Theta function allow to derive that γ (τ ) =

e

− τπ |j τ −k|2 I

(j,k)∈Z2

and prove (see [3]) that τ → γ (τ ) is minimized for τ = j = e2iπ/3 , which corresponds to the hexagonal lattice. The minimum is b = γ (j ) ≈ 1.1596.

(1.32)

2 We shall see that κ ≈ ν + ε in the sense that the ratio κ/(ν + ε) is bounded above and below by some fixed positive constants, so that the weakly anisotropic case is indeed ν ε 1/3 .


763

The function uτ allows us to construct the vortex lattice and we multiply it by the proper inverted parabola to get a good upper bound: Theorem 1.1. We have for I (ε, κ) defined in (1.24), b given in (1.32), κ in (1.20), 2 3

2 2g0 εκ < I (ε, κ) π 3

3 1/8 √ 2g0 bεκ κ , +O εκ π ε

(1.33)

when ( , κ −1/3 ) → (0, 0). Moreover, the following function provides the upper bound: v = Π0 (uτ ρ), where uτ is defined by (1.28) with τ = e ρ(x) = 2

2

√ π bR1 R2

2iπ 3

(1.34)

and

x2 x2 1− √ 1 2 − √ 2 2 bR1 bR2

,

+

R1 =

4g0 κ πε 3

1/4

, R2 =

4g0 ε πκ 3

1/4 .

We expect v to be a good approximation of the minimizer and the energy asymptotics to match the right-hand side of (1.33). Thus, the lower bound is not optimal (it does not include b). In addition, the test function (1.34) (with a general τ = j a priori) gives the upper bound of (1.33) with γ (τ ) instead of b. The proof is a refinement of that in [3]. 1.4.2. Strong anisotropy In the case where the rotation is fast enough in the sense that κ ε 1/3

(1.35)

we have found a regime unknown by physicists where vortices disappear and the problem can be reduced in fact to a 1D energy. Theorem 1.2. For I (ε, κ) defined in (1.24), b given in (1.32), κ in (1.20), we have lim

( , 1/3 κ −1 )→(0,0)

I (ε, κ) − ε 2/3

κ2 8π

= J,

(1.36)

where 1 2 g0 J = inf t p(t)2 + p(t)4 , p real-valued ∈ L2 (R) ∩ L4 (R), pL2 (R) = 1 . (1.37) 2 2 In addition, if u is a minimizer of I (ε, κ), then x1 1 −→ 21/4 e−πx22 p(x1 ), u , x 2 1/3 2/3 ε ε in L2 (R2 ) ∩ L4 (R2 ), where p is the minimizer of J.

(1.38)

764


Note that the minimizer p of (1.37) is explicit: p(t)2 =

3 t2 1− 2 , 4R R +

R=

3g0 2

1/3 .

A few words about the proof of Theorem 1.2. The first point is that the operator Π0 x22 Π0 (see (1.22), (1.23)) is bounded from below by a positive constant: ∀u ∈ Λ0 ,

x22 |u|2

1 4π

R2

|u|2 . R2

This is proven in Lemma 4.4 below. Actually, the spectrum of this operator is purely continuous, and any Weyl sequence associated with the value 1/(4π) converges (up to renormalization) to the function u0 (x1 , x2 ) = exp −πx22 + iπx1 x2 , which satisfies the equation Π0 (u0 ) =

1 4π u0 .

(1.39)

This gives the lower bound

I (ε, κ)

κ2 , 8π

and indicates that in order to be close to this lower bound, a test function should be close to (1.39). Thus, the second point is to construct a test function having the same behaviour as (1.39) in x2 , and a large extension in x1 . This is done by using the function u1 (x1 , x2 ) =

1 21/4

e

− π2 x22

π

e− 2 ((x1 −y1 )

2 −2iy x ) 1 2

ρ(y1 ) dy1 ,

R

which is equal to Π0 (ρ(x1 )δ0 (x2 )), where δ0 is the Dirac delta function and ρ any real-valued 2 function of one variable. This test function is then proved to be close to 21/4 e−πx2 ρ(x1 ), which allows to compute its energy, and gives the upper bound, provided that ρ(t) = ε1/3 p(ε 2/3 t), where p is the minimizer of (1.37). Finally, in order to prove the lower bound, we first extract bounds on the minimizer from the energy, which allow to pass to the limit in the equation (after rescaling as in (1.38)), hence prove that the limit is the right-hand side of (1.38). This uses the fact that the energy appearing in (1.37) is strictly convex, hence that any critical point is the unique minimizer. The paper is organized as follows: in Section 2, we review some standard facts on positive definite quadratic forms in a symplectic space. This allows us, in Section 3, to construct a symplectic mapping χ , which yields a simplification of the quadratic form q. In Section 4, quantizing that symplectic mapping in a metaplectic transformation, we find the expression of the LLL and manage to reach the reduced form of the energy (Proposition 4.5). Section 5 is devoted to the proof of Theorem 1.1 and Section 6 to Theorem 1.2.


765

Open questions. We have no information on the intermediate regime where, for instance, 4/3 ε 1/3 /κ converges to some constant R0 (in that case, R1 ≈ ε −2/3 , R2 ≈ R0 ). We expect that the extension in the x2 direction depends on R0 and wonder whether the condensate has a finite number of vortex lines. We have not determined the limiting problem. 2. Quadratic Hamiltonians We first review some standard facts on positive definite quadratic forms in a symplectic space. 2.1. On positive definite quadratic forms on symplectic spaces We consider the phase space Rnx × Rnξ , equipped with its canonical symplectic structure: the symplectic form σ is a bilinear alternate form on R2n given by σ (x, ξ ); (y, η) = ξ · y − η · x = σ X, Y , x y 0 X= , Y= , σ= ξ η −In

with In , 0

(2.1) (2.2)

where the form σ is identified with the 2n × 2n matrix above given in n × n blocks. The symplectic group Sp(n) (a subgroup of Sl(2n, R)), is defined by the equation on the 2n × 2n matrix χ , χ ∗ σ χ = σ,

i.e. ∀X, Y ∈ R2n ,

σ χX, χY = σ X, Y .

(2.3)

The following lemma is classical (see e.g. the chapter XXI in [9], or [12]). Lemma 2.1. Let B ∈ GL(n, R) and let A, C be n×n real symmetric matrices. Then the matrix Ξ , given by n × n blocks ΞA,B,C =

B −1 AB −1

−B −1 C B ∗ − AB −1 C

=

I A

0 I

B −1 0

0 B∗

I 0

−C I

(2.4)

belongs to Sp(n). Any element of Sp(n) can be written as a product ΞA1 ,B1 ,C1 ΞA2 ,B2 ,C2 . N.B. The first statement is easy to verify directly and we shall not use the last statement, which is nevertheless an interesting piece of information. For a symplectic mapping Ξ , to be of the form above is equivalent to the assumption that the mapping x → pr1 Ξ (x ⊕ 0) is invertible from Rn to Rn . Given a quadratic form Q on R2n , identified with a symmetric 2n × 2n matrix, we define its fundamental matrix F by the identity F = −σ −1 Q = σ Q,

so that for X, Y ∈ R2n

σ Y, F X = QY, X.

The following proposition is classical (see e.g. Theorem 21.5.3 in [9]).

766


Proposition 2.2. Let Q be a positive definite quadratic form on the symplectic Rnx × Rnξ . One can find χ ∈ Sp(n) such that with R2n X = χY,

Y = (y1 , . . . , yn , η1 , . . . , ηn ), QX, X = QχY, χY = ηj2 + μ2j yj2 , μj > 0. 1j n

The {±iμj }1j n are the 2n eigenvalues of the fundamental matrix, related to the 2n eigenvectors {ej ± iεj }1j n . The {ej , εj }1j n make a symplectic basis of R2n : σ (εj , ek ) = δj,k ,

σ (εj , εk ) = σ (ej , ek ) = 0,

and the symplectic planes Πj = Rej ⊕ Rεj are orthogonal for Q. N.B. A one-line-proof of these classical facts: on C2n equipped with the dot-product given by Q, diagonalize the sesquilinear Hermitian form iσ . 2.2. Generating functions We define on Rn × Rn the generating function S of the symplectic mapping of the form ΞA,B,C given in Lemma 2.1 by the identity S(x, η) =

1 Ax, x + 2Bx, η + Cη, η . 2

(2.5)

We have

∂S ∂S , η = x, ΞA,B,C . ∂η ∂x ∈Rn ×Rn

(2.6)

∈Rn ×Rn

In fact, we see directly

I A

0 I

B −1 0

0 B∗

I 0

−C I

Bx + Cη η

=

I 0 A I

x B ∗η

=

x . Ax + B ∗ η

Given a positive definite quadratic form Q on R2n , identified with a symmetric 2n × 2n matrix, we know from Proposition 2.2 that there exists χ ∈ Sp(n) such that χ ∗ Qχ =

μ2 0

0 In

,

μ2 = diag μ21 , . . . , μ2n .

Looking for χ = ΞA,B,C given by a generating function S as above, we end-up (using the notation q(X) = QX, X with X ∈ R2n ) with the equation q(x, ∂x S ) = μ∂η S2 + η2 , Rn ×Rn

μ∂η S = (μj ∂ηj S)1j n ∈ Rn ,


767

where · stands for the standard Euclidean norm on Rn . This means 2 q(x, Ax + B ∗ η) = μ(Bx + Cη) + η2 .

(2.7)

We want now to go back to the study of our quadratic form (1.12). 2.3. Effective diagonalization Lemma 2.3. Let q be the quadratic form on R4 given by (1.12), where ω, ν, ε are nonnegative parameters such that ω2 + ν 2 + ε 2 = 1. The eigenvalues of the fundamental matrix are ±iμ1 , ±iμ2 with (2.8) 0 μ21 = 1 + ω2 − α μ22 = 1 + ω2 + α, α = ν 4 + 4ω2 , μ21 =

2ν 2 ε 2 + ε 4 . μ22

(2.9)

In the isotropic case ν = 0, we recover μ1 = 1 − ω, μ2 = 1 + ω. When ε > 0, we have 0 < μ21 μ22 and q is positive-definite. When ε = 0, we have μ1 = 0 < μ2 , and q is positive semi-definite with rank 2 if ν = 0 and with rank 3 if ν > 0. Proof. The matrix Q of q is thus ⎞ 0 0 −ω 1 + ν2 ω 0 ⎟ ⎟ , and ω 1 0 ⎠ 0 0 1 ⎞ 0 ω 1 0 ⎜ −ω 0 0 1⎟ ⎟. F = σQ = ⎜ ⎝ ν2 − 1 0 0 ω⎠ 0 −ν 2 − 1 −ω 0 ⎛

1 − ν2 ⎜ 0 Q=⎜ ⎝ 0 −ω ⎛

(2.10)

The characteristic polynomial p of F is easily seen to be even and we calculate 2 2 p(λ) = det(F − λI4 ) = λ4 + 2 1 + ω2 λ2 + 1 − ω2 − ν 4 = λ2 + 1 + ω2 − ν 4 + 4ω2 . √ The four eigenvalues of F are thus ±i 1 + ω2 ± ν 4 + 4ω2 , proving the first statement of the lemma. Since (1 + ω2 )2 − α 2 = (1 − ω2 )2 − ν 4 = ε 2 (2ν 2 + ε 2 ), we get μ21 = ε 2 (2ν 2 + ε 2 )/μ22 . The statements on the cases ν = 0, ε > 0 are now obvious. When ε = 0 = ν, we have ω = 1, and rank q = 2 as it is obvious on (1.17). When ε = 0, ν > 0, we consider the following minor determinant in F , cofactor of f31 ω 1 0 0 0 1 = (−1) −ω2 + ν 2 + 1 = −2ν 2 = 0, −ν 2 − 1 −ω 0 so that rank Q = rank F = 3 in that case.

2

768


N.B. We may note here that the condition ω2 + ν 2 1 is an iff condition on the real parameters ν, ω for the quadratic form (1.12) to be positive semi-definite. This is obvious on the expression (1.17) in the isotropic case ν = 0, and more generally, the (non-symplectic) decomposition in independent linear forms q = (ξ1 + ωx2 )2 + (ξ2 − ωx1 )2 + x12 1 − ν 2 − ω2 + x22 1 + ν 2 − ω2 , shows that q has exactly one negative eigenvalue when ω2 + ν 2 > 1 ω2 − ν 2 , and exactly two negative eigenvalues when ω2 − ν 2 > 1. As a result, when ω2 + ν 2 > 1, the operator q w is unbounded from below. Using now Eqs. (2.7), (1.12) and assuming that we may find a linear symplectic transformation given by a generating function (2.5), we have to find A, B, C like in Lemma 2.1 with n = 2, so that for all (x, η) ∈ R2 × R2 , 2 Ax + B ∗ η2 + x2 + ν 2 x22 − x12 − 2ω x ∧ (Ax + B ∗ η) = μ(Bx + Cη) + η2 , with x ∧ ξ = x1 ξ2 − x2 ξ1 , μ = diag(μ1 , μ2 ). At this point, we see that the previous identity forces some relationships between the matrices A, B, C. However, the algebra is somewhat complicated and assuming that B is diagonal, A, C are (symmetrical) with zeroes on the diagonal lead to some simplifications and to the following results. We introduce first some parameters: β1 =

β2 =

γ=

2ωμ1 α − 2ω2 − ν 2 = 2ωμ1 α − 2ω2 + ν 2 2 2 4 since α − 2ω − ν = 4ω2 + 4ω4 − 4ω2 α = 4ω2 μ21 , + 2ω2

(2.11)

− ν2

2ωμ2 α = 2 2 2ωμ2 α + 2ω + ν 2 2 4 since α + 2ω − ν = 4ω2 + 4ω4 + 4ω2 α = 4ω2 μ22 , 2α , ω

(2.12) (2.13)

λ21 =

μ1 1 = = β μ1 + β1 β2 μ2 1 + 1 β2 μ2 1+ μ1

λ22 =

μ2 1 = = μ2 + β1 β2 μ1 1 + β1 β2 μ1 1+ μ2

1 α+2ω2 −ν 2 α−2ω2 +ν 2

1 α−2ω2 −ν 2 α+2ω2 +ν 2

=

α − 2ω2 + ν 2 , 2α

(2.14)

=

α + 2ω2 + ν 2 , 2α

(2.15)

and we have λ21 + λ22 = 1 +

ν2 , α

λ21 λ22 =

(α + ν 2 )2 − 4ω4 . 4α 2

(2.16)

We define also d=

γ λ1 λ2 , 2

c=

λ21 + λ22 2λ1 λ2

which gives cd =

2α(1 + ν 2 /α) α + ν 2 = . 4ω 2ω

(2.17)


769

Lemma 2.4. We define the 2 × 2 matrices B=

λ−1 1 0

0 λ−1 2

C=

,

d −1 0

0 d −1

A=

,

d λ1 λ2

0 − cd

d λ1 λ2

− cd 0

.

The 4 × 4 matrix given with 2 × 2 blocks by χ = ΞA,B,C =

0 I2

I2 A

B −1 0

0 B∗

−C I2

I2 0

belongs to Sp(2) and ⎛ ⎜ ⎜ χ =⎜ ⎝ ⎛

λ1

0

0

0

λ2

− λd2

d λ2

− λ2 cd

d λ1

0 − λ1 cd

− λd1 ⎞ 0 ⎟ ⎟ ⎟, 0 ⎠

cλ2

0

0

cλ1 λ2 d

⎞

cλ2

0

0

0

cλ1

λ1 d

+ λ1 cd

λ1

0⎟ ⎟ ⎟. 0⎠

0

0

λ2

⎜ ⎜ χ −1 = ⎜ ⎝

− λd2

0 − λd1 + λ2 cd

(2.18)

(2.19)

Proof. Lemma 2.1 gives that χ ∈ Sp(2) and we have also χ

−1

=

I2 0

C I2

B 0

0

B ∗ −1

0 I2 −A I2

.

The remaining part of the proof depends on the formula giving ΞA,B,C in Lemma 2.1 and a direct computation whose verification is left to the reader. 2 Lemma 2.5. Let χ be the symplectic matrix given by (2.18) and Q be the matrix given in (2.10). Then, with μj given by (2.8), we have χ ∗ Qχ = diag μ21 , μ22 , 1, 1 .

(2.20)

The (tedious) proof of that lemma is given in Appendix A.3.1. Using the expression of χ −1 in (2.18), defining ⎛ ⎞ y1 ⎜ ⎜ y2 ⎟ ⎜ ⎝ ⎠=⎜ η1 ⎝ η2 ⎛

cλ2 0 0 − λd1 + λ2 cd

we get from Lemma 2.5 the following result.

− λd2

0

0

cλ1

λ1 d

+ λ1 cd

λ1

0

0

⎞⎛

⎞ x1 ⎟ 0 ⎟ ⎜ x2 ⎟ ⎟⎜ ⎟, 0 ⎠ ⎝ ξ1 ⎠ ξ2 λ2 λ2 d

(2.21)

770


Lemma 2.6. For (x1 , x2 , ξ1 , ξ2 ) ∈ R4 , (y1 , y2 , η1 , η2 ) ∈ R4 given by (2.21), we have the following identity, 2 2 μ21 y12 + μ22 y22 + η12 + η22 = μ21 cλ1 x2 + λ2 d −1 ξ2 + μ22 cλ2 x1 + λ1 d −1 ξ1 2 2 −1 + −dλ−1 2 + λ1 cd x2 + λ1 ξ1 + −dλ1 + λ2 cd x1 + λ2 ξ2 = ξ12 + ξ22 + 1 − ν 2 x12 + 1 + ν 2 x22 − 2ω(x1 ξ2 − x2 ξ1 ), where the parameters c, λ2 , d, λ1 are defined above (note that all these parameters are welldefined when (ω, ν) are both positive with ω2 + ν 2 < 1). We have achieved an explicit diagonalization of the quadratic form (1.12) and, most importantly, that diagonalization is performed via a symplectic mapping. That feature will be of particular importance in our next section. Expressing the parameters in terms of α, ω, ν, ε (cf. Appendix A.2), we obtain 1/2 1/2 2 α − ν 2 x2 ξ1 − 2−3/2 ω−1 α −1/2 α − 2ω2 + ν 2 q = 2−1/2 α −1/2 α − 2ω2 + ν 2 α + 2ω2 − ν 2 1/2 (2ν 2 ε 2 + ε 4 )1/2 + 2−1/2 α −1/2 ξ2 μ2 2ν 2 + ε 2 2 α + 2ω2 − ν 2 1/2 (2ν 2 ε 2 + ε 4 )1/2 1/2 α + ν 2 α −1/2 2−3/2 ω−1 + x 1 μ2 2ν 2 + ε 2 −1/2 1/2 1/2 −1/2 + 1 + ω2 + α 2 α ω α + 2ω2 + ν 2 ξ1 −1/2 2 1/2 1 + α −1 ν 2 2−1/2 α 1/2 α + 2ω2 + ν 2 + 1 + ω2 + α x2 1/2 2 −1/2 −1/2 1/2 α + 2ω2 + ν 2 + 2 α ξ2 − 2−3/2 ω−1 α −1/2 α − ν 2 α + 2ω2 + ν 2 x1 , so that η12

μ21 y12

2 2 α − 2ω2 + ν 2 α + 2ω2 − ν 2 2 α − ν2 α + ν2 ε ξ2 + ξ1 − x2 + x1 q= 2α 2ω 2ω 2αμ22 2 α + ν2 1 + ω2 + α + 2ω2 + x2 ξ 1 2ω α(α + 2ω2 + ν 2 ) +

α + 2ω2 + ν 2 2α

μ22 y22

ξ2 −

2 α − ν2 x1 . 2ω

(2.22)

η22

Eq. (2.22) encapsulates most of our previous work on the diagonalization of q. In Appendix A.3.2, we provide another way of checking the symplectic relationships between the linear forms, yj , ηl .


771

We have seen in Lemma 2.3 that when ε = 0, ν > 0, the rank of q is 3, whereas its symplectic rank is 2. Indeed, ε = 0 and ν > 0, we have η12

2 α − 2ω2 + ν 2 α − ν2 ξ1 − x2 q= 2α 2ω 2 1 + ω2 + α α + ν2 x2 ξ + 2ω2 + 1 2ω α(α + 2ω2 + ν 2 ) μ22 y22

α + 2ω2 + ν 2 + 2α

2 α − ν2 ξ2 − x1 . 2ω

(2.23)

η22

3. Quantization 3.1. The Irving E. Segal formula Let a be defined on Rnx × Rnξ (say a tempered distribution on R2n ). Its Weyl quantization is the operator, acting for instance on u ∈ S (Rn ), w a u (x) =

e2iπ(x−x )ξ a

x + x , ξ u(x ) dx dξ. 2

(3.1)

In fact, the weak formula a w u, v = R2n a(x, ξ )H(u, v)(x, ξ ) dx dξ makes sense for a ∈ S (R2n ), u, v ∈ S (Rn ) since the Wigner function H(u, v) defined by H(u, v)(x, ξ ) =

e

−2iπx ξ

x x v¯ x − dx u x+ 2 2

belongs to S (R2n ) for u, v ∈ S (R n ) . Note also our definition of the Fourier transform u(ξ ˆ )=

−2iπx·ξ e u(x) dx (so that u(x) = e2iπx·ξ u(ξ ˆ ) dξ ) and ξjw u =

1 ∂u = Dj u, 2iπ ∂xj

xjw u = xj u,

1 (xj ξj )w = (xj Dj + Dj xj ). 2

Let χ be a linear symplectic transformation χ(y, η) = (x, ξ ). The Segal formula (see e.g. Theorem 18.5.9 in [9]) asserts that there exists a unitary transformation M of L2 (Rn ), uniquely determined apart from a constant factor of modulus one, which is also an automorphism of S (Rn ) and S (Rn ) such that, for all a ∈ S (R2n ), (a ◦ χ)w = M ∗ a w M,

(3.2)

772


providing the following commutative diagrams S (Rnx )

aw

S (Rnx ) M∗

M

S (Rny )

(a◦χ)w

aw

L2 (Rnx ) and if a w ∈ L L2 Rn

S (Rny )

L2 (Rnx ) M∗

M

L2 (Rny )

(a◦χ)w

L2 (Rny )

3.2. The metaplectic group and the generating functions For a given χ , how can we determine M? We shall not need here the rich algebraic structure of the two-fold covering Mp(n) (the metaplectic group in which live the transformations M) of the symplectic group Sp(n). The following lemma is classical (and also easy to prove directly using the factorization of Lemma 2.1) and provides a simple expression for M when the transformation χ has a generating function. Lemma 3.1. Let χ = ΞA,B,C be the symplectic mapping given by (2.4). Then the Segal formula (3.2) holds with (Mv)(x) = e2iπS(x,η) v(η) ˆ dη| det B|1/2 , (3.3) where S is given by (2.5). 3.3. Explicit expression for M Lemma 3.2. Let χ be the symplectic transformation of R4 given by (2.18). Then the Segal formula (3.2) holds with M given by −1

(Mv)(x1 , x2 ) = (λ1 λ2 )−1/2 e2iπd((λ1 λ2 ) −c)x1 x2 −1 −1 −1 × e2iπd η1 η2 v(η ˆ 1 , η2 )e2iπ(λ1 x1 η1 +λ2 x2 η2 ) dη1 dη2 , (Mv)(x1 , x2 ) = (λ1 λ2 )−1/2 e2iπd((λ1 λ2 )

−1 −c)x x 1 2

2iπd −1 D D −1 −1 1 2v λ e 1 x1 , λ2 x2 .

(3.4) (3.5)

Proof. We apply Lemmas 3.1 and 2.4, along with the fact that the mapping Mp(n) M → χ ∈ Sp(n) is an homomorphism or more elementarily that (3.2) implies for χj ∈ Sp(n), (a ◦ χ2 ◦ χ1 )w = M1∗ (a ◦ χ2 )w M1 = M1∗ M2∗ a w M2 M1 . The factorization of Lemma 2.4 implies that iπAx,x e2iπBx,η eiπCη,η v(η) ˆ dη, (Mv)(x) = e R2

which gives readily the formulas above.

2


773

Summing-up, we have proven the following result. Theorem 3.3. Let q be the quadratic form on R4 given by (1.12). We define the symplectic mapping χ by (2.18) and the metaplectic mapping M by (3.5). We have the μ2j are given by (2.8) ,

(q ◦ χ)(y, η) = μ21 y12 + μ22 y22 + η12 + η22

∗ w

(q ◦ χ) = M q M. w

(3.6) (3.7)

We can also explicitly quantize the formulas of Lemma 2.6, to obtain3 (η12 )w

μ21 (y12 )w

2 2 −1 w 2 −1 q = λ1 cd − dλ2 x2 + λ1 Dx1 + μ1 λ2 d Dx2 + cλ2 x1 2 2 2 −1 + λ2 cd − dλ−1 1 x1 + λ2 Dx2 + μ2 λ1 d Dx1 + cλ1 x2 . (η22 )w

(3.8)

μ22 (y22 )w

4. The Fock–Bargmann space and the anisotropic LLL 4.1. Nonnegative quantization and entire functions Definition 4.1. For X, Y ∈ R2n we set Π(X, Y ) = e− 2 |X−Y | e−iπ[X,Y ] , π

2

(4.1)

where [X, Y ] = σ X, Y is the symplectic form (2.1). For v ∈ L2 (Rn ), we define (W v)(y, η) = v, ϕy,η L2 (Rn ) ,

y

with ϕy,η (x) = 2n/4 e−π(x−y) e2iπ(x− 2 )η . 2

(4.2)

We define also − π2 |z|2 , z = η + iy, f entire . Λ0 = u ∈ L2 R2n y,η such that u = f (z)e

(4.3)

Proposition 4.2. The operator Π0 with kernel Π(X, Y )is the orthogonal projection in L2 (R2n ) on Λ0 , which is a proper closed subspace of L2 (R2n ), canonically isomorphic to L2 (Rn ). We have π (4.4) Λ0 = ran W = L2 R2n ∩ ker ∂¯ + z , 2 ∗ W W = IdL2 (Rn ) reconstruction formula u(x) = (W u)(Y )ϕY (x) dY , (4.5) R2n

∗

W W = Π0

W is an isomorphism from L2 Rn onto Λ0 .

3 Note that for a linear form L on R2n , Lw Lw = (L2 )w .

(4.6)

774


Proof. These statements are classical (see e.g. [10]); however, since we shall need some extension of that proposition, it is useful to examine the proof. We note that e−iπyη (W v)(y, η) is the partial Fourier transform w.r.t. x of Rn × Rn (x, y) → v(x)2n/4 e−π(x−y) , 2

whose L2 (R2n )-norm is vL2 (Rn ) so that W is isometric from L2 (Rn ) into L2 (R2n ), thus with a closed range. As a result, we have W ∗ W = IdL2 (Rn ) , W W ∗ is selfadjoint and such that W W ∗ W W ∗ = W W ∗ : W W ∗ is indeed the orthogonal projection on ran W (ran W W ∗ ⊂ ran W and W u = W W ∗ W u). The straightforward computation of the kernel of W W ∗ is left to the reader. Let us prove that Λ0 = ran W is indeed defined by (4.3). For v ∈ L2 (Rn ), we have (W v)(y, η) =

y

v(x)2n/4 e−π(x−y) e−2iπ(x− 2 )η dx 2

Rn

=

π

v(x)2n/4 e−π(x−y+iη) dxe− 2 (y 2

2 +η2 )

π

e− 2 (η+iy)

2

(4.7)

Rn

and we see that W v ∈ L2 (R2n ) ∩ ker(∂¯ + π2 z). Conversely, if Φ ∈ L2 (R2n ) ∩ ker(∂¯ + π2 z), we π

have Φ(x, ξ ) = e− 2 (x

2 +ξ 2 )

(W W ∗ Φ)(x, ξ ) =

f (ξ + ix) with Φ ∈ L2 (R2n ) and f entire. This gives

π

= e− 2 (ξ =e

2 +x 2 )

− π2 (ξ 2 +x 2 ) π

= e− 2 (ξ =e

π

e− 2 ((ξ −η)

2 +x 2 )

− π2 |z|2 π

= e− 2 |z|

2

2 +(x−y)2 +2iξy−2iηx)

e− 2 (η

π

2 −2ξ η+y 2 −2xy+2iξy−2iηx)

π

2 +y 2 +2iy(ξ +ix)−2η(ξ +ix))

e− 2 (η e−π(y

2 +η2 )

¯

2

f (ζ )

= e− 2 |z| f (ζ ) 2

1j n

π

∂ ∂ ζ¯j

Φ(y, η) dy dη

Φ(y, η) dy dη

eπ(η−iy)(ξ +ix) f (η + iy) dy dη

e−π|ζ | eπ ζ z f (ζ ) dy dη

1j n π

Φ(y, η) dy dη

(ζ = η + iy, z = ξ + ix)

∂ −π|ζ |2 π ζ¯ z 1 e dy dη e π(zj − ζj ) ∂ ζ¯j

! 1 2 ¯ , e−π|ζ | eπ ζ z π(ζj − zj ) S (R2n ),S (R2n )

= e− 2 |z| f (z), 2

since f is entire. This implies W W ∗ Φ = Φ and Φ ∈ ran W . The proof of the proposition is complete. 2


775

Proposition 4.3. Defining π ¯ K = ker ∂ + z ∩ S R2n , 2

(4.8)

the operator W given by (4.2) can be extended as a continuous mapping from S (Rn ) onto K " defined by (the L2 (Rn ) dot-product is replaced by a bracket of (anti)duality). The operator Π its kernel Π given by (4.1) defines a continuous mapping from S (R2n ) into itself and can be extended as a continuous mapping from S (R2n ) onto K . It verifies "2 = Π, " Π

"|K = IdK . Π

(4.9)

Proof. As above we use that e−iπyη (W v)(y, η) is the partial Fourier transform w.r.t. x of the tempered distribution on R2n x,y v(x)2n/4 e−π(x−y) . 2

Since e±iπyη are in the space OM (R2n ) of multipliers of S (R2n ), that transformation is continuous and injective from S (Rn ) into S (R2n ). Replacing in (4.7) the integrals by brackets of duality, we see that W (S (Rn )) ⊂ K . Conversely, if Φ ∈ K , the same calculations as above give (4.9) and (4.8). 2 For a Hamiltonian a defined on R2n , for instance a bounded function on R2n , we define = W ∗ aW :

a Wick

L2 (R2n )

a (multiplication by a)

L2 (R2n ) W∗

W

L2 (Rn )

a Wick

L2 (Rn )

we note that a(x, ξ ) 0 ⇒ a Wick = W ∗ aW 0, as an operator. There are many useful applications of the Wick quantization due to that non-negativity property, but for our purpose here, it will be more important to relate explicitely that quantization to the usual Weyl quantization (as given by (3.1)) for quadratic forms. Lemma 4.4. Let q(X) = QX, X be a quadratic form on R2n (Q is a 2n×2n symmetric matrix). Then we have q Wick = q w +

1 trace Q. 4π

(4.10)

Let L(y, η) = τ · y − t · η be a real linear form on R2n ; then, for all Φ ∈ Λ0 , we have

2 |τ |2 + |t|2 Φ2L2 (R2n ) . L(y, η)2 Φ(y, η) dy dη 4π

(4.11)

776


Proof. A straightforward computation shows that where Γ (X) = 2n e−2π|X|

2

q Wick = (q ∗ Γ )w ,

X ∈ R2n .

(4.12)

2 By Taylor’s formula, we have (q ∗ Γ )(X) = q(X) + R2n 2n e−2π|Y | QY, Y dY, we can use the

1/2 2 −2πt 2 1 dt = 4π to get the first result. For Φ ∈ Λ0 , we have Φ = W u with formula R 2 t e u ∈ L2 (Rn ) and thus LΦ2L2 (R2n ) = L2 W u, W u L2 (R2n ) = W ∗ L2 W u, u L2 (Rn ) Wick w trace(L2 ) u2L2 (Rn ) , = L2 u, u L2 (Rn ) = L2 u, u L2 (Rn ) + 4π and since Lw Lw = (L2 )w for a linear form, we get since L is real-valued, 2 |τ |2 + |t|2 LΦ2L2 (R2n ) = Lw uL2 (Rn ) + Φ2L2 (R2n ) , 4π which implies (4.11).

2

N.B. The inequality (4.11) looks like an uncertainty principle related to the localization in R2n for the functions of Λ0 . Moreover the equality (4.10) provides a simple way to saturate approximately the inequality (4.11); for instance if L(y, η) = y1 , we consider the sequence Φε = W uε with uε (x) = ϕ(x1 /ε)ε −1/2 ψ(x ), ϕL2 (R) = ψL2 (Rn−1 ) = 1, and we get, provided xϕ(x) ∈ L2 (R), 2 2 1 1 = O ε2 + . y12 Φε (y, η) dy dη = x12 ϕ(x1 /ε) ε −1 dx1 + 4π 4π R

4.2. The anisotropic LLL Going back to the Gross–Pitaevskii energy (1.11), with q given by (1.13), we see, using Theorem 3.3 and (3.8) that, with u = Mv, 2EGP (u) = q w u, u L2 (R2 ) + g

|u|4 dx 4 = M ∗ q w Mv, v L2 (R2 ) + g (Mv)(x) dx = Dy21 + μ21 y12 + Dy22 + μ22 y22 v, v L2 (R2 ) + g =

(Mv)(x)4 dx

2 2 2 −1 λ1 cd − dλ−1 2 x2 + λ1 Dx1 u + μ1 λ2 d Dx2 + cλ2 x1 u, u 2 2 2 −1 + λ2 cd − dλ−1 1 x1 + λ2 Dx2 u, u + μ2 λ1 d Dx1 + cλ1 x2 u, u + g |u|4 dx.


777

The question at hand is the determination of infuL2 =1 EGP (u), which is equal to infvL2 =1 EGP (Mv). Since μ1 = 0 at ε = 0 (see (2.9)) and μ2 ∈ [1, 4] (see (A.1)), it is natural to modify our minimization problem, and in the (y, η) coordinates, to restrict our attention to the Lowest Landau Level, i.e. the groundspace of Dy22 + μ22 y22 , that is the subspace of L2 (R2 ) 2 1/4 LLLy = v1 (y1 ) ⊗ 21/4 μ2 e−πμ2 y2 v

2 1 ∈L (R)

= ker(Dy2 − iμ2 y2 ) ∩ L2 R2 .

(4.13)

If we want to stay in the physical coordinates (x, ξ ) we reach the following definition, obtained by using Segal’s formula (3.2) with M, χ given in Lemma 3.1 so that LLLx = M(LLLy ). Proposition 4.5. Let q be the quadratic form on R4 given by (1.13). We define the LLL as LLL = (ker L) ∩ L2 R2 ,

with −1 w w L = λ2 cd − dλ−1 1 x1 + λ2 Dx2 − iμ2 λ1 d Dx1 − iμ2 cλ1 x2 = η2 − iμ2 y2 .

(4.14) (4.15)

The LLL is the subspace of L2 (R2 ) of functions of type ν2 ν2 πν 2 γ γπ 2 2 x 1− (4.16) + (β2 x2 ) 1 + exp −i x1 x2 F (x1 + iβ2 x2 ) exp − 4β2 1 2α 2α 4α where F is entire on C, and the parameters γ , β2 , ν, α are given in Appendix A.2. The real part of the phase of the Gaussian function multiplying F (x1 + iβ2 x2 ) is a negative definite quadratic form when (ω, ν) = (0, 0). Proof. We have μ2 y2

η2

− λ cd x1 iL = μ2 λ1 d −1 Dx1 + μ2 cλ1 x2 +i λ2 Dx2 − dλ−1 2 1 1 μ2 λ1 d −1 ∂1 + iλ2 ∂2 + 2iπμ2 cλ1 x2 + 2π dλ−1 − λ2 cd x1 1 2iπ 1 1 1 . μ2 λ1 d −1 ∂1 + i λ2 ∂2 + iπμ2 cλ1 x2 + π dλ−1 − λ cd x = 2 1 1 iπ 2 2 =

We set −1 t1 = μ−1 2 λ1 dx1 ,

t2 = λ−1 2 x2 ,

and we get for z = t1 + it2 , ∂ + iπμ2 cλ1 λ2 t2 + π dλ−1 − λ2 cd μ2 λ1 d −1 t1 1 ∂ z¯ z − z¯ z + z¯ ∂ + iπμ2 cλ1 λ2 + π dλ−1 = − λ2 cd μ2 λ1 d −1 1 ∂ z¯ 2i 2

(4.17)

778


∂ ∂ μ2 μ2 μ2 μ2 + zπ + z¯ π (1 − 2λ1 λ2 c) = + zπ − z¯ π ν 2 α −1 ∂ z¯ 2 2 ∂ z¯ 2 2 μ2 ν 2 μ2 μ2 ν 2 μ2 ∂ 2 2 = e−π 2 z¯z eπ 4α (¯z) eπ 2 z¯z e−π 4α (¯z) . ∂ z¯ =

As a consequence, the LLL is the subspace of L2 (C) of functions f (z)e−π

μ2 z 2 z¯

eπ

ν 2 μ2 z )2 4α (¯

,

with f holomorphic.

We note that the real part of the exponent is −

πμ2 2 2α + ν 2 πμ2 2 2α − ν 2 ν2 2 t1 + t22 − t1 − t22 = − t1 + t22 2 2α 2 2α 2α

and that 2α − ν 2 > 0

⇐⇒

(ω, ν) = (0, 0).

Leaving the t-coordinates for the original x-coordinates, we get with f entire, 2 −1 −1 πμ2 2 2α − ν 2 −1 2 2α + ν f μ2 λ1 dx1 + iλ2 x2 exp − t1 + t2 2 2α 2α 2 πμ2 ν t1 t2 , × exp −i 2α i.e. 2 −1 −1 πμ2 2 2 2α − ν 2 −1 2 2α + ν f μ2 λ1 dx1 + iλ2 x2 exp − + x2 x1 d 2 2αλ21 μ22 2αλ22 πμ2 ν 2 d × exp −i x1 x2 , 2αλ1 λ2 μ2 and since −1 −1 −1 −1 −1 μ2 λ1 d −1 λ−1 λ1 λ2 λ2 = μ2 2γ −1 λ−2 γβ2 (2μ2 )−1 = β2 , 2 = μ2 λ1 2γ 2 = μ2 2γ γ −2 −1 2 −1 2 −2 −1 2 −1 −1 −1 −2 2−1 μ2 d 2 λ−2 β2 μ2 = , 1 μ2 = 2 μ2 γ 4 λ2 μ2 = 2 μ2 γ 4 2μ2 γ 4β2 −1 2−1 μ2 λ−2 2 = 2 μ2

γβ2 γβ2 , = 2μ2 4

πμ2 ν 2 d πν 2 d πν 2 γ , = = 2αλ1 λ2 μ2 2αλ1 λ2 2α2 we obtain


779

$ −1 # −1 −1 f μ−1 2 λ1 d x1 + i μ 2 λ1 d λ2 x2 =β2

2 πμ2 ν 2 d πμ2 2 2 2α − ν 2 2 2α + ν + x2 exp −i x1 d × exp − x1 x2 , 2 2αλ1 λ2 μ2 2αλ21 μ22 2αλ22 that is, with F entire on C, ν2 ν2 γπ 2 x1 1 − + (β2 x2 )2 1 + F (x1 + iβ2 x2 ) exp − 4β2 2α 2α 2 πν γ x1 x2 . × exp −i 4α The proof of the proposition is complete.

(4.18)

2

Remark 4.6. We note that in the isotropic case ν = 0, we have β2 = 1, γ = 4, recovering (1.15) 2 2 (f (x1 + ix2 )e−π(x1 +x2 ) ) for ω = 1. On the other hand, the reader may have noticed that it seems difficult to guess the above definition without going through the explicit computations on the diagonalization of q of the previous sections. 4.3. The energy in the anisotropic LLL Lemma 4.7. The LLL is defined by Proposition 4.5 and the Gross–Pitaevskii energy by (1.11). For u ∈ LLL, we have EGP (u) =

1 2

R2

+

g 2

2 2α 2α(2ν 2 + ε 2 ) 2 2 2 dx1 dx2 u(x ε x + x , x ) 1 2 1 2 α + 2ω2 + ν 2 α − ν 2 + 2ω2

R2

u(x1 , x2 )4 dx1 dx2 + μ2 − μ1 β1 β2 + 1 . 4π 8π β1 β2

(4.19)

Proof. In the LLL, one can simplify the energy. We define A2 = M(η2 − iμ2 y2 )w M ∗ = μ2 λ1 d −1 Dx1 + cλ1 x2 + i λ2 Dx2 − dλ−1 1 − λ2 cd x1 , A1 = M(η1 − iμ1 y1 )w M ∗ = μ1 λ2 d −1 Dx2 + cλ2 x1 + i λ1 cd − dλ−1 2 x2 + λ1 Dx1 , which satisfy the canonical commutation relations: [Aj , A∗j ] = μj /π, while all other commutators vanish. We have proven that q w = A∗1 A1 + A∗2 A2 +

μ1 + μ2 = (Re A1 )2 + (Im A1 )2 + (Re A2 )2 + (Im A2 )2 2π

and the LLL is defined by the equation A2 u = 0. On the other hand, we have −1 dμ−1 1 Re A1 − Im A2 = dλ1 x1 ,

dμ2 −1 Re A2 − Im A1 = dλ−1 2 x2 ,

780


and thus for u ∈ LLL, since A2 u = 0 , using the commutation relations of the Aj ’s, one gets 2 −1 2 2 −2 2 ∗ ∗ d 2 λ−2 1 x1 = d μ1 (Re A1 ) + A2 − A2 /2i + 2dμ1 (Re A1 ) A2 − A2 /2i μ2 2 = d 2 μ−2 1 (Re A1 ) + 4π , and similarly, 2 2 2 −2 d 2 λ−2 A2 + A∗2 /2 + (Im A1 )2 2 x2 = d μ 2 = (Im A1 )2 +

d2 . 4πμ2

As a result, we get on the LLL, 2 2 −2 2 2 2 μ21 λ−2 1 x1 + d λ2 x2 = (Re A1 ) + (Im A1 ) + 2 2 −2 2 and q w = μ21 λ−2 1 x1 + d λ2 x2 −

γ 2EGP (u) = 2

d2 4πμ2

−

μ2 μ21 4πd 2

μ1 β1 x12 R2

+g

+

μ2 2π ,

μ2 μ21 d2 + , 4πμ2 4πd 2

so that

2 μ1 2 + x2 u(x1 , x2 ) dx1 dx2 β1

u(x1 , x2 )4 dx1 dx2

R2

μ1 μ2 1 − + , β1 β2 + 2π 4π β1 β2 for any u ∈ LLL, that is, satisfying (4.16). We note that 2α γ μ1 β1 = ε2 2 α + 2ω2 + ν 2 2α(2ν 2 + ε 2 ) γ μ1 = 2β1 α − ν 2 + 2ω2

coefficient of x12 , coefficient of x22 .

Definition 4.8. For u ∈ LLL (see Proposition 4.5), we define 1 ELLL (u) = 2

2 2 2 g1 ε x1 + κ12 x22 u(x1 , x2 ) dx1 dx2 + 2

R2

u(x1 , x2 )4 dx1 dx2 , (4.20)

R2

with κ12 =

(α + 2ω2 + ν 2 )(2ν 2 + ε 2 ) , α − ν 2 + 2ω2

g1 = g

α + 2ω2 + ν 2 , 2α

α=

ν 4 + 4ω2 . (4.21)


781

We note that, from (4.19), EGP (u) =

2α μ2 1 μ1 E (u) + β + . − β LLL 1 2 4π 8π β1 β2 α + 2ω2 + ν 2

Remark 4.9. Since α 2 = ν 4 + 4ω2 , we see that 2 (α + 2ω2 + ν 2 )(2ν 2 + ε 2 ) 2ν 2 2ν + ε 2 1 + 2ν 2 + ε 2 , = κ2 = 2 2 α − ν + 2ω α − ν 2 + 2ω2

(4.22)

(4.23)

and κ 2 = ε 2 ⇐⇒ ν = 0. Remark 4.10. We stay away from the case where ω = 0 and shall always assume ω > 0. In the case ω = 0, the quadratic part of the energy is diagonal and the LLL is, 1/8 −π(2−ε2 )1/2 x 2 2, e v1 (x1 ) ⊗ 21/4 2 − ε 2 and we get a 1D problem on the function v1 . 4.4. The (final) reduction to a simpler lowest Landau level Given the fact that in (4.16), we can write F (x1 + iβ2 x2 ) as a holomorphic function 2 times e−δz , with δ = γ πν 2 /(8β2 α), and that the energy ELLL depends only on the modulus of u and not on its phase, it is equivalent to minimize ELLL on the LLL or on the space γπ 2 2 x + (β2 x2 ) , with f entire. f (x1 + iβ2 x2 ) exp − 4β2 1 A rescaling in x1 and x2 yields the space of the introduction with u(x1 , x2 ) =

γ v(y1 , y2 ), 2

y1 = x1

γ , 2β2

y2 = x2

γβ2 , 2

(4.24)

and, with Λ0 given by (4.3), the mapping LLL u → v ∈ Λ0 is bijective and isometric. With κ1 , g1 given in Definition 4.8, β2 in (2.12), γ in (2.13), we introduce κ=

κ1 , β2

g0 =

g1 γ 2 , 4β2

(4.25)

and E(v) =

1 2

2 2 2 g0 ε y1 + κ 2 y22 v(y1 , y2 ) dy1 dy2 + v4L4 (R2 ) . 2

(4.26)

R2

Using the transformation (4.24), we have ELLL (u) =

2β2 E(v), γ

(4.27)

782


so that, via Definition 4.8, we are indeed reduced to the minimization of (1.21) in the space Λ0 (given in (1.22)) under the constraint uL2 (R2 ) = 1. We note also that the quantities 2α , α + 2ω2 + ν 2 β2 ,

γ2 , β2

2β2 γ

α + 2ω2 + ν 2 2α

factors of ELLL (u) in (4.22) and E(v) in (4.27) ,

and (4.28)

factors of κ in (4.25), of g1 in (4.25), of g in (4.21) ,

(4.29)

are bounded and away from zero as long as ω stays away from zero, a condition that we shall always assume, say 0 < ω0 ω 1. 5. Weak anisotropy This section is devoted to the proof of Theorem 1.1. We assume ε κ ε 1/3 . The isotropic case is recovered by assuming κ = ε. We first give some approximation results in Section 5.1, and prove the theorem in Section 5.2. We recall that the space Λ0 , the operator Π0 , the energy E and the minimization problem I (ε, κ) are defined by (1.22), (1.23), (1.21) and (1.24), respectively. An important test function will be (1.28), namely π

uτ (x1 , x2 ) = e 2 (z for τ = τR + iτI = e

2iπ 3

2 −|z|2 )

√ Θ( τI z, τ ),

z = x1 + ix2 ,

(5.1)

.

5.1. Approximation results π

Lemma 5.1. Let u(x) = f (x1 + ix2 )e− 2 |x| ∈ L∞ (R2 ), with f holomorphic. Assume 0 β 1 and let p ∈ C 0,β (R2 ) be such that supp(p) ⊂ BS the Euclidean ball of radius S > 0 and of center 0. Define x1 x2 1 . (5.2) p , ρ(x) = √ R1 R2 R1 R 2 2

Then, for any r 1, there exists a constant CS,r > 0 depending only on S and r such that, setting R = min(R1 , R2 ), we have, 1

1

r −2 Π0 (ρu) − ρu r 2 CS,r u ∞ 2 p 0,β 2 (R1 R2 ) . L (R ) C (R ) L (R ) Rβ

(5.3)

Proof. We first prove the lemma in the case β = 0. For this purpose, we write Π0 (ρu) e− π2 |x−y|2 u(y)ρ(y) dy. R2

Young’s inequality implies, for any r 1 and any p, q 1 such that 1/p + 1/q = 1 + 1/r, Π0 (ρu) r e− π2 |x|2 p uρLq uL∞ e− π2 |x|2 p ρLq . L L L


783

Fixing q = r, hence p = 1, we find Π0 (ρu)

1

Lr

1

2uL∞ ρLr = 2uL∞ (R1 R2 ) r − 2 pLr .

(5.4)

This proves (5.3) for β = 0. Next, we assume β = 1. We use a Taylor expansion of ρ(y) = ρ(x + y − x) around x: ρ(y) = ρ(x) 1 +√ R1 R 2

1 0

x1 y1 − x1 x2 y2 − x2 ∇p +t , +t R1 R1 R2 R2

y1 − x1 y2 − x2 · dt. , R1 R2

We then notice that, although u ∈ / Λ0 a priori, it belongs to K (see Proposition 4.3) and we have Π0 (u) = u since u ∈ L∞ and u(x) = f (x1 + ix2 ) exp(−π|x|2 /2) with f holomorphic. Hence, we have Π0 (ρu) − ρu π 2 e− 2 |x−y| +iπ(x2 y1 −y2 x1 ) u(y1 , y2 ) = R ,R2

1 BS+1

1 ×√ R1 R 2

1 ∇p 0

− ρ(x)

x1 y1 − x1 x2 y2 − x2 +t , +t R1 R1 R2 R2

u(y)e− 2 |x−y| π

2 +iπ(x y −y x ) 2 1 2 1

y1 − x1 y2 − x2 · dt dy , R1 R2

dy,

R ,R

1 2 )c (BS+1

R1 ,R2 is where the set BS+1

R1 ,R2 BS+1 = (y1 , y2 ) = (R1 t1 , R2 t2 ), t ∈ BS+1 .

(5.5)

We thus have, with R = min(R1 , R2 ),

Π0 (ρu) − ρu ∇pL∞

R ,R2

1 π |y − x| 2 dy e− 2 |x−y| u(y) √ R1 R 2 R

1 BS+1

+ ρ(x)

u(y)e− π2 |x−y|2 dy.

(5.6)

R1 ,R2 c (BS+1 )

We bound the first term of the right-hand side of (5.6) using Young’s inequality, while for the second term, we have, ∀x ∈ supp(ρ) ⊂ BSR1 ,R2 ,

784


u(y)e− π2 |x−y|2 dy uL∞ e− π4 R 2

e− 4 |x−y| dy π

2

R2

R ,R

1 2 )c (BS+1 π

= 4uL∞ e− 4 R uL∞ 2

C , R

where C is a universal constant. Hence, we have Π0 (ρu) − ρu

Lr

π 1 R1 ,R2 1/r 1 2 B ∇pL∞ |y|e− 2 |y| L1 uL∞ √ R R1 R2 S+1

C uL∞ ρLr R √ 1 1 1 = ∇pL∞ 2uL∞ (R1 R2 ) r − 2 |BS+1 |1/r R +

+

1 1 C uL∞ pL∞ (R1 R2 ) r − 2 |BS |1/r . R

This gives (5.3) for β = 1. We then conclude by a real interpolation argument between C 0 and C 0,1 . 2 A comment is in order here: we have chosen to state Lemma 5.1 with a general function p. 1/2 However, since our aim is to apply the above result with the special case p(x) = (1 − |x|2 )+ , it is also possible to use explicitly this value of p in order to give a simpler proof of the above result. The method would then be to prove the estimate for r = +∞ first, then for r = 1, and then use an interpolation argument between L1 and L∞ . For instance, the proof of the r = +∞ case would go as follows: Π0 (ρu)(x) − ρ(x)u(x) = e− π2 |x−y|2 +iπ(x2 y1 −y2 x1 ) ρ(y)u(y) − ρ(x)u(y) dy R2

uL∞

π 2 e− 2 |x−y| ρ(y) − ρ(x) dy

R2

uL∞

e

− π2 |x−y|2

|x − y| dy R

R2

uL∞ = √ R

π

e− 2 |y|

2

|y| dy.

R2

The proof of the case r = 1 is slightly more involved, but is based on the same idea.


785

We now prove Lemma 5.2. With the same hypotheses as in Lemma 5.1, we have, for any s 1,

2 x12s Π0 (ρu) − ρu

1/2

1 + R1s S s , Rβ

(5.7)

(1 + R2s S s ) , Rβ

(5.8)

CS,s uL∞ (R2 ) pC 0,β (R2 )

R2

and

2 x22s Π0 (ρu) − ρu

1/2 CS,s uL∞ (R2 ) pC 0,β (R2 )

R2

where CS,s depends only on S and s. Proof. Here again, we first deal with the case β = 0. For this purpose, we write: |x1 |s Π0 (ρu) 2s−1

R2

π 2 |x1 − y1 |s e− 2 |x−y| u(y)ρ(y) dy

+ 2s−1

π 2 |y1 |s e− 2 |x−y| u(y)ρ(y) dy,

(5.9)

R2

where we have used the inequality (a + b)s 2s−1 (a s + bs ), valid for any a, b 0, s 1. The first line of (5.9) is dealt with exactly as in the proof of Lemma 5.1, leading to (5.4) with r = 2, which reads here |x1 − y1 |s e− π2 |x−y|2 u(y)ρ(y) dy uL∞ |x|s e− π2 |x|2 1 ρ 2 L L L2

R2

Cs uL∞ pL2 ,

(5.10)

where Cs depends only on s. The second line of (5.9) is treated in the same way, but ρ(y) is replaced by |y1 |s ρ(y), that is, p(y) is replaced by R1s |y1 |s p(y). Hence, we have |y1 |s e− π2 |x−y|2 u(y)ρ(y) dy R2

L2

2R1s uL∞ |y1 |s p L2 .

(5.11)

Collecting (5.9), (5.10) and (5.11), we find |x1 |s Π0 (ρu)

L2

Cs 1 + R1s S s uL∞ pC 0 |BS |1/2 .

This proves (5.7) for β = 0. Next, we consider the case β = 1. Here again, we use a Taylor expansion to obtain (5.6). This implies

786


∇pL∞ |x1 | Π0 (ρu) − ρu 2s−1 R

s

1 π 2 e− 2 |x−y| u(y) √ |y − x||y1 − x1 |s dy R1 R 2

R ,R2

1 BS+1

+2

s−1 ∇pL∞

R R ,R2

1 π 2 e− 2 |x−y| u(y) √ |y − x||y1 |s dy R1 R 2

1 BS+1

+ |x1 |s ρ(x)

u(y)e− π2 |x−y|2 dy,

R ,R

1 2 )c (BS+1

R1 ,R2 where BS+1 is defined by (5.5). We use Young’s inequality again, finding

|x1 |s Π0 (ρu) − ρu

L2

2

s−1 ∇pL∞

π 2 |y|s+1 e− 2 |y| L1

R1 ,R2 |BS+1 |

1/2

uL∞ R1 R2 1/2 |y1 |2s s−1 ∇pL∞ − π2 |y|2 |y|e +2 dy uL∞ L1 R R1 R2 R

R ,R2

1 BS+1

+

C uL∞ |x1 |s ρ L2 , R

where C is a universal constant. Hence, |x1 |s Π0 (ρu) − ρu 2 CS,s pC 1 1 + R s S s uL∞ . 1 L R This gives (5.7) in the case β = 1. Here again, we conclude with a real interpolation argument. The proof of (5.8) follows the same lines. 2 5.2. Energy bounds Proposition 5.3. Let τ ∈ C \ R, let p ∈ C 0,1/2 (R2 ) be such that supp(p) ⊂ K for some compact set K, and |p|2 = 1. Consider uτ as defined by (1.28), and define −1 (5.12) v = Π0 (ρuτ )L2 (R2 ) Π0 (ρuτ ), where ρ is given by 1 x1 x2 ρ(x) = √ , p , R1 R2 R1 R 2

R1 =

4g0 κ πε 3

1/4

,

R2 =

4g0 ε πκ 3

1/4 .

Then we have, with E(u) defined by (1.21) 3 1/8 2 πγ (τ ) √ 2gεκ 1 2 κ p(x)4 + O , |x| p(x) + E(u) = εκ π 2 4 ε R2

for (ε, κε −1/3 ) → (0, 0), where γ (τ ) is given by (1.31).

(5.13)

(5.14)


787

N.B. The L∞ function ρuτ does not belong to Λ0 since it is compactly supported and not identically 0; as a result, Π0 (ρuτ )L2 = 0 and v makes sense. Proof. First note that R = min(R1 , R2 ) = R2 , and that Lemma 5.1 with r = 2 implies 3 1/8 κ − ρuτ L2 CR −1/2 = C . ε

Π0 (ρuτ )

L2

(5.15)

We then apply Lemma 5.2 for s = 1, β = 1/2, finding x 2 Π0 (ρuτ )2 − x 2 |ρ|2 |uτ |2 C x1 Π0 (ρuτ ) 2 + x1 ρuτ 2 1 + R1 L 1 1 L R 1/2 R2

R2

1 + R1 1 + R1 C 2x1 ρuτ L2 + C 1/2 . R R 1/2

We also compute

2 2 x12 ρ(x) uτ (x) dx

R12 uτ 2L∞

R2

2 x12 p(x) dx CR12 .

R2

Hence, we get 3 1/8 2 2 √ ε 2 κ 2 2 2 2 2 1 + R1 x1 Π0 (ρuτ ) − x1 |ρ| |uτ | Cε C εκ . 1/2 2 ε R R2

(5.16)

R2

A similar argument allows to show that 3 1/8 2 2 √ κ 2 κ 2 2 2 2 2 1 + R2 x2 Π0 (ρuτ ) − x2 |ρ| |uτ | Cκ C εκ . 1/2 2 ε R R2

(5.17)

R2

Turning to the last term of the energy, we apply Lemma 5.1 again, with r = 4, β = 1/2, finding Π0 (ρuτ )4 − |ρuτ |4 2 Π0 (ρuτ )3 4 + ρuτ 3 4 Π0 (ρuτ ) − ρuτ 4 L L L R2

R2

Cρuτ 3L4 (R1 R2 )−1/4 R −1/2 . In addition, we have

|ρuτ |4 uτ 4L∞

R2

R2

|ρ|4 = uτ 4L∞ (R1 R2 )−1

p4 . R

788


Hence, we obtain 3 1/8 √ Π0 (ρuτ )4 − |ρuτ |4 C(R1 R2 )−1 R −1/2 C εκ κ . ε R2

(5.18)

R2

Combining (5.16), (5.17) and (5.18), we have 3 1/8 κ . E Π0 (ρuτ ) = E(ρuτ ) 1 + O ε Hence, with the help of (5.15), we get

ρuτ E(v) = E ρuτ L2

3 1/8 κ 1+O . ε

Finally, we estimate the terms of E(ρuτ /ρuτ L2 ): using real interpolation between C 0 and C 0,1 , we obtain ρuτ 2L2 =

p(x)2 uτ (R1 x1 , R2 x2 )2 dx

R2

3 1/8 1 κ 2 2 = – |uτ | + O . = – |uτ | + O ε R 1/2

(5.19)

Moreover, we have ε2 2

x12 |ρ|2 |uτ |2 = R2

κ2 2

3 1/8 2 ε2 2 κ R1 – |uτ |2 + O x12 p(x) dx, 2 ε

(5.20)

R2

3 1/8 2 κ2 2 κ 2 2 2 2 x2 |ρ| |uτ | = R2 – |uτ | + O x22 p(x) dx, 2 ε

R2

g 2

3 1/8 g κ 4 |ρ| |uτ | = |p|4 . – |uτ | + O 2R1 R2 ε 4

R2

(5.21)

R2

4

R2

Thus, collecting (5.19), (5.20), (5.21) and (5.22),

ε2 2 R E(u) = 2 1

2 x12 p(x) dx

R2

κ2 + R22 2

R2

2 x22 p(x) dx

3 1/8 – |uτ |4 κ g0 4 1+O |p| +

2 2 ε ( – |uτ | ) 2R1 R2 R2

(5.22)


=

2g0 εκ π

2 πγ (τ ) 4 1 2 x1 + x22 p(x) + |p| 2 4

789

R2

3 1/8 κ × 1+O ε 2 πγ (τ ) 4 2g0 εκ 1 2 2 x + x2 p(x) + |p| = π 2 1 4 R2

3 1/8 √ κ . εκ +O ε

2

Proof of Theorem 1.1. We first prove the lower bound in (1.33): this is done by noticing that J (ε, κ) I (ε, κ), where 2 2 2 2 4 2 J (ε, κ) = inf E(u), u ∈ L R , 1 + |x| dx ∩ L R , |u| = 1 . R2

In addition, the minimizer of J (ε, κ) may be explicitly computed (up to the multiplication by a complex function of modulus one): % x12 x22 1/2 2 1− 2 − 2 u(x) = , (5.23) πR1 R2 R1 R2 + with R1 , R2 defined by (5.13). Inserting (5.23) in the energy, one finds the lower bound of (1.33). In addition, the inverted parabola (5.23) is compactly supported, so it cannot be in Λ0 . Hence, the inequality is strict. In order to prove the upper bound, we apply Proposition 5.3, with % |x|2 1/2 2 1− √ , p(x) = √ π γ (τ ) γ (τ ) + and τ = j. This corresponds

to minimizing the leading order term of (5.14) with respect to τ and p, with the constraint |p|2 = 1. 2 6. Strong anisotropy We give in this section the proof of Theorem 1.2. We deal here with the strongly asymmetric case that is, (1.35), which we recall here: κ ε 1/3

(6.1)

We first prove an upper bound for the energy in Section 6.1, then a lower bound in Section 6.2, and conclude the proof in Section 6.3

790


6.1. Upper bound for the energy Lemma 6.1. Assume that ρ ∈ L2 (R). Then the function π 2 π 1 2 u(x1 , x2 ) = 1/4 e− 2 x2 e− 2 ((x1 −y1 ) −2iy1 x2 ) ρ(y1 ) dy1 , 2

(6.2)

R

satisfies u ∈ Λ0 . Proof. We first write π

u(x1 , x2 )e 2 (x1 +x2 ) = 2

2

1 21/4

π

e− 2 (y1 −2(x1 +ix2 )y1 ) ρ(y1 ) dy1 , 2

R

which is a holomorphic function of x1 + ix2 . In addition, we have u(x1 , x2 ) 1 e− π2 x22 ρ ∗ e− π2 y12 (x1 ). 1/4 2 Hence, using Young’s inequality, we get uL2 (R2 ) hence u ∈ L2 (R2 ).

− π y2 ρ = 21/4 ρL2 (R) , 2 (R) e 2 1 1 L L (R) 1/4 1

2

2

Lemma 6.2. Let p ∈ C 2 (R) have compact support with supp(p) ⊂ (−T , T ), and consider the function t 1 ρ(t) = √ p . (6.3) R R Then, for any r 1, there exists a constant Cr depending only on r such that the function u defined by (6.2) satisfies, for R 1, u(x1 , x2 ) − 21/4 ρ(x1 )e−πx22 +iπx1 x2 − i21/4 x2 ρ (x1 )e−πx22 +iπx1 x2 r 2 L (R ) Cr T 1/r

p L∞ (R) . R 5/2−1/r

(6.4)

Proof. We use a Taylor expansion of p( yR1 ) around

y1 p R

x1 =p R

x1 R,

that is,

1 x1 + p (y1 − x1 ) R R

1 + 2 (x1 − y1 )2 R

1 0

(1 − t)p

x1 t (y1 − x1 ) + dt. R R

(6.5)


791

In addition we have x1 x1 1/4 −πx 2 +iπx1 x2 1 1 − π x2 − π2 ((x1 −y1 )2 −2iy1 x2 ) 1 2 2 2 dy 2 e p p e e = , √ √ 1 1/4 R R 2 R R R

and 1

2

π

e− 2 x2 1/4

2

π

e− 2 ((x1 −y1 )

R

=

1 R

i21/4 x2 p 3/2

1

2 −2iy x ) 1 2

R

p 3/2

x1 (y1 − x1 ) dy1 R

x1 −πx 2 +iπx1 x2 2 e . R

Setting v(x1 , x2 ) = u(x1 , x2 ) − 21/4 ρ(x1 )e−πx2 +iπx1 x2 − i21/4 x2 ρ (x1 )e−πx2 +iπx1 x2 , 2

2

(6.6)

we infer v(x1 , x2 )

1

π 2 1 e− 2 x2 21/4 R 5/2

R 0

1

π 2 p L∞ 1/4 5/2 e− 2 x2 2 R

π 2 x1 y1 dt dy1 +t y12 e− 2 y1 (1 − t)p R R π

y12 e− 2 y1 (1 − t)1(−T R,T R) (x1 + ty1 ) dt dy1 . 2

R 0

Hence, using Jensen’s inequality, we see that there is a constant Cr depending only on r such that r v(x1 , x2 )r Cr p L∞ e−r π2 x22 R 5r/2

1

π

y12 e− 2 y1 (1 − t)1(−T R,T R) (x1 + ty1 ) dt dy1 , 2

R 0

whence vrLr

p rL∞ Cr R 5r/2

= Cr

1

π

2

2

R R 0

p rL∞ (2T R) R 5r/2

p rL∞ = Cr T R, R 5r/2 which implies (6.4).

π

e−r 2 x2 y12 e− 2 y1 (1 − t)

2

1(−T R,T R) (x1 + ty1 ) dx1 dt dx2 dy1 R

1

R R 0

π

π

e−r 2 x2 y12 e− 2 y1 (1 − t) dt dx2 dy1 2

2

792


Lemma 6.3. Under the same assumptions as Lemma 6.2, let u be defined by (6.2). Then, there exists a constant CT > 0 depending only on T such that u satisfies

2 2 2 x12 u(x1 , x2 ) − 21/4 ρ(x1 )e−πx2 +iπx1 x2 − i21/4 x2 ρ (x1 )e−πx2 +iπx1 x2 dx

R2

CT

p 2L∞ (R) R2

(6.7)

,

and

2 2 2 x22 u(x1 , x2 ) − 21/4 ρ(x1 )e−πx2 +iπx1 x2 − i21/4 x2 ρ (x1 )e−πx2 +iπx1 x2 dx

R2

CT

p 2L∞ (R) R4

(6.8)

.

Proof. Here again, we use the Taylor expansion (6.5). Hence, v being defined by (6.6), we have p L∞ π 2 |x1 |v(x1 , x2 ) 1/4 5/2 |x1 |e− 2 x2 2 R π 2 p L∞ 1/4 5/2 e− 2 x2 2 R

1

y12 e− 2 y1 (1 − t)1(−T R,T R) (x1 + ty1 ) dt dy1 π

2

R 0

1

π

y12 e− 2 y1 (1 − t)|x1 + ty1 |1(−T R,T R) (x1 + ty1 ) dt dy1 2

R 0

π 2 p L∞ + 1/4 5/2 e− 2 x2 2 R

1

π

|y1 |3 e− 2 y1 t (1 − t)1(−T R,T R) (x1 + ty1 ) dt dy1 . 2

R 0

Hence, using Jensen’s inequality and arguing as in the proof of Lemma 6.2, we have x1 vL2 (R2 ) C

√ p L∞ 3/2 (RT ) + RT , R 5/2

where C is a universal constant. This implies (6.7). A similar computation gives x2 vL2 (R2 ) C which proves (6.8).

p L∞ √ RT , R 5/2

2

6.2. Lower bound for the energy We first recall an important result by Carlen [6] about wave functions in Λ0 (defined by (1.22)):


Lemma 6.4. (See E.A. Carlen [6].) For any u ∈ Λ0 , ∇u ∈ L2 , and we have ∇|u|2 = π |u|2 . R2

793

(6.9)

R2

Remark 6.5. The result of Carlen is actually much more general than the one we cite here, but the special case (6.9) is the only thing we need. Lemma 6.4 implies the following decomposition of the energy in Λ0 : Lemma 6.6. Let u ∈ Λ0 be such that uL2 = 1. Then, we have E(u) = −

κ2 1 κ2 ∂2 |u|2 + x 2 |u|2 + 2 8π 2 4π 2

+

κ2 8π 2

R2 2 ∂1 |u|2 + ε 2

R2

R2

x12 |u|2 +

g0 2

R2

|u|4 .

(6.10)

R2

Proof. We write

κ2 κ2 κ2 + + E(u) = − 8π 8π 2

x22 |u|2

ε2 + 2

R2

Hence, applying (6.9), we find (6.10).

x12 |u|2

g0 + 2

R2

|u|4 .

(6.11)

R2

2

Note that the first line is easily seen to be bounded from below by the first eigenvalue of the corresponding harmonic oscillator, namely κ 2 /(4π). Hence, (6.10) readily implies E(u) This explains why we chose the constant gives the highest lower bound in (6.12).

κ2 8π

κ2 . 8π

(6.12)

in the decomposition (6.11): it is the constant which

6.3. Proof of Theorem 1.2 Step 1. Upper bound for the energy. We pick a real-valued function p such that p ∈ C 2 (R),

supp(p) ⊂ (−T , T ),

p 2 = 1, R

and define u by (6.2), where ρ is defined by (6.3), with R = ε −2/3 .

(6.13)

794


Hence, setting v =

1 uL2 u,

we know by Lemma 6.1 that v is a test function for I (ε, κ). Hence, I (ε, κ) E(v).

(6.14)

Next, we set v1 = 21/4 ρ(x1 )e−πx2 +iπx1 x2 + i21/4 x2 ρ (x1 )e−πx2 +iπx1 x2 , 2

2

and point out that, applying Lemma 6.2 with r = 2, u2L2

= v1 2L2

+ O ε 4/3 = 1 + 21/2

= 1 + Cε 4/3

ρ (x1 )2

R

2 x22 e−2πx2 dx2 + O ε 4/3

R

p 2 + O ε 4/3 ,

R

where we have used that the two terms defining v1 are orthogonal to each other. Hence, uL2 = 1 + O ε 4/3 , where the term O(ε 4/3 ) depends only on p L2 , p L∞ and T . According to (6.14) and the definition of v, we thus have $ # I (ε, κ) E(u) 1 + O ε 4/3 ,

(6.15)

where the term O(ε 4/3 ) is independent of κ. We now compute the energy of u: applying Lemma 6.3, we have x 2 |u|2 − x 2 |v1 |2 Cε 2/3 x1 u 2 + x1 v1 2 Cε 2/3 2x1 v1 2 + Cε 2/3 . L L L 1 1 R2

R2

Moreover, we have, since ρ is real-valued, 1 2 2 2 2 2 2 −4/3 x1 |v1 | dx = x1 ρ(x1 ) dx1 + x1 ρ (x1 ) dx1 = ε t 2 p(t)2 dt + O(1). 4π R2

R

R

R

Hence, we have x12 |u|2

=ε

−4/3

t 2 p(t)2 dt + O(1).

The same kind of argument allows us to prove that 1 + O ε 4/3 . x22 |u|2 = x22 v12 + O ε 4/3 = 4π R2

(6.16)

R

R2

R2

(6.17)


795

Next, we apply Lemma 6.2 with r = 4: |u|4 − |v1 |4 2u − v1 4 u3 4 + v1 3 4 Cε 3/2 u3 4 + v1 3 4 . L L L L L R2

R2

Moreover, we have uL4 v1 L4 + Cε 2/3 , hence |u|4 − |v1 |4 Cε 3/2 v1 3 4 . L R2

R2

We also have 2 2 2 4 |v1 | = 2ρ(x1 )4 e−4πx2 + 4ρ(x1 )2 ρ (x1 )2 x22 e−4πx2 + 2x24 ρ (x1 )4 e−4πx2 R2

R2

= ε 2/3

p4 + ε2 R

1 4π

p(t)2 p (t)2 dt + ε 10/3

R

3 64π 2

p 4 .

R

Hence, we obtain

|u| = ε 4

2/3

p(t)4 dt + O ε 2 .

(6.18)

R

R2

Collecting (6.16), (6.17) and (6.18), we thus have ' & 2 4/3 κ2 1 2 g0 2/3 2 4 E(u) = +O κ ε t p(t) dt + +ε p(t) dt + O ε 2 . 8π 2 2 R

R

Recalling (6.15), this implies I (ε, κ) − ε 2/3

κ2 8π

1 2

t 2 p(t)2 dt + R

g0 2

p(t)4 dt + O κ 2 ε 2/3 + O ε 4/3 .

R

As a conclusion, we have lim sup 1/3 ε→0, ε κ →0

I (ε, κ) − ε 2/3

κ2 8π

1 2

g0 t p(t) dt + 2 2

R

2

p(t)4 dt, R

for any real-valued p ∈ C 2 (R) having compact support, and such that pL2 = 1. A density argument allows to prove that lim sup 1/3

ε→0, ε κ →0

I (ε, κ) − ε 2/3

κ2 8π

J,

796


where J is defined by (1.37). Thus, we get I (ε, κ) − ε 2/3

κ2 8π

ε 1/3 , = J + c ε, κ

with lim (t,s)→(0,0) c(t, s) = 0. t,s>0

Step 2. Convergence of minimizers. Let u be a minimizer of I (ε, κ). Then, according to the first step, we have ε 1/3 κ2 2/3 2/3 + J ε + ε c ε, , E(u) 8π κ with lim (t,s)→(0,0) c(t, s) = 0. Hence, applying Lemma 6.6, we obtain t,s>0

2 2 κ2 1 g0 2 2 ∂2 |u|2 + x 2 |u|2 + κ ∂1 |u|2 + ε x |u| + |u|4 2 1 2 4π 2 2 2 8π 2 R2

R2

R2

ε 1/3 2/3 2/3 + J ε + ε c ε, . 4π κ

R2

R2

κ2

(6.19)

We set x1 1 v(x1 , x2 ) = 1/3 u 2/3 , x2 , ε ε

(6.20)

so that vL2 = uL2 = 1, v 0, and (6.19) becomes κ2 κ 2 ε 4/3 1 ε 2/3 2 2 2 2 2 2 4 |∂2 v| + x2 v + |∂1 v| + x 1 v + g0 v 2 4π 2 2 8π 2 R2

R2

R2

1/3

κ2 ε + J ε 2/3 + ε 2/3 c ε, 4π κ

R2

R2

(6.21)

.

This implies that

|∂2 v| +

x22 v 2 C,

2

R2

(6.22)

R2

where C does not depend on (ε, κ). Moreover, since the first eigenvalue of the operator 2 − 4π1 2 d 2 + x22 is equal to 1/(2π), (6.21) implies that dx2

x12 v 2 + g0

R2

v 4 C, R2

(6.23)


797

where C does not depend on (ε, κ). Hence, up to extracting a subsequence, v converges weakly in L4 and weakly in L2 to some limit v0 0. Using (6.22) and (6.23), we see that |x|2 v 2 C, R2

hence v converges strongly in L2 . Since in addition ∂2 v converges weakly in L2 , we have: ⎧ v −−−−−−−−−−→ v0 strongly in L2 (R2 ), ⎪ ⎪ ⎪ (ε,ε 1/3 κ −1 )→(0,0) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ x1 v −−−−−−−−−−→ x1 v0 weakly in L (R ), (ε,ε 1/3 κ −1 )→(0,0)

(6.24)

⎪ v −−−−−−−−−−→ v0 weakly in L4 (R2 ), ⎪ ⎪ ⎪ (ε,ε 1/3 κ −1 )→(0,0) ⎪ ⎪ ⎪ ⎪ ⎩ ∂2 v −−−−−−−−−−→ ∂2 v0 weakly in L2 (R2 ). (ε,ε 1/3 κ −1 )→(0,0)

Hence, we may pass to the liminf in the two first terms of (6.21), getting 1 4π 2

|∂2 v0 |2 + R2

x22 v02 R2

lim inf

(ε,ε 1/3 κ −1 )→(0,0)

We use that the first eigenvalue of the operator − 4π1 2

1 4π 2

|∂2 v|2 + R2

d2 dx22

1 . x22 v 2 2π

(6.25)

R2

+ x22 on L2 (R) is equal to 1/(2π), is

simple, with an eigenvector equal to 21/4 exp(−πx22 ). Thus, v0 (x1 , x2 ) = ξ(x1 )21/4 e−πx2 , 2

(6.26)

with ξ 0. Next, (6.21) and (6.24) also imply 1 2

x12 v02 R2

g0 + 2

v04

1 g0 2 2 4 lim inf x1 v + v J. 1/3 2 ε→0, ε →0 2 κ

R2

R2

(6.27)

R2

Using (6.26), we infer 1 2

x12 ξ 2 + R

g0 2

ξ 4 J. R

Hence, recalling that, in view of (6.24) and (6.26), we have ξ 2 = 1, the definition of J implies that ξ is the unique nonnegative minimizer of (1.37). This proves (1.38), with strong convergence in L2 and weak convergence in L4 . Moreover, using (6.27) again and the fact that ξ is a minimizer of (1.37), we have 4 2 2 2 4 lim v0 − v = 0. x 1 v 0 − v + g0 (ε,ε 1/3 κ −1 )→(0,0)

R2

R2

798


Next, using the explicit formula giving v0 , a simple computation gives 2 2 v − v02 , x12 v 2 − v02 + g0 v 4 − v04 g R2

R2

hence v 2 converges to v02 strongly in L2 (R2 ). Thus,

v 4 −→ R2

v04 . R2

The space L4 (R2 ) being uniformly convex, this implies strong convergence in L4 , hence (1.38). Step 3. Lower bound for the energy. Using Lemma 6.6, we have ε 2/3 κ2 + E(u) 4π 2

x12 v 2

+ g0

R2

v . 4

R2

In addition, we already proved (1.38), which implies 1 g0 1 g0 2 2 4 2 2 x1 v + v −→ x1 v 0 + v04 = J, 2 2 2 2 R2

R2

R2

which implies the lower bound for the energy.

R2

2

Acknowledgments We would like to thank A.L.Fetter and J. Dalibard for very useful comments on the physics of the problem. We also acknowledge support from the French ministry grant ANR-BLAN-0238, VoLQuan and express our gratitude to our colleagues participating to this ANR-project, in particular T. Jolicœur and S. Ouvry. Appendix A A.1. Glossary A.1.1. The harmonic oscillator The operator

w π ξj2 + λ2j xj2 = π Dx2j + λ2j xj2 ,

1j n

λj > 0, Dxj =

1j n

1 ∂x , 2iπ j

(A.1)

has a discrete spectrum 1 λj + αj λj , 2 (α1 ,...,αn )∈Nn 1j n

1j n

(A.2)


799

and its ground state is one-dimensional generated by the Gaussian function

ϕλ (x) = 2n/4

λj e−πλj xj . 2

1/4

(A.3)

1j n

A.1.2. Degenerate harmonic oscillator Let r ∈ {1, . . . , n}. Using the identity Hr u, u =

(Dx − iλj xj )u2 2 + λj u2 2 , (A.4) Dx2j + λ2j xj2 u, u = j L L 2π

1j r

1j r

we can define the ground state Er of the operator Hr as Er = L2 Rn ∩1j r ker(Dxj − iλj xj ) = ϕ(λ1 ,...,λr ) (x1 , . . . , xr ) ⊗ v(xr+1 , . . . , xn ) v∈L2 (Rn−r ) . The bottom of the spectrum of πHr is

1 2

,

1j r

(A.5)

λj .

A.2. Notations for the calculations of Section 2.3

α=

ν 2 + ω2 + ε 2 = 1, ν 2 + ω2 1, ν 4 + 4ω2 = 4ω2 + (1 − ω2 − ε 2 )2 (if ν = 0, α = 2ω).

μ21 = 1 + ω2 − α =

(1 + ω2 )2

− α2

1 + ω2 + α

μ22 = 1 + ω2 + α

=

(1 − ω2 )2

− ν4

μ22

=

2ν 2 ε 2

+ ε4

μ22

(if ν = 0, μ2 = 1 + ω).

(A.6) (A.7) (A.8) (A.9)

Remark A.1. If ν = 0, μ1 = O(ε 2 ) and if ν = 0, μ1 = O(ε). Moreover, for ν 2 + ω2 1, μ22 ∈ [1, 4] and for ν 2 + ω2 = 1, μ22 ∈ [2, 4]: we have indeed 1/2 1 1 + ω2 + ν 4 + 4ω2 4

(A.10)

since ν 4 + 10ω2 (1 − ω2 )2 + 10ω2 = 8ω2 + 1 + ω4 9 + ω4 , implying (3 − ω2 )2 ν 4 + 4ω2 and (A.10). If ν 2 + ω2 = 1, we have (1 − ω2 )2 = ν 4 ν 4 + 4ω2 ⇒ 2 1 + ω2 + (ν 4 + 4ω2 )1/2 . We define the following set of parameters, 2ωμ1 α − 2ω2 − ν 2 = 2 2 2ωμ1 α − 2ω + ν 2 since α − 2ω2 − ν 4 = 4ω2 + 4ω4 − 4ω2 α = 4ω2 μ21 ,

β1 =

(A.11)

800


2ωμ2 α + 2ω2 − ν 2 = 2 2 2ωμ2 α + 2ω + ν 2 since α + 2ω2 − ν 4 = 4ω2 + 4ω4 + 4ω2 α = 4ω2 μ22 ,

β2 =

γ= λ21 = λ22 =

μ1 1 = = β μ1 + β1 β2 μ2 1 + 1 β2 μ2 1+ μ1

μ2 1 = = β μ2 + β1 β2 μ1 1 + 1 β2 μ1 1+ μ2 λ21 + λ22 = 1 + d=

2α , ω

γ λ1 λ2 , 2

c=

ν2 , α

λ21 + λ22 2λ1 λ2

1 α−2ω2 −ν 2 α+2ω2 +ν 2

λ21 λ22 =

(A.12) (A.13)

1 α+2ω2 −ν 2 α−2ω2 +ν 2

=

α − 2ω2 + ν 2 , 2α

(A.14)

α + 2ω2 + ν 2 , 2α

and we have (A.15)

(α + ν 2 )2 − 4ω4 , 4α 2

(A.16)

=

so that cd =

2α(1 + ν 2 /α) α + ν 2 = . 4ω 2ω

(A.17)

We have also 2μ1 α − 2ω2 + ν 2 α − 2ω2 + ν 2 = = = λ21 , γβ1 ωγ 2α 2μ2 α + 2ω2 + ν 2 α + 2ω2 + ν 2 = = λ22 , = γβ2 ωγ 2α and λ21 + λ22 21/2 α 1/2 = 1 + ν 2 α −1 2−1 √ 2λ1 α − 2ω2 + ν 2 √ 21/2 α 1/2 α + 2ω2 − ν 2 = 1 + ν 2 α −1 2−1 √ √ α − 2ω2 + ν 2 α + 2ω2 − ν 2 √ α + 2ω2 − ν 2 2 −1 −1 1/2 1/2 = 1+ν α 2 2 α α 2 − (2ω2 − ν 2 )2 −1/2 = 1 + ν 2 α −1 2−1/2 α 1/2 α + 2ω2 − ν 2 (2ω)−1 2ν 2 + ε 2 .

cλ2 =

Moreover, we have % α + 2ω2 − ν 2 cλ2 = 2−3/2 α 1/2 + ν 2 α −1/2 ω−1 2ν 2 + ε 2 if ν = 0, cλ2 = 2−1/2 (1 − ω)−1/2 , % α + 2ω2 − ν 2 2ω cλ 2 = 2−3/2 α 1/2 + ν 2 α −1/2 ω−1 λ2 d −1 = , cd 2ν 2 + ε 2 α + ν 2

(A.18)


λ2 d

−1

=2

−1/2

% λ2 d

−1

−1/2

= (2α) cλ1 =

801

% 1/2 α + 2ω2 − ν 2 −1 2 −1/2 α +ν α α + ν2 , 2 2 2ν + ε

α + 2ω2 − ν 2 2ν 2 + ε 2

if ν = 0, λ2 d −1 = 2−1/2 (1 − ω)−1/2 ,

−1/2 λ21 + λ22 = 1 + α −1 ν 2 2−1/2 α 1/2 α + 2ω2 + ν 2 2λ2 if ν = 0, cλ1 = 2−1/2 (1 + ω)−1/2 ,

(A.19)

(A.20)

−1/2 −1 λ1 d −1 = λ1 c(cd)−1 = 1 + α −1 ν 2 2−1/2 α 1/2 α + 2ω2 + ν 2 2ω α + ν 2 , −1/2 λ1 d −1 = 21/2 α −1/2 ω α + 2ω2 + ν 2 if ν = 0, λ1 d −1 = 2−1/2 (1 + ω)−1/2 ,

(A.21)

1/2 −1/2 −1/2 λ1 cd = α + ν 2 2−1 ω−1 α − 2ω2 + ν 2 2 α , 1/2 λ1 cd = 2−3/2 α + ν 2 ω−1 α −1/2 α − 2ω2 + ν 2 if ν = 0, λ1 cd = 2−1/2 (1 − ω)1/2 , 1/2 −1/2 −1/2 d γ λ1 = αω−1 α − 2ω2 + ν 2 = 2 α λ2 2 1/2 = 2−1/2 α 1/2 ω−1 α − 2ω2 + ν 2 , λ1 cd −

1/2 −3/2 d 2 = α − 2ω2 + ν 2 (α + ν 2 )ω−1 α −1/2 − 2−1/2 α 1/2 ω−1 , λ2

λ1 cd −

1/2 d α + ν 2 − 2α , = 2−3/2 ω−1 α −1/2 α − 2ω2 + ν 2 λ2

λ1 cd −

1/2 d α − ν2 , = −2−3/2 ω−1 α −1/2 α − 2ω2 + ν 2 λ2 if ν = 0, λ1 cd − λd2 = −2−1/2 (1 − ω)−1/2 ,

1/2 λ1 = 2−1/2 α −1/2 α − 2ω2 + ν 2

if ν = 0, λ1 = 2−1/2 (1 − ω)1/2 ,

d d −1 = λ1 λ2 λ1 cd − λ2 cd − λ1 λ2 1/2 = −2−3/2 ω−1 α −1/2 α − 2ω2 + ν 2 α − ν2 1/2 −1/2 × α + 2ω2 + ν 2 α − 2ω2 + ν 2 1/2 = −2−3/2 ω−1 α −1/2 α − ν 2 α + 2ω2 + ν 2 ,

(A.22) (A.23)

802


1/2 d = −2−3/2 ω−1 α −1/2 α − ν 2 α + 2ω2 + ν 2 λ1 if ν = 0, λ2 cd − λd1 = −2−1/2 (1 + ω)1/2 , 1/2 if ν = 0, λ2 = 2−1/2 (1 + ω)1/2 , λ2 = 2−1/2 α −1/2 α + 2ω2 + ν 2 λ2 cd −

(A.24) (A.25)

γ μ1 4αω(2ν 2 + ε 2 ) = . 2β1 α − ν 2 + 2ω2

2α γ μ1 β1 = ε2 , 2 α + 2ω2 + ν 2

(A.26)

A.3. Some calculations A.3.1. Proof of Lemma 2.5 We have to calculate ⎛

1 − ν2 ⎜ " = χ ∗ Qχ = χ ∗ ⎜ 0 Q ⎝ 0 −ω ⎛ ⎜ ⎜ =χ ⎜ ⎝

(1 − ν 2 )λ1 −

−ωλ1 +

⎛

λ1 ⎜ ⎜ 0 =⎜ ⎜ ⎝ 0 − λd1 ⎜ ⎜ ×⎜ ⎜ ⎝

(1 + ν 2 )λ2 + ωλ2 + − λ1 cd

0

0 d λ1

λ2 − λd2 0

(1 − ν 2 )λ1 −

ωd λ2

−ωλ1 +

− λ1 cd 0

cλ2

0

0

cλ1

+ λ1 cdω

0

0

λ2

− λd2

d λ1

0 d λ2

− λ2 cd

− λ1 cd

0

− λ2 cdω

− λ2 cd

2 )λ

− (1+νd

2

− λd1 ⎞ 0 ⎟ ⎟ ⎟ 0 ⎠

cλ2 0

cλ1

− λ1 (1−ν d

2)

+ ωcλ2

0

⎞

− cλ1 ω

0

2 − ωλ d + cλ2

0 ωλ1 d

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

+ cλ1

⎟ ⎟ ⎟ ⎟ ⎠ 0

ωλ2 + − λ1 cd

0

0 ωd λ1

d λ1

(1 + ν 2 )λ2 +

0

λ1

0 d λ2

− λ2 cd

0 d λ2

⎞⎛ −ω ⎜ 0 ⎟ ⎟⎜ ⎜ 0 ⎠⎝ 1 0

0 d λ2

0 ω 1 0

+ λ1 cdω

0

∗⎜

⎛

ωd λ2

0 1 + ν2 ω 0

d λ1

0

− λ1 (1−ν d

0 ωd λ1

− λ2 cdω

− λ2 cd

2 )λ 2

− (1+νd

+ ωcλ2

− cλ1 ω

0

2 − ωλ d + cλ2

0

2)

0 ωλ1 d

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

+ cλ1

˜ is diagonal, it is We get easily q˜12 = q˜13 = 0 = q˜24 = q˜34 . To prove that the symmetric matrix Q thus sufficient to prove that q˜14 = 0 = q˜23 . We have λ21 λ1 cdλ1 1 − ν 2 − ωcλ21 + ω + − λ21 cω − c2 λ21 d d λ2 λ2 λ21 ωd cd 2 2 2 2 −1 + ν − 2ωcd + + −c d = d λ2 λ1 λ2 λ1 λ2 (α + ν 2 ) α (α + ν 2 )2 = 1 −1 + ν 2 − α − ν 2 + α + − d 2ω ω 4ω2

q˜14 = −


λ21 −ω2 + dω2 λ2 = 12 −ω2 + dω λ2 = 12 −ω2 + dω λ2 = 12 −ω2 + dω

=

(α 2 + ν 2 α) (α + ν 2 )2 − 2 4

803

(ν 4 + 4ω2 + ν 2 α) (α 2 + ν 4 + 2αν 2 ) − 2 4

(ν 4 + 4ω2 + ν 2 α) (ν 4 + 4ω2 + ν 4 + 2αν 2 ) − 2 4 (2ν 4 + 8ω2 + 2ν 2 α) (2ν 4 + 4ω2 + 2αν 2 ) − = 0. 4 4

Moreover we have λ22 ωλ2 cdλ2 1 + ν 2 + ωcλ22 − + + λ22 ωc − λ22 c2 d d λ1 λ1 λ2 ωd cd 2 + − c2 d 2 = 2 −1 − ν 2 + 2ωcd − d λ2 λ1 λ2 λ1 2 λ (α + ν 2 ) α (α + ν 2 )2 = 2 −1 − ν 2 + α + ν 2 − α + − d 2ω ω 4ω2 λ2 (α 2 + ν 2 α) (α + ν 2 )2 = 22 −ω2 + − = 0, from the previous computation. 2 4 dω

q˜23 = −

˜ is indeed diagonal. We calculate We know now that Q λ21 (1 − ν 2 ) 2cλ21 ω λ21 2 2 2 2 2 1 − ν + c + λ = + 2ωcd + c d 1 d d2 d2 λ2 (α + ν 2 )2 , = 12 1 − ν 2 + α + ν 2 + d 4ω2 λ2 λ2 (ν 4 + 4ω2 + ν 4 + 2αν 2 ) (ν 4 + αν 2 ) q˜44 = 2 1 2 ω2 + αω2 + = 2 1 2 2ω2 + αω2 + . 4 2 ω d ω d

q˜44 =

Since

λ21 ω2 d 2

=

4 γ 2 λ22 ω2

q˜44 =

=

1 α 2 λ22

=

2α , α 2 (α+2ω2 +ν 2 )

we have

# 2 $ α 2 + 2αω2 + αν 2 1 2 4 2 4ω = 2 + 2αω + ν + αν = 1. α(α + 2ω2 + ν 2 ) α + 2αω2 + αν 2

Analogously, we have λ22 (1 + ν 2 ) 2cλ22 ω λ22 2 2 2 2 2 1 + ν − λ = − 2ωcd + c d + c 2 d d2 d2 λ2 (α + ν 2 )2 = 22 1 + ν 2 − α − ν 2 + , d 4ω2

q˜33 =

804


λ22 (ν 4 + 4ω2 + ν 4 + 2αν 2 ) 2 2 ω − αω + 4 ω2 d 2 λ2 (ν 4 + αν 2 ) . = 2 2 2 2ω2 − αω2 + 2 ω d

q˜33 =

Since

λ22 ω2 d 2

=

4 γ 2 λ21 ω2

q˜33 =

=

1 α 2 λ21

=

2α , α 2 (α−2ω2 +ν 2 )

we have

# 2 $ α 2 − 2αω2 + αν 2 1 2 4 2 4ω = 2 − 2αω + ν + αν = 1. α(α − 2ω2 + ν 2 ) α − 2αω2 + αν 2

We calculate ωdλ1 d2 cd 2 λ1 + 2λ21 cdω + 2 − 2 + λ21 c2 d 2 , q˜11 = λ21 1 − ν 2 − 2 λ2 λ2 λ2 2ωd d2 cd 2 2 2 2 2 q˜11 = λ1 1 − ν − + 2cdω + 2 2 − 2 +c d , λ1 λ2 λ1 λ2 λ1 λ2 α2 α + ν 2 α (α + ν 2 )2 2 2 2 q˜11 = λ1 1 − ν − 2α + α + ν + 2 − 2 , + 2ω ω ω 4ω2 λ21 # (α + ν 2 )2 $ 2 2 2 2 , (1 − α)ω + α − α − αν + 4 ω2 α − 2ω2 + ν 2 2 ν 4 + 4ω2 + ν 4 + 2αν 2 2 2 q˜11 = − αω − αν + ω , 4 2αω2 α − 2ω2 + ν 2 1 2 ν4 2 2 2ω − αω − αν + . q˜11 = 2 2 2αω2 q˜11 =

More calculations: ν4 ν2 α − 2ω2 + ν 2 2ω2 + − α ω2 + 2 2 4 ν ν2 − ν 4 + 4ω2 ω2 + = ν 2 − 2ω2 2ω2 + 2 2 4 2 ν ν 2 + ω2 + 2ω − ν 2 + α 2ω2 + 2 2 4 4 2 4 = −8ω − 2ω ν + α 2ω + 2ω2 which is equal to 2αω2 1 + ω2 − α = α 2ω4 + 2ω2 − 2α 2 ω2 = α 2ω4 + 2ω2 − 2ω2 ν 4 + 4ω2 , proving thus that q˜11 = 1 + ω2 − α. The previous calculations and (2.8) give ϕ q˜22 = 1 + ω2 + α, completing the proof of the lemma.


805

A.3.2. On the symplectic relationships in Lemma 2.6 The reader is invited to check the following formulas,4 with the notations of Lemma 2.6: α − ν2 α + ν2 x 2 , ξ2 + x1 = αω−1 , ξ1 − 2ω 2ω α − ν2 α + ν2 ξ2 − x 1 , ξ1 + x2 = αω−1 , 2ω 2ω α − ν2 α + ν2 ξ1 − x 2 , ξ1 + x2 = 0, 2ω 2ω α − ν2 α − ν2 ξ1 − x 2 , ξ2 − x1 = 0, 2ω 2ω α + ν2 α + ν2 ξ2 + x 1 , ξ1 + x2 = 0, 2ω 2ω α + ν2 α − ν2 x 1 , ξ2 − x1 = 0, ξ2 + 2ω 2ω as well as

α + 2ω2 − ν 2 2 1/2 −1 ε αω 2αμ22 2 1/2 −1 2 α − 2ω2 − ν 2 = 2−1 εμ−1 2 ω 2 1/2 −1 2 2 1/2 4ω − 4ω4 + 4ω2 ν 2 = 2−1 εμ−1 = εμ−1 2 ω 2 1−ω +ν 2 2 1/2 = εμ−1 = μ1 2 2ν + ε

α − 2ω2 + ν 2 2α

1/2

and

α + 2ω2 + ν 2 2α

1/2

2

1/2

1 + ω2 + α ω α(α + 2ω2 + ν 2 )

1/2

1/2 αω−1 = 1 + ω2 + α = μ2 .

References [1] A.A. Abrikosov, On the Magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174–1182. [2] A. Aftalion, Vortices in Bose Einstein Condensates, Progr. Nonlinear Differential Equations Appl., vol. 67, Birkhäuser, 2006. [3] A. Aftalion, X. Blanc, F. Nier, Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates, J. Funct. Anal. 241 (2) (2006) 661–702, MR MR2271933 (2008c:82052). [4] A. Aftalion, X. Blanc, Reduced energy functionals for a three-dimensional fast rotating Bose Einstein condensates, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2) (2008) 339–355 (in English), MR MR2400105. [5] V. Bretin, S. Stock, Y. Seurin, J. Dalibard, Fast rotation of a Bose–Einstein condensate, Phys. Rev. Lett. 92 (2004) 050403. 4 This is indeed double-checking since those formulas are proven in Section 2.

806


[6] Eric A. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal. 97 (1) (1991) 231–249, MR MR1105661 (92i:46025). [7] A.L. Fetter, Lowest-Landau-level description of a Bose–Einstein condensate in a rapidly rotating anisotropic trap, Phys. Rev. A 75 (2007) 013620. [8] T.L. Ho, Bose–Einstein condensates with large number of vortices, Phys. Rev. Lett. 87 (2001) 060403. [9] Lars Hörmander, The analysis of linear partial differential operators. III, Classics Math., Springer, Berlin, 2007, Pseudo-differential operators, reprint of the 1994 edition, MR MR2304165 (2007k:35006). [10] Nicolas Lerner, The Wick calculus of pseudo-differential operators and some of its applications, Cubo Mat. Educ. 5 (1) (2003) 213–236, MR MR1957713 (2004a:47058). [11] K. Madison, F. Chevy, V. Bretin, J. Dalibard, Vortex formation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 84 (2000) 806. [12] Dusa McDuff, Dietmar Salamon, Introduction to Symplectic Topology, second ed., Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1998, MR MR1698616 (2000g:53098). [13] M.Ö. Oktel, Vortex lattice of a bose-einstein condensate in a rotating anisotropic trap, Phys. Rev. A 69 (2) (2004) 023618. [14] C.J. Pethick, H. Smith, Bose Einstein Condensation in Dilute Gases, Cambridge Univ. Press, 2002. [15] L. Pitaevskii, S. Stringari, Bose Einstein Condensation, Internat. Ser. Monogr. Phys., vol. 116, Oxford Science Publications, 2003. [16] P. Sanchez-Lotero, J.J. Palacios, Vortices in a rotating bose-einstein condensate under extreme elongation, Phys. Rev. A (Atomic, Molecular, and Optical Physics) 72 (4) (2005) 043613. [17] S. Sinha, G.V. Shlyapnikov, Two-dimensional bose-einstein condensate under extreme rotation, Phys. Rev. Lett. 94 (15) (2005) 150401. [18] G. Watanabe, G. Baym, C.J. Pethick, Landau levels and the Thomas–Fermi structure of rapidly rotating Bose– Einstein condensates, Phys. Rev. Lett. 93 (2004) 190401.


On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary J.I. Díaz a,∗ , J.M. Rakotoson b a Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias No. 3,

28040 Madrid, Spain b Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie,

Téléport 2, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France Received 7 December 2008; accepted 8 March 2009 Available online 1 April 2009 Communicated by H. Brezis

Abstract We study the differentiability of very weak solutions v ∈ L1 (Ω) of (v, L ϕ)0 = (f, ϕ)0 for all ϕ ∈ C 2 (Ω) vanishing at the boundary whenever f is in L1 (Ω, δ), with δ = dist(x, ∂Ω), and L∗ is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function f of f . © 2009 Elsevier Inc. All rights reserved. Keywords: Very weak solutions; Distance to the boundary; Regularity; Linear PDE; Monotone rearrangement; Lorentz spaces

1. Introduction The origin of this paper starts with an originally unpublished manuscript by H. Brezis (personal communication of him to the first author [4]), later mostly in the paper by Brezis et al. [5] (see also the mention made in [17]). In his note, when f is given in L1 (Ω, dist(x, ∂Ω)) * Corresponding author.

E-mail address: [email protected] (J.I. Díaz). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.002

808

J.I. Díaz, J.M. Rakotoson / Journal of Functional Analysis 257 (2009) 807–831

(Ω bounded smooth open set of RN ), H. Brezis shows the existence and uniqueness of a very weak solution v ∈ L1 (Ω) of − Ω vϕ dx = Ω f ϕ dx, ∀ϕ ∈ V1 (Ω), GD(Ω) = with V1 (Ω) = {ϕ ∈ C 2 (Ω), ϕ = 0 on ∂Ω}, and also that |v|L1 (Ω) c|f |L1 (Ω,dist(x,∂Ω)) . ∂v Therefore, the question of the integrability of the generalized derivative ∂i v = ∂x arises in a i natural way and was raised already in the note by H. Brezis. To give some answer to the above question, we shall note δ(x) = dist(x, ∂Ω) and introduce the following spaces

L1 Ω, δ α = f : Ω → R Lebesgue measurable: f (x)δ(x)α dx is finite , 0 α 1,

L Ω, δ 1 = L1 (Ω, δ), 1

Ω

L1 Ω, δ 0 = L1 (Ω),

L1 Ω, δ| Ln δ| = f : Ω → R Lebesgue measurable such that |f |(x)δ(x) Ln δ(x) dx < +∞ . Ω

One has, for α ∈ [0, 1[ L1 Ω, δ α L1 Ω, δ| Ln δ| L1 (Ω, δ). One of our results contains in particular the following statements: 1,q

(i) The very weak solution v ∈ W0 (Ω) for some q > 1 if and only if f ∈ L1 (Ω, δ α ) for some α ∈ [0, 1[, for nonnegative f . N (ii) If f ∈ L1 (Ω, δ α ), 0 α < 1 then |∇v| belongs to the Lorentz space L N−1+α ,∞ (Ω). 1

The above result contains the result given in [12] since Lp (Ω, δ) L1 (Ω, δ p ) for p > 1. We also improve the result of Cabré and Martel [6], by showing that if f is only in L1 (Ω, δ) then N the function v is in L N−1 ,∞ (Ω). Moreover, we can show that |∇v| ∈ Lq (Ω, δ) for some q > 1, 1,q in particular v ∈ Wloc (Ω). As a matter of fact, all our results in the first four sections are valid when we replace the Laplacian operator by a linear elliptic second order operator L with variable coefficients. In Section 5, we consider the case of L∗ = − and Ω being the unit ball. Our aim is to study if we may have the W 1,1 -regularity whenever f ∈ L1 (Ω, δ) − L1 Ω, δ 1− ,

L1 Ω, δ 1− = L1 Ω, δ α . 0α 0 |∇ω|L1 (Ω) c(Ω) · |f |L1 (Ω,δ) . Under our assumptions on Ω, 1

c(Ω) =

1+ N1

.

N αN

We shall restate the necessary and sufficient conditions to ensure that ω ∈ W 1,q (Ω) for q > 1 and we shall show that

ω

1,q ∈ W0 (Ω)

if and only if

q |Ω| |Ω| f∗ (t) dt dσ is finite. 0

σ

We also remark that the usual comparison technique based on the decreasing rearrangement f∗ (t) of f 0, is inefficient in the case where f ∈ L1 (Ω, δ) \ L1 (Ω). Indeed, the function |Ω| U (x) = cN

t αN |x|N

2 N −2

t f∗ (σ ) dσ 0

is in L1 (Ω) if and only if f ∈ L1 (Ω). In any case the pointwise comparison v U and the comparison in mass (see, e.g. the results and references presented in Section 1.3 of [7]) are still true (but they do not give any information on the integrability of v). We end the paper by giving two applications of our differentiability results to two special data f which are in L1 (Ω, δ) but not in L1 (Ω) nor in L1 (Ω, δ α ), respectively. The application to the existence, uniqueness and qualitative properties of the very weak solution of some associate semilinear problem will be the object of a separate paper by the authors (Díaz and Rakotoson [8]). 2. Notation – preliminary results We shall always consider Ω ⊂ RN , N 2, a bounded open set of class C 2,1 . For any measurable set E ⊂ RN we shall denote by |E| its Lebesgue measure.

810


We shall consider a linear operator L:

Lu = −

N

N ∂i aij (x)∂j u + bi (x)∂i u + c0 (x)u

i,j =1

i=1

under the same assumptions as in [9], say aij ∈ C 0,1 (Ω), bi ∈ C 0,1 (Ω), c0 ∈ L∞ (Ω), c0 0, ∀ξ = (ξ1 , . . . , ξN ) ∈ RN

1 i ∂i b (x) 0 a.e. in Ω. 2 N

aij (x)ξi ξj α|ξ |2

c0 (x) −

for some α > 0,

i,j

i=1

We shall use the adjoint operator associated to L, that is L∗ ϕ = −

N ∂j aij (x)∂i ϕ − ∂i bi ϕ + c0 (x)ϕ.

i,j

i=1

Remark 1. The case of unbounded term c0 (x), blowing up on the boundary, will be considered in a subsequent paper by the authors (Díaz and Rakotoson [8]) where in fact the general framework will concern the case of semilinear equations. We recall that: – the decreasing rearrangement of a measurable function u is given by u∗ : Ω∗ = 0, |Ω| → R,

u∗ (s) = inf t ∈ R: |u > t| s , u∗ |Ω| = ess inf u; u∗ (0) = ess sup u, Ω

Ω

having the – the decreasing radial rearrangement of the function u is defined, on the ball Ω same measure as Ω, by → R, u:Ω

u(x) = u∗ αN |x|N ;

– the increasing rearrangement of a measurable function u is given by u∗ : Ω∗ → R,

u∗ (s) = u∗ |Ω| − s ,

s ∈ 0, |Ω| ;

– the increasing radial rearrangement of the function u is defined by → R, u:Ω

u(x) = u∗ αN |x|N .


811

We shall use the following Lorentz spaces (see [14,1] for example), for 1 < p < +∞, 1 q +∞ L

p,q

(Ω) = v : Ω → R measurable

q |v|Lp,q

|Ω| 1 q dt t p |v|∗∗ (t) = < +∞ , t 0

for q < +∞ 1 Lp,∞ (Ω) = v : Ω → R measurable |v|Lp,∞ = sup t p |v|∗∗ (t) < +∞ , t|Ω|

χE is the characteristic function of a set E ⊂ Ω and |v|∗∗ (t) = 1t ]0, |Ω|[. 2 We denote by ∂i = ∂x∂ i , ∂ij = ∂x∂i ∂xj . We define the following sets

t 0

|v|∗ (s) ds for t ∈ Ω∗ =

W 1 Ω, | · |p,q = v ∈ W 1,1 (Ω): |∇v| ∈ Lp,q (Ω) and W 2 Ω, | · |p,q = v ∈ W 2,1 (Ω): ∂ij v ∈ Lp,q (Ω) for (i, j ) ∈ {1, . . . , N }2 . We shall denote by c various constants depending only on the data. The notation ≈ stands for equivalence of nonnegative quantities, that is Λ1 ≈ Λ2

⇐⇒

∃c1 > 0, c2 > 0 such that c1 Λ2 Λ1 c2 Λ2 .

We first extend the Agmon–Douglis–Nirenberg theorem to Lorentz spaces. Lemma 1. Consider L∗ the above linear operator. There exists a constant c(Ω, L∗ ) > 0 such that ∀g ∈ LN,1 (Ω) there exists a function ϕ ∈ W 2 (Ω, | · |N,1 ) ∩ H01 (Ω) satisfying L∗ ϕ = g, and |ϕ|H 1 + Max |∂ij ϕ|LN,1 c(Ω, L∗ )|g|LN,1 . i,j

Proof. For g ∈ L2 (Ω), we know (see [9]) that there exists a unique function ϕ ∈ H 2 (Ω) ∩ H01 (Ω) such that L∗ ϕ = g. This defines a continuous linear operator A from L2 (Ω) into H 2 (Ω) by setting Ag = ϕ. Let (i, j ) ∈ {1, . . . , N}2 , x ∈ Ω, we define Tij g(x) = ∂ij Ag(x),

812


Tij is a linear map acting continuously from Lp (Ω) into Lp (Ω) for all p ∈ [2, +∞[ according to Agmon–Douglis–Nirenberg’s theorem [9], we derive from Marcinkiewicz’s interpolation theorem (see [1]) that it maps continuously LN,1 (Ω) into LN,1 (Ω):

|Tij g|LN,1 c(Ω, L∗ )|g|LN,1 .

This shows the lemma with the fact that |∇Ag|L2 c|g|L2 c|g|LN,1 (Ω) .

2

3. General result for f ∈ L1 (Ω, δ) The following existence theorem follows the idea of H. Brezis [4] and the regularity improves the one obtained in [6] for the case L = δ and in [17] for the case of a general operator L. Theorem 1. Let f ∈ L1 (Ω, δ) and N = satisfying DGL (Ω) :

N N −1 . Then there exists a unique function v

∗

vL ϕ dx = Ω

f ϕ dx,

∈ LN ,∞ (Ω)

∀ϕ ∈ W 2 Ω, | · |N,1 ∩ H01 (Ω).

Ω

Moreover, there exists a constant c(Ω, L) > 0 such that |v|LN ,∞ c(Ω, L)|f |L1 (Ω,δ) .

(1)

Proof. For k 1, we define the usual truncation Tk (σ ) =

σ if |σ | k, k sign(σ ) otherwise,

σ ∈ R.

We set fk = Tk (f ) ∈ L1 (Ω, δ) ∩ L∞ (Ω) and fk → f in L1 (Ω, δ). By standard result there exists a unique function vk ∈ W 2,p (Ω) ∩ H01 (Ω),

∀p ∈ [1, +∞[:

Lvk = fk .

Next we want to show that vk is a Cauchy sequence in LN ,∞ (Ω). For n 1, k 1, we set v nk = vn − vk , f nk = fn − fk . Then Lv nk = f nk which implies that ∀ϕ ∈ H 2 (Ω) ∩ H01 (Ω) Ω

v nk L∗ ϕ dx =

f nk ϕ dx.

(2)

Ω

For any E measurable in Ω, there exists a function ϕE ∈ W 2,p (Ω) ∩ H01 (Ω) such that L∗ ϕE = χE sign v nk .

(3)


813

From Sobolev embedding associated to Lorentz spaces (see [14]), we have |∇ϕE |∞ c(Ω) Max |∂ij ϕE |LN,1 + |ϕE |H 1 i,j

(4)

and using Lemma 1, we derive that 1

|∇ϕE |∞ c|χE |LN,1 c|E| N .

(5)

ϕE (x) δ(x) c|∇ϕE |∞ ,

(6)

Since ∀x ∈ Ω, we have

and from relations (5) and (6), we get 1 |ϕE (x)| c|E| N , δ(x)

∀x ∈ Ω.

(7)

From relations (2) and (3), we derive |vn − vk | dx = f nk ϕE dx.

(8)

Ω

E

From relations (7) and (8), we have 1 N |vn − vk | dx c|E| |fn − fk |(x)δ(x) dx

(9)

Ω

E

for all E measurable sets in Ω. Using the Hardy–Littlewood inequality (see [14,1]), we have 1 sup t 1− N |vn − vk |∗∗ (t) c|fn − fk |L1 (Ω,δ) .

(10)

t|Ω|

This shows the result. Knowing that LN ,∞ is the dual and associate space of LN,1 we pass to the limit in relation that vk L∗ ψ dx = fk ψ dx, ∀ψ ∈ W 2 Ω, | · |N,1 ∩ H01 (Ω) (11) Ω

as k → +∞ to derive the result.

Ω

2

Next, we want to show that the solution is in W 1,q (Ω, δ) for 0 = bi (x) and aij = aj i .

2N 2N −1

> q provided that c(x) =

814


Lemma 2. Assume L is the self-adjoint uniformly elliptic operator L = − i,j ∂i (ai,j (·)∂j ). Then there exists a function ϕ1 ∈ W 2,p (Ω) ∩ H01 (Ω) and λ1 > 0, ∀p ∈ ]1, +∞[ satisfying

Lϕ1 = λ1 ϕ1 ϕ1 = 0.

in Ω,

Moreover, there are two constants c1 > 0, c2 > 0 such that c1 δ(x) ϕ1 (x) c2 δ(x)

∀x ∈ Ω.

Proof. The proof of the existence is classical (see [15,5]). The estimate is a consequence of Hopf lemma and can be proved as in the case L = − (see [12,2]). 2 Theorem 2. Under the same assumptions as for Lemma 2, the unique generalized function v given in Theorem 1 belongs to W 1,q (Ω, δ) for 1 q < 2N2N−1 . Proof. Let us show that the sequence vk ∈ W 2,p (Ω) ∩ H01 (Ω) ∀p ∈ [1, +∞[ solution of Lvk = fk is a Cauchy sequence in W 1,q (Ω, δ). For η > 0 (that we shall choose later), we consider σ φη (σ ) = 0

dt (1 + t 2 )

1+η 2

σ ∈ R,

,

the function ψk = φη (vk )ϕ1 , with ϕ1 the first eigenfunction associated to L. Then,

aij (x)∂i vk ∂j ψk dx =

i,j Ω

(12)

fk ψk dx. Ω

Using the coercivity condition on aij , we have α Ω

|∇vk |2 (1 + vk2 )

1+η 2

ϕ1 dx +

vk

i,j Ω

φη (t) dt dx

aij (x)∂j ϕ1 ∂i

fk ψk dx.

(13)

Ω

0

We have

i,j Ω

vk

φη (t) dt dx =

aij (x)∂j ϕ1 ∂i 0

vk

φη (t) dt Lϕ1 dx Ω

0

vk

= λ1

ϕ1 Ω

φη (t) dt dx.

0

(14)


815

From relations (13) and (14), we derive using Lemma 2 Ω

|∇vk |2 (1 + vk2 )

1+η 2

δ(x) dx c|φη |∞ Ω

c(η)

ϕ1 |vk | dx +

fk ϕ1 dx Ω

fk (x)δ(x) dx.

(15)

Ω

We conclude as in [13] (see also [3] for another proof), using Hölder inequality, with q ∈ [1, 2N2N−1 [, we have 1− q 2 m |∇vk | (x)δ(x) dx c|fk |L1 (Ω,δ) 1 + |vk | (x)δ(x) dx

q 2

q

Ω

with m = Since

(1+η)q 2−q

(16)

Ω

0 |∇v|

N

L N−1+α

,∞

c(Ω, L)|f |L1 (Ω,δ α ) .

816


The proof relies on the following result which is a consequence of Simader’s result [16]. Lemma 3. For u ∈ L1loc (Ω), we consider a measurable vector field H (u) ∈ L∞ (Ω)N . 1,p For any function g ∈ Lp (Ω), 2 p < +∞ there exists a unique function ϕ ∈ W0 (Ω) such that . B(ϕ, ψ) = aij (x)∂i ϕ∂j ψ dx + bi (x)ϕ∂i ψ dx + c0 ϕψ dx i,j Ω

Ω

=

g(x)H (u) · ∇ψ dx,

i

Ω 1,p

∀ψ ∈ W0

(Ω),

1 1 + = 1. p p

Ω

Moreover, there exists a constant c = c(Ω, L, p) > 0 (independent of ϕ) such that |ϕ|W 1,p (Ω) cH (u)g Lp (Ω) .

(18)

0

Proof. If p = 2 it is a consequence of Lax–Milgram theorem. We notice that B(ϕ, ϕ) α|∇ϕ|2L2

+ Ω

N 1 ∂bi 2 c0 − ϕ dx α|∇ϕ|2L2 2 ∂xi i=1

for ϕ ∈ H01 (Ω) and then 2 |∇ϕ|2L2 + |ϕ|2L2 cH (u)g L2 (Ω) .

(19)

If p > 2, we apply Simader’s result to derive the regularity of the above unique function 1,p ϕ ∈ W0 (Ω). Moreover, there exists a constant γ (Ω, p, L) > 0: |ϕ|W 1,p γ H (u)g Lp + |ϕ|Lp .

(20)

|ϕ|L2 cH (u)g L2 cH (u)g Lp ,

(21)

Since

and |ϕ|p c|ϕ|θL2 |ϕ|1−θ 1,p W0

for some θ ∈ ]0, 1[,

one derives via Young’s inequality, relations (20) to (22) |ϕ|W 1,p cH (u)g Lp . We shall use a corollary of Lemma 3.

2

(22)


817

Corollary 3.1 of Lemma 3. Under the same assumptions as for Lemma 3, for all p 2, all r ∈ [1, +∞] if g ∈ Lp,r (Ω) then the unique solution ϕ of B(ϕ, ψ) =

1,p

∀ψ ∈ W0

gH (u)∇ψ dx,

(23)

(Ω),

Ω

belongs to W 1 (Ω, | · |p,r ) and |∇ϕ|Lp,r (Ω) cH (u)g Lp,r (Ω) .

(24)

Moreover, for p N , for all x ∈ Ω 1− N |ϕ(x)| cp H (u)g Lp,1 · δ(x) p .

(25)

Proof. To deduce the relation (24), we apply the Marcinkiewicz’s interpolation theorem (see [1]) with T g = |∇Ag|, where the map A is defined as A(g) = ϕ with ϕ the unique solution of (23). T maps Lp (Ω) into Lp (Ω) continuously and then from Lp,r (Ω) into itself. Therefore, we have relation (24) thanks to Lemma 3. While for relation (25), we use the Sobolev embedding W 1 (Ω, | · |p,1 ) ⊂> C 0,β (Ω) with β = 1 − Np if p > N and W 1 (Ω, | · |N,1 ) ⊂> C(Ω) ∩ L∞ (Ω) if p = N (see [14]). We combine these results with relation (24) to derive the result. 2 Proof of Theorem 3. We shall consider vk ∈ W 2,p (Ω) ∩ H01 (Ω) ∀p ∈ [1, +∞[ satisfying Lvk = fk = Tk (f ). We want to show that (vk )k1 is a Cauchy sequence in W 1 (Ω, | · |qα ,∞ ) with qα = We introduce ∇vk H (vk ) = |∇vk | if ∇vk = 0, 0 otherwise.

N N −1+α .

Then, for any E measurable ⊂ Ω, we have from Lemma 3 and its corollary a function ϕE ∈ W 1 (Ω, | · |p,1 ) ∀p ∈ [2, +∞[ such that B(ϕE , ψ) =

H (vk ) · ∇ψ dx

∀ψ ∈ W

1,p

(Ω)

1 1 + =1 . p p

E

Choosing ψ = vk , we have

|∇vk | dx = B(ϕE , vk ) =

ϕE Lvk dx =

Ω

E

fk ϕE dx.

(26)

Ω

From relation (25), we know that ϕE (x) c|χE |

Lp,1

1− N p

· δ(x)

.

(27)

818


So let us fix α ∈ [0, 1[ and choose p so that 1 1−α = . p N

(28)

|∇vk | c|χE |Lp,1 · |fk |L1 (Ω,δ α ) .

(29)

α=1−

N p

that is

Therefore, from relations (26) and (27) one has E 1

Since |χE |L1,p cp |E| p , one has from relation (29) 1− 1 sup t p |∇vk |∗∗ (t) c|fk |L1 (Ω,δ α )

with 1 −

t|Ω|

N −1+α 1 1 = = . p N qα

By linearity, relation (30) implies that (vk )k1 is a Cauchy sequence in W 1 (Ω, | · |qα ,∞ ).

(30) 2

Now, we are able to prove Theorem 4. Let v be the unique solution of the generalized Dirichlet problem (DGL (Ω)), f 0. 1,q Then v ∈ W0 (Ω) for some q > 1 if and only if there exists α ∈ [0, 1[ such that f ∈ L1 (Ω, δ α ). Proof. From Theorem 3, we know that 1,q if f ∈ L1 Ω, δ α then v ∈ W 1 Ω, | · |qα ,∞ ⊂ W0 (Ω) for all 1 q < qα . For the converse, we use that f 0. 1,q If v ∈ W0 (Ω), q > 1 then we have for all ϕ ∈ Cc∞ (Ω) −

v∂i aij (x) · ∂j ϕ dx =

Ω

(31)

∂i v∂j ϕaij dx. Ω

We deduce from the equation satisfied by v that ∀ϕ ∈ Cc∞ (Ω), ϕ 0 f ϕ dx = Ω

aij (x)∂i v∂j ϕ dx +

i,j Ω

bi v∂i ϕ dx +

Ω

c0 vϕ dx = B(v, ϕ).

(32)

Ω 1,q

Using a density argument and Fatou’s lemma, the relation (32) implies ∀ϕ ∈ W0 1 1 q + q = 1

(Ω), ϕ 0,

f ϕ dx B(v, ϕ). Ω

(33)


We choose 1 > α = 1 − β > q1 , we have 1,q

δ α ∈ W0

Ω

(Ω)

819

δ(x)−q β dx < +∞, therefore the function

α q ∇δ dx

and Ω

δ −βq (x) dx < +∞.

Ω

Choosing as a test function ϕ = δ α in relation (32)

0

f δ dx c Ω

|v| dx +

q

Ω

which shows the result.

|∇v| dx +

α

|v| dx

q

Ω

(34)

Ω

2

Next, we want to analyze some specific case, namely when we “symmetrize” the equation. Unfortunately, the usual trick consisting to compare v (when f 0) with a radial decreasing function U associated to f radial decreasing rearrangement of f , does not give any information for the integrability of v either its gradient (see Lemma 6). The following remark explains partly this fact. having the same measure |Ω| than Ω, then Remark 2. In general, when we consider the ball Ω the distance to the boundary δ(x) = δΩ (x) is given by

−1

1

δΩ = αN N |Ω| N − |x|, x ∈ Ω. −1

δΩ ), f 0 then f ∈ L1 (Ω). Indeed Setting R = αN N |Ω| N , if f ∈ L1 (Ω, 1

f dy =

f (x) dx

Ω

Ω

f (x) dx +

{|x| R2 }

2 R

f (x)δΩ (x) dx + f∗

Ω

f (x) dx

{ R2 0 such that δ(y) |∇G|(x, y) c|x − y|1−N min 1, . (39) |x − y|

822


By Fubini’s theorem, one has

|∇vk |(x) dx

Ω

fk (y)

Ω

|∇G|(x, y) dx dy

(40)

Ω

and by the estimates (39), we have

|∇G|(x, y) dx c

|x − y|1−N dx + cδ(y)

(41)

{x: |x−y|>δ(y)}

{x: |x−y|δ(y)}

Ω

|x − y|−N dx.

Thus

|∇G|(x, y) dx cδ(y) + cδ(y)Ln δ(y).

(42)

Ω

From relations (40) and (42), we deduce

|∇vk |(x) dx c

Ω

f (y)δ(y)Ln δ(y) dy.

(43)

Ω

This shows that (vk )k1 is a Cauchy sequence in W01,1 (Ω). Thus v ∈ W01,1 (Ω). 2 We recall the following result which can be obtained by some direct integrations (see for instance [11,7]). . Lemma 5. Let f ∈ L1 (Ω, δ), f 0 and let for n ∈ N, Tn (f ) = min(f, n) = fn . Then the sequence (Un )n0 defined on Ω by

Un (x) =

αN σ 1 σ −2(1− N ) fn∗ (t) dt dσ,

1 2

N 2 αNN α

N N |x|

0

is the unique solution of

−Un (x) = fn (x) = fn∗ αN |x|N ,

x ∈ Ω,

Un = 0 on ∂Ω, 1,q

Un ∈ W0 (Ω), ∀q < +∞. Another lemma that shall explain the difference between the results when f ∈ L1 (Ω) and f ∈ L1 (Ω, δ) is the following necessary and sufficient condition.


823

Lemma 6. Under the same assumptions as in Lemma 5, we have f ∈ L1 (Ω) if and only if limn→+∞ Un = U is in L1 (Ω). And in this case (i.e. f ∈ L1 (Ω)), the function U is the unique solution of ⎧ ⎨ −U = f in Ω, N ⎩ U ∈ W01,q (Ω), 1 q < . N −1 Proof. We first note that fn∗ = Tn (f∗ ) = min(f∗ , n), thus by Beppo–Levi monotone convergence limn→+∞ Un (x) = U (x) exists in [0, +∞] for a.e. x, since Un (x) Un+1 (x)

∀x ∈ Ω.

The first part is well known since if f ∈ L1 (Ω) then f ∈ L1 (Ω) and therefore, the unique solution of the Dirichlet problemis U . Conversely, assume that 0 Ω U (x) dx < +∞, one has for all n 1, using integration by parts

U (x) dx Ω

Un (x) dx Ω

αN =

s

−1+ N1

s

fn∗ (t) dt ds

0

0

αN 1 N 1 =N αN − s N fn∗ (s) ds.

(44)

0 1

1

From relation (44), one has for 0 < ε N

αNN 2

ε fn∗ (s) ds 0

2 1

|U |L1 (Ω)

(45)

N αNN

and αN fn∗ (s) ds fn∗ (ε) · (αN − ε) f∗ (ε)(αN − ε).

(46)

ε

We note that since f ∈ L1+ (Ω, δ) then for all s ∈ ]0, |Ω|], 0 f∗ (s) < +∞, in particular f∗ (ε) < +∞. Thus, for all n 0 2 fn (x) dx f∗ (ε)(αN − ε) + |U |L1 (Ω) . 1 N N α Ω N

824


This implies, by Fatou’s lemma that f (x) dx f∗ (ε)(αN − ε) +

2 1

|U |L1 (Ω) .

2

(47)

N αNN

Ω

Next, we want to prove the following theorem. Theorem 5. Let h ∈ L1 (Ω, δ), h 0. Then, the unique solution ω ∈ L1 (Ω) of

−ω(x) = h(x) = h∗ αN − αN |x|N , ω(x) = 0, x ∈ ∂Ω,

in Ω,

in the very weak sense given above belongs to W01,1 (Ω). Moreover we have |ω|L1 (Ω)

h∗ (σ )σ

1 1+ 1 αN N

N 1

|∇ω|L1 (Ω)

1 N

L1 (Ω∗ )

1 1

|h|L1 (Ω,δ) ,

αNN

h∗ (σ )σ 1 |h|L1 (Ω,δ) . L (Ω ) ∗

N αN

For this, we shall prove the following more general theorem which shows merely that for radial solution ω, one has the W 1,1 -regularity. Theorem 6. Let f0 be a given nonnegative measurable function on the interval Ω∗ with σf0 (σ ) ∈ L1 (Ω∗ ). Then, f ∈ L1 (Ω, δΩ ), with f (x) = f0 (αN − αN |x|N ), and the unique generalized function ω ∈ L1 (Ω) of −ω = f0 (αN − αN |x|N ) belongs to W01,1 (Ω). Moreover we have |ω|L1 (Ω)

f0 (σ )σ

1 1+ 1 αN N

|∇ω|L1 (Ω)

N 1 1 N

N αN

L1 (Ω∗ )

1 1

|f |L1 (Ω,δ) ,

αNN

f0 (σ )σ 1 |f |L1 (Ω,δ) . L (Ω ) ∗

N Proof. Consider for f 0, with f (x) = f0 (αN − ααNN |x| ). We first remark, arguing as in Lemma 4, that Ω (1 − |x|)f (x) dx is equivalent to 0 σf0 (σ ) dσ and we have precisely

αN f (x)δ(x) dx

αN Ω

σ f0 (σ ) dσ N αN

0

f (x)δ(x) dx. Ω

Thus, under the condition on f0 , one deduces that f ∈ L1 (Ω, δ).


825

The function αN

1

ω(x) =

σ

2

−2(1− N1 )

σ

f0 (αN − t) dt dσ

N 2 αNN α

N N |x|

0

is L1 (Ω). Indeed, considering f0 n = Tn (f0 ) and the function αN ωn (x) = αN

σ

−2(1− N1 )

s

f0 n (αN − t) dt ds,

|x|N

x ∈ Ω,

0

one has by change of variables

ωn (x) dx =

Ω

αN αN σ

−2(1− N1 )

f0 n (αN − t) dt

s

0

σ

dσ

0

αN 1 N 1 =N αN − s N f0 n (αN − s) ds 0

αN =N

(αN − s)f0 n (αN − s) ds PN (s)

(48)

0

with PN (s) =

αN −s 1

1

αNN −s N

, s ∈ [0, αN [. Thus from (48), we deduce

ωn (x) dx

Ω

N PN (0)

αN f0 (σ )σ dσ.

(49)

0

Letting n → +∞, in relation (49), we deduce from Fatou’s lemma

ω(x) dx

1

f0 (σ )σ dσ.

1+ N1

N αN

Ω

αN 0

The same analysis shows that for (j, n) ∈ N2 , one has

1 ∇(ωn − ωj )(x) dx α N

αN s

N

Ω

From the latter, we derive

0

−1+ N1

s (f0 n − f0 j )(αN − t) dt ds. 0

(50)

826


Ω

1

N ∇(ωn − ωj )(x) dx N αN PN (0)

αN (αN − s)|f0 n − f0 j |(αN − s) ds 0

2 N −1

αN

N αN

σ (f0 n − f0 j )(σ ) dσ.

0

By Lebesgue dominate theorem, we deduce that αN lim

n→+∞ j →+∞ 0

σ (f0 n − f0 j )(σ ) dσ = 0.

Thus ωn is a Cauchy sequence in W01,1 (Ω). Therefore ω is W01,1 (Ω) and so is ω. Moreover, we have the identity

∇ω(x) dx = N αN N1

Ω

αN

(αN − t)f0 (αN − t) dt. PN (t)

(51)

0

From the latter, we derive |∇ω|L1 (Ω) c3N |f |L1 (Ω,δ)

with c3N =

1 . N αN

(52)

Since one has −ωn = f0 n αN − αN |x|N , this implies that ω is a solution of DG(Ω).

ωn ∈ H01 (Ω),

2

As a complement for Theorem 4, we can make precise the necessary and sufficient condition for radial solution as in the above theorem. This will allow us to construct easily some examples for the applications. Lemma 7. Let q ∈ [1, N [. Then the function ω given in Theorem 6 is in W0 (Ω) if and only if we have 1,q

αN

αN σ f0 (σ )

0

q−1 dσ =

f0 (t) dt σ

q

αN αN f0 (t) dt 0

σ

Proof. Assume first that f0 ∈ L∞ (Ω∗ ). One has for any q ∈ [1, N [ Ω

∇ω(x)q dx = γN

αN s 0

− Nq

q

s f0 (αN − t) dt 0

dx

dσ

is finite.


827

s q−1 αN q q 1− 1− αN N − s N f0 (αN − s) = γN f0 (αN − t) dt ds 0

= γN

0

αN

αN q−1 q q αN 1− N − (αN − σ )1− N σ f0 (σ ) f0 (t) dt dσ. σ σ

0

One has two constants c1 > c0 > 0 such that ∀σ ∈ [0, αN ] 1− Nq

c0

αN

q

− (αN − σ )1− N c1 . σ

Indeed 1− q αN N

1− Nq

− (αN − σ )

1− q = αN N

σ 1− 1− αN

1−

q N

.

The last function is equivalent to − q αN N

q 1− σ N

as σ → 0.

Therefore the quotient 1− Nq

αN

q

− (αN − σ )1− N σ

−

q

≈ αN N

σ →0

1−

q . N

Thus, one has the equivalence

∇ω(x)q dx ≈

Ω

αN

αN

σ f0 (σ )

q−1 f0 (t) dt

dσ.

σ

0

Since f0 (t), t ∈ L1 (0, αN ), then by integration by parts, the last integral is equal to q

αN αN f0 (t) dt 0

dσ = I (f ).

σ

We have shown that there are two constants k1N and k2N depending only on N and q such that k1N I (f )

|∇ω|q dx k2N I (f ). Ω

(53)

828


If f ∈ L1 (Ω, δ), f 0, fn = Tn (f ) ∈ L1 (Ω, δ) and with the expression of ωn and ω, we have by Beppo–Levi monotone convergence lim |∇ωn |q dx = |∇ω|q dx (54) n→+∞

Ω

Ω

and q

αN αN lim

dσ =

f0 n (t) dt

n→+∞ σ

0

q

αN αN f0 (t) dt

dσ.

σ

0

These numbers might be infinite. Thus (53) is true for f ∈ L1 (Ω, δ). This gives the equivalence. 2 We shall end this section by a few examples of applications of the above results. Corollary 7.1. Let Ω be the unit ball of RN and q ∈ [1, NN−1 [ for γ ∈ [1, 2[, we consider f (x) =

1 . (1 − |x|N )γ

Then f ∈ L1 (Ω, δ)

and f ∈ / L1 (Ω).

Moreover 1,q

• if γ ∈ [1 + q1 , 2[ then the function ω given in Theorem 5 is not in W0 (Ω); 1,q

• if γ ∈ [1, 1 + q1 [ then the function ω ∈ W0

1 with q ∈ [1, min( γ −1 , NN−1 )[.

Proof. One has f∗ (σ ) =

βN , σγ

σ ∈ [0, αN ].

The necessary and sufficient condition can be written as αN 1−γ 1−γ q I (γ ) = σ − αN dσ 0 1,q

is finite if and only if ω ∈ W0 (Ω). And αN σ (1−γ )q dσ is finite,

I (γ ) is finite if and only if 0

(55)


829

and αN σ (1−γ )q dσ is finite if and only if

γ 1 we set f (σ ) = −g (σ ), σ ∈ ]0, αN [. Then

N

(1) g ∈ L1 (]0, αN [) and g ∈ / Lq (]0, αN [) for all q > 1. (2) Setting h(x) = f (αn − αN |x|N ), x ∈ Ω, h ∈ L1 (Ω, δ| Ln δ|), but h is not in L1 (Ω, δ α ), for any α ∈ [0, 1[. (3) The generalized function ω ∈ L1 (Ω) solution of −ω = h belongs to W01,1 (Ω) but not to 1,q W0 (Ω) for q > 1. α +∞ Proof. (1) 0 N g(σ )q dσ ≈ Ln 4 exp((q − 1)σ )σ −γ dσ from which we have the result. γ (2) We set X(σ ) = | Ln 4ασN | = Ln 4ασN for σ ∈ ]0, αn ], Y (σ ) = 1 − X(σ ). By a straightforward computation, we have g (σ ) = − Thus f (σ ) =

g(σ ) σ Y (σ )

g(σ ) Y (σ ). σ

and for α ∈ [0, 1] we have:

Jα =

α |h|(x) 1 − |x| dx =

Ω

αN

σ α |f |(σ ) dσ −1

0

1

,

PNα (αN N (αN − σ ) N )

N

α with PNα (t) = ( 1−t 1−t αN ) , t ∈ [0, 1]. α Since inft∈[0,1] PN (t) > 0, we deduce that

αN Jα ≈

σ f (σ ) dσ ≈

α

0

αN Y (σ )g(σ )σ α−1 dσ. 0

Let us introduce σN = 4αN exp(−γ ), then 0 < Y (σ ) 1 if σ ∈ [0, σN [

and Y (σ ) < 0

for σ > σN ,

If α = 1 since Maxσ ∈[0,αN ] |Y (σ )| is finite then αN J1 c

g(σ ) dσ < +∞. 0

Y (σN ) = 0.

830


If 0 α < 1, then we have εαN Y (σ )g(σ )σ α−1 dσ, Jα c 0

with 0 < ε < 1, 0 < εαN < 12 inf(αN , σN ). From the latter, we have +∞ Jα c exp (1 − α)σ σ −γ dσ = +∞. Ln 4ε

To show that h ∈ L1 (Ω, δ| Ln δ|), we start with the case γ Ln 4 then f (σ ) 0 for all σ ∈ ]0, αN [, then f (σ ) = f∗ (σ ). Since

|h|(x)δ(x)Ln δ(x) dx c

Ω

αN f (σ )σ dσ 0

αN f∗ (σ )σ dσ c

=c 0

f (x)δ(x) dx < +∞, Ω

if γ > 2, we have (arguing as before):

h(x)δ(x)Ln δ(x) dx

Ω

c Ω

h(x)δ(x) dx + c

αN 0

+∞ g(σ ) d(σ ) + c σ (1−γ ) dσ < +∞, Ln 4

thus h ∈ L1 (Ω, δ| Ln δ|). (3) According to Theorem 6 or Proposition 1, we then have that the very weak solution 1,q / L1 (Ω, δ α ) then ω does not belong to W0 (Ω) −ω = h, ω ∈ L1 (Ω) is in W01,1 (Ω). Since h ∈ for all q > 1. 2 Acknowledgments The research of the first author was partially supported by projects MTM2008-06208 of the Ministerio de Ciencia e Innovación, Spain and CCG07-UCM/ESP-2787 of the DGUIC of the CAM and the UCM.


831

References [1] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, London, 1983. [2] I. Birindelli, F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6) 13 (2) (2004) 261–287. [3] L. Boccardo, T. Gallouët, Non-linear elliptic and parabolic equations involving measure as data, J. Funct. Anal. 87 (1989) 149–169. [4] H. Brezis, Personal communication to J.I. Díaz: Une équation semi-linéaire avec conditions aux limites dans L1 , unpublished. [5] H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa, Blow up for ut − u = g(u) revisited, Adv. Differential Equations 1 (1996) 73–90. [6] X. Cabré, Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156 (1998) 30–56. [7] J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Res. Notes Math., vol. 106, Pitman, London, 1985. [8] J.I. Díaz, J.M. Rakotoson, On very weak solutions of semilinear elliptic equations with right hand side data integrable with respect to the distance to the boundary, in preparation. [9] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. [10] H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Nachr. 307 (1997) 89–102. [11] L. Korkut, M. Pasic, D. Zubrinić, A singular ODE related to quasilinear elliptic equations, Electron. J. Differential Equations 12 (2000) 1–37. [12] P. Quittner, P. Souplet, Superlinear Parabolic Problems, Birkhäuser, Basel, 2007. [13] J.M. Rakotoson, Quasilinear elliptic problems with measure as data, Differential Integral Equations 4 (3) (1991) 449–457. [14] J.M. Rakotoson, Réarrangement Relatif: un instrument d’estimation dans les problèmes aux limites, SpringerVerlag, Berlin, 2008. [15] J.E. Rakotoson, J.M. Rakotoson, Analyse fonctionnelle appliquée aux équations aux dérivées partielles, P.U.F., Paris, 1999. [16] C. Simader, On Dirichlet’s Boundary Value Problem and Lp Theory Based on Generalization of Gårding’s Inequality, Lecture Notes in Math., vol. 268, Springer-Verlag, New York, 1972. [17] L. Veron, Singularities of Solutions of Second Order Quasilinear Equations, Longman, Edinburgh Gate, Harlow, 1995, pp. 176–180.


Almost periodically forced circle flows ✩ Wen Huang a , Yingfei Yi b,∗ a Department of Mathematics, University of Science and Technology of China, Hefei Anhui 230026, PR China b School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 8 December 2008; accepted 8 December 2008 Available online 14 January 2009 Communicated by J. Bourgain

Abstract We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li–Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2, R)-valued, continuous or discrete cocycle. Continuous almost periodically forced circle flows are among the simplest non-monotone, multifrequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete

✩ The first author is partially supported by NSFC grant 10531010, 973 project 2006CB805903, and FANEDD grant 200520. The second author is partially supported by NSFC grant 10428101 and NSF grants DMS0204119, DMS0708331. * Corresponding author. E-mail addresses: [email protected] (W. Huang), [email protected] (Y. Yi).

0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2008.12.005

W. Huang, Y. Yi / Journal of Functional Analysis 257 (2009) 832–902

833

counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems. Published by Elsevier Inc. Keywords: Almost automorphic dynamics; Almost periodically forced circle flows; Forced nonlinear oscillators; minimal sets; Topological dynamics

1. Introduction Through out the paper, we let T = R or Z. Consider askew-product circle flow (SPCF for short) (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ) with a compact base (or forcing) flow (Y, T) = (Y, {σt }t∈T ), i.e., for each t ∈ T the following diagram S1 × Y

Λt

S1 × Y

π

Y

π σt

Y

commutes, where π : S 1 × Y → Y denotes the natural projection. Let y0 · t = σt (y0 ), ψ(s0 , y0 , t) = ΠΛt (s0 , y0 ), where Π : S 1 × Y → S 1 denotes the natural projection. The SPCF can be expressed more explicitly as Λt : S 1 × Y → S 1 × Y : Λt (s0 , y0 ) = ψ(s0 , y0 , t), y0 · t ,

t ∈ T.

(1.1)

In particular, when T = Z, the discrete flow Λt is generated by the iteration of the skew-product circle map Λ : S 1 × Y → S 1 × Y : Λ(s0 , y0 ) = f (s0 , y0 ), T (y0 ) ,

(1.2)

where f (s0 , y0 ) = ψ(s0 , y0 , 1) and T (y0 ) = y0 · 1. Using angular coordinate φ0 ∈ R 1 (mod 1), the SPCF Λt can be also expressed as Λˆ t (φ0 , y0 ) = φ(φ0 , y0 , t), y0 · t ,

t ∈ T,

(1.3)

where e2πiφ(φ0 ,y0 ,t) = ψ(s0 , y0 , t) and e2πiφ0 = s0 . The present paper is mainly devoted to the study of dynamical and topological properties of minimal sets in an almost periodically forced circle flow (APCF for short), i.e., a SPCF (S 1 × Y, T) with an almost periodic base (or forcing) flow (Y, T). We refer an APCF as continuous APCF if T = R and as discrete APCF if T = Z. In the case that the SPCF is discrete, we assume that the generating skew-product circle map Λ is homotopic to the identity and fiber-wise monotone, i.e., there is a homeomorphism Λ˜ : R 1 × Y → R 1 × Y : ˜ 0 , y0 ) = f˜(x0 , y0 ), T (y0 ) , Λ(x

834


where for each y0 ∈ Y , f˜(·, y0 ) is monotone and f˜(x0 + 1, y0 ) ≡ f˜(x0 , y0 ) + 1, such that when both f˜ and x0 are identified modulo 1 the identified map Λ˜ is homeomorphic to Λ. In the case that the SPCF (1.1) is either discrete and unforced (i.e., a circle map) or continuous but periodically forced, it follows from the classical Poincaré–Birkhoff–Denjoy theory that a minimal set of the SPCF is periodic if the rotation number is rational, and is either almost periodic or of Denjoy type if the rotation number is irrational. It is also known that a Denjoy type of minimal set has a Cantor structure and is almost automorphic. But even if the SPCF (1.1) becomes an APCF, its minimal dynamics can be far more complex, though it always admits zero topological entropy. Taking the continuous case for example, while the majority existence of quasi-periodic dynamics in a parameter family of quasi-periodically forced ordinary differential equations on the circle is asserted by the Arnold–Moser theorem [2,43] when the forcing frequencies are Diophantine and the forcing functions are sufficiently small, smooth perturbations of a constant plus a deformation parameter, almost periodic dynamics are however not generally expected in a continuous APCF. Even in the continuous APCF generated from the projective bundle flow of a continuous, almost periodic, sl(2, R)-valued cocycle, it was shown by Johnson [29] that if the cocycle is not uniformly hyperbolic and admits two minimal sets, then both minimal sets are almost automorphic which are not necessarily almost periodic, and moreover, if only one minimal set exists, then dynamics of the minimal set can be much more complicated than being almost automorphic. For discrete APCFs, it was recently shown in [27,28] that even with one forcing frequency the flows can be topologically transitive. Besides the presence of multiple forcing frequencies and the topological complexity of the forcing space, much of the dynamical complexity in an APCF (1.1) is governed by the loss of the so-called mean motion property. Let ˜ φ˜ 0 , y0 , t) φ( t→∞ t

ρ = lim

˜ φ˜ 0 , y0 , t) denotes the be the rotation number associated with the APCF (1.1) or (1.3), where φ( 1 ˜ ˜ ˜ ˜ lift of φ(φ0 , y0 , t) in R satisfying φ(φ0 + 1, y0 , t) ≡ φ(φ0 , y0 , t) + 1. The limit exists and is independent of orbits. Differing from the unforced discrete case and periodically forced continuous case, there are general cases in which ˜ φ˜ 0 , y0 , t) − ρt = +∞ (1.4) supφ( t∈T

for some (φ˜ 0 , y0 ) ∈ R 1 × Y . We say that the APCF (1.1) admits mean motion (or bounded mean motion) if ˜ φ˜ 0 , y0 , t) − ρt < +∞ (1.5) supφ( t∈T

for all (φ˜ 0 , y0 ) ∈ R 1 × Y . In the opposite case, we say that the APCF (1.1) admits no mean motion (or unbounded mean motion). It is well known that if the APCF (1.1) has an almost periodic orbit, then it always admits mean motion. In fact, as suggested by works [27,28,54,62], dynamics of an APCF (1.1) with mean motion should resemble more closely to the unforced discrete case or the periodically forced continuous case, while dynamics of an APCF (1.1) without mean motion should be considerably different than those with mean motion. Indeed, in the case that an APCF (1.1) is continuous in time and admits mean motion, a translation x = φ˜ − ρt will


835

readily transform the APCF into an almost periodically forced totally monotone system and an application of results in [56] shows that each of its minimal set is almost automorphic. In the present paper, we will employee techniques from topological dynamics and ergodic theory to study minimal sets in an APCF (1.1) with particular attentions paying to their general (dynamical and topological) complexities, structures, and topological classifications. We will also investigate in other fundamental dynamics issues for an APCF such as the existence of almost automorphic minimal sets and sufficient conditions for the validity of the mean motion property. We will obtain a set of general results for an APCF including the following: (a) Each minimal set is either point-distal or residually Li–Yorke chaotic; (b) Any minimal set is either an almost finite to one extension of the base, or the entire phase space, or a Cantorian; (c) If the flow admits more than one minimal set, then each minimal set is an almost finite cover of the base; (d) If the flow admits mean motion, then each minimal set is almost automorphic; (e) If the flow admits no mean motion, then each minimal set is either the entire phase space or is everywhere non-locally connected. In the case that the forcing is quasi-periodic, the following more concrete results will be obtained: (f) If the rotation number is rationally independent of the forcing frequencies, then the flow admits a unique minimal set and the minimal set is either the entire phase space or is everywhere non-locally connected; (g) If the flow admits no mean motion, then it is positively transitive and admits a unique minimal set, and consequently, if the flow has more than one minimal sets, then it must admit mean motion and each minimal set is almost automorphic. We remark that, except in those involving rotation number and mean motion, our results above actually hold for a general discrete APCF without assuming its generating skew-product map to be homotopic to identity. Based on our general results and works [7,29], we will also give a complete classification of minimal sets for the projective bundle flow (P 1 × Y, T) of an almost periodic, sl(2, R)-valued, continuous (i.e., T = R) or discrete (i.e., T = Z) cocycle. APCFs are among the simplest but fundamental models of non-monotone, multi-frequency systems in which interactions of (both internal and external) frequencies are expected to generate rather complicated dynamics. First of all, APCFs arise naturally in the study of almost periodically forced nonlinear oscillators and their discretizations. Consider an almost periodically forced, damped, nonlinear oscillator x + F (x, x , y · t) = 0,

x ∈ R 1 , y ∈ Y.

(1.6)

Such an oscillator admits both internal and external frequencies, and due to damping, its oscillations all lie in a compact global attractor. According to the classical oscillation theory, not only are almost periodic oscillations rare in a such system, but also the global attractor can become complicated especially when the damping is weak. In particular, even in quasi-periodically forced nonlinear oscillators as simple as Van der Pol and Josephson junction, numerical stud-

836


ies have discovered the existence of so-called strange, non-chaotic attractors (SNAs) which are geometrically strange but admit no positive Lyapunov exponent (see e.g., [22,51]). There have been many theoretical and numerical studies on SNAs with respect to both quasi-periodically forced nonlinear oscillators and their discretizations. It is well believed that a SNA typically lies in the intermittency during the transition from order to chaos. An almost periodically forced, damped, nonlinear oscillator (1.6) can be often reduced through an integral manifolds reduction to an almost periodically forced scalar ordinary differential equation of the form φ = f (φ, y · t),

φ ∈ R 1 , y ∈ Y,

(1.7)

where f (φ + 1, y) ≡ f (φ, y) (see [54,62] for the case of almost periodically forced Van der Pol and Josephson junction oscillators). When both the solution φ(φ0 , y0 , t) and the initial value φ0 corresponding to y = y0 are identified modulo 1, Eq. (1.7) clearly generates an APCF containing the SNA. In fact, much of the study on SNAs has been made with respect to such reduced quasiperiodically forced circle flows and their discrete counterparts (we refer the readers to [14,21, 27,28,35] for recent progresses on the subject). But besides the case of (1.7) with mean motion property in which minimal sets are known to be almost automorphic [54,62], dynamical and topological structures of SNAs in general situation are yet to be understood. In the case that an almost periodically forced nonlinear oscillator is damping-free, it becomes an almost periodically forced, one-degree-of-freedom Hamiltonian system of the form x + Vx (x, y · t) = 0,

x ∈ R 1 , y ∈ Y,

(1.8)

where V (x + 1, y) ≡ V (x, y). Due to the conservative nature, oscillations of a such system spread over the entire phase space. It is known that if (1.8) is quasi-periodically forced with Diophantine frequencies, then associated with high “energy” the system becomes nearly integrable and an application of KAM theory shows the existence of a positive Lebesgue measure set of quasi-periodic invariant tori with Diophantine frequencies. But it is also known that these quasiperiodic tori tend to disappear if either the system become less integrable or the frequencies are close to resonance. Instead, the so-called Mather sets (or Cantori) supporting minimizing measures can be shown to exist in the phase space R 1 × S 1 × Y based on the Mather theory [39,41]. An application of the Mather theory further shows that dynamics on each projected Mather set in S 1 × Y is topologically conjugated to that of the corresponding Mather set (see e.g., [26,40]). 2 More precisely, consider the Lagrangian L = p2 − V (x, y · t) associated with (1.8). Then for each η ∈ R 1 , minimizing measures μη exist, i.e., each μη is an invariant measure for the skew-product flow (R 1 × S 1 × Y, R) generated from (1.8) and satisfies (L − η) dμη = inf (L − η) dμ, μ

R 1 ×S 1 ×Y

R 1 ×S 1 ×Y

where the infimum is taken over all Borel probability measures on R 1 × S 1 × Y . The set Mη = 1 1 μη supp μη is called Mather set, which is a compact invariant set of (R × S × Y, R). Let π : R 1 × S 1 × Y → S 1 × Y be the natural projection and M˜ η = πMη be the projected Mather set. Then π : Mη → M˜ η is a homeomorphism. It follows that the projected flow on M˜ η , as a subflow of an APCF, is topologically conjugated to that defined on Mη . When (1.8) is periodically forced, the projected Mather sets are just the well-known Aubry–Mather sets which have the


837

basic structure of Denjoy sets supporting 2-frequency almost automorphic dynamics. However, not much is known on dynamical and topological properties of Mather sets in situations involving more than two frequencies. The second, APCFs arise naturally in the spectral theory of 2D linear systems with almost periodic coefficients. For instance, as the projective bundle flows of sl(2, R)-valued, almost periodic, continuous or discrete cocycles, they play an essential role in 2D almost periodic Floquet theory and the spectral problem of linear Schrödinger equations/operators and their discrete counterparts (such as Harper’s equations and almost Mathieu operators) with almost periodic potentials [6,29,34]. In particular, for the 2D almost periodic Floquet problem, it was a remarkable observation due to Johnson [29] that, with the general unavailability of an almost periodic strong Perron transformation which transforms an almost periodic linear differential system into a canonical form, one can often introduce an almost automorphic transformation instead, provided that an almost automorphic minimal set exists in the reduced continuous APCF. The third, as recently shown by Pliss and Sell [47], continuous APCFs arise in oceanic dynamics and climate models through invariant manifolds reductions and high frequency averaging. Hence they can be used as basic models to explain complicated oceanic dynamics in particular the nature of turbulence. In addition, in the case that the forcing flow is quasi-periodic, (1.1) becomes a toral flow or map whose rotation set is a singleton. Dynamics of toral flows and maps have been extensively studied for cases with convex rotation sets, but the case with “thin” rotation set is more or less open. Linking to these important problems and applications, our primary goal for the present study is to make a preliminary understanding of frequency-driven dynamical irregularity and complexity in non-monotone, multi-frequency systems. It is our hope that the present study on APCFs will lead to some deep insights on dynamics and structures of SNAs and Mather sets in almost periodically forced, nonlinear oscillators in the damped and damping-free case respectively, on the spectral problem of almost periodic Schrödinger-like operators, on dynamics of toral flows and maps with “thin” rotation sets, on the nature of turbulence in oceanic flows, and on intermittency phenomenon during the transition from order to chaos. We remark that smooth dynamical systems theory plays a less role in the general problems which we are studying. First of all, due to the general almost periodic time dependence, the forcing space of an APCF need not be smooth. Secondly, even if the forcing space is smooth, APCFs arising in applications need not be smooth at all. For instance, for a quasi-periodically forced, damped nonlinear oscillator, it is well known that the weaker the damping is, the less smoother an integral manifold becomes [61]. While for a quasi-periodically forced, damping-free nonlinear oscillator, the torus which a projected Mather set is embedded into is only Lipschiz in general even in the periodically forced case [13,39]. The rest of the paper is organized as follows. In Section 2, we will give precise statements of our main results with respect to APCFs along with some discussions. Section 3 is a preliminary section in which we will review basic concepts and results from topological dynamics and ergodic theory to be used in later sections. Our main results will be proved in Sections 4–8 based on some general results which we will prove for compact flow extensions, as well as for general SPCFs. More precisely, in Section 4, we will give an ordering condition under which a compact flow extension preserves topological entropy. In Section 5, we will show that if a minimal flow is a proximal extension of another minimal flow which is not almost 1–1, then it must be Li– Yorke chaotic. In Section 6, we will classify the general topological structures of minimal sets in a SPCF. In Section 7, we will study the nature of minimal sets of a SPCF which are almost finite to one extensions of the base space. In Section 8, we will study dynamical and topological

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behaviors of minimal sets in a SPCF in connection with the validity of the mean motion property. In particular, we will show that if a SPCF is positively transitive, then it has a unique minimal set. We then consider the case with locally connected base and show that if the SPCF admits no mean motion, then it must be positively transitive. In Section 9, we generalize results in [4,7,29] to give a complete classification of minimal dynamics of the projective bundle flow generated from a (continuous or discrete) almost periodic, sl(2, R)-valued cocycle of all four basic types: elliptic, parabolic, partially hyperbolic, and hyperbolic. 2. Main results In this section, we will state our main results with respect to APCFs which are the main motivations for the present study, though most of these results actually hold for more general SPCFs (see Sections 4–8 for more details). Dynamical and topological structures of minimal sets of an APCF (S 1 × Y, T) seem to depend on various factors: the number of minimal sets, local connectivity of minimal sets, validity of the mean motion property, and the topological nature of the forcing space. Hence our main results lie in several categories which particularly include cases for general Y as well as for Y being locally connected (e.g., Y is a torus). A special example of the later case is when (1.1) is quasiperiodically forced. As to be seen from the proofs of these results, both the validity of mean motion property and the local connectivity of Y seem to be essential for a minimal set of an APCF to be better behaved in general. Let π : S 1 × Y → Y denote the natural projection. 2.1. General dynamical complexities General dynamical complexities of an APCF is characterized in the following two theorems. Theorem 1. An APCF always has zero topological entropy. Theorem 1 may be proved by using an entropy inequality due to Bowen [9] for compact flow extensions. In Section 4 we will give a self-contained proof of this theorem by providing a general result on the preservation of topological entropy between flow extensions under an ordering condition. Such a general result on preservation of topological entropy will also be useful to other skew-product flows having certain fiber-wise order preserving properties. Theorem 2. Let M be a minimal set of an APCF. Then precisely one of the following holds: (a) M is point-distal; (b) M is residually Li–Yorke chaotic. Theorem 2 will be proved in Section 5 following a general result which says that if a minimal flow is a proximal extension of another minimal flow that is not almost 1–1, then it must be Li–Yorke chaotic.


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The notion of Li–Yorke chaos is introduced based on the well-known work of Li and Yorke [37]. A compact metric flow (X, T) is called Li–Yorke chaotic if X contains an uncountable scrambled set S—set in which any pair of distinct points {x, y} ⊂ S is a Li–Yorke pair, i.e., lim sup d(x · t, y · t) > 0 and t→+∞

lim inf d(x · t, y · t) = 0, t→+∞

where d denotes the metric on X. It is known that if (X, T) admits positive topological entropy, then it is necessarily Li–Yorke chaotic, but not vice versa [8]. Residual Li–Yorke chaos is a stronger notion than Li–Yorke chaos. A compact metric flow extension π : (X, T) → (Y, T) is said to be residually Li–Yorke chaotic if there exists a residual (i.e., dense Gδ ) subset Yc of Y such that each fiber π −1 (y), y ∈ Yc , admits an uncountable scrambled set. Remark 1. (1) A point-distal minimal set (including almost automorphic minimal set), though cannot be residually Li–Yorke chaotic, it can well be Li–Yorke chaotic. (2) Minimal sets in a circle map or a periodically forced continuous circle flow can never be Li–Yorke chaotic. (3) Using Theorem 2, one can show that many APCFs admit Li–Yorke chaos. Consider a 2D almost periodic linear system x˙ = A(y · t)x,

x ∈ R 2 , y ∈ Y,

(2.1)

where tr A = 0 and (Y, R) is an almost periodic minimal flow. The system naturally generates a 1 continuous APCF ΛA t on the projective bundle P × Y . Let S0 be the set of continuous matrixvalued function A whose respective linear system (2.1) can be reduced to a system with skewsymmetric coefficient matrix B(y) of zero mean via a strong Perron transformation. It was shown by Johnson [31] (see also [42]) that there is a residual subset S of S¯0 such that for each A ∈ S, the entire phase space of ΛA t is minimal, strictly ergodic, and a proximal extension of Y . Now, for each A ∈ S, it follows from Theorem 2 that all minimal sets of ΛA t are residually Li–Yorke chaotic. We note that with the non-existence of almost automorphic dynamics, the APCFs ΛA t with A ∈ S admit no mean motion. We refer the readers to a recent work of Bjerklov and Johnson [7] for more concrete discussions on Li–Yorke chaos in continuous, almost periodically forced projective bundle flows and to Section 9 of this paper for some general discussions in this regard. (4) We expect that point-distality of minimal sets stated in Theorem 2 can be replaced by almost automorphy in a generic sense. However, there are minimal sets of APCFs which are pointdistal but not almost automorphic. An easy example is as follows. Let (Y, R) be an non-periodic, periodic minimal flow and let a be a continuous function on Y with zero mean such that almost t a(y 0 · s) ds is unbounded for some y0 ∈ Y (such functions largely exist, see [30]). Then the 0 t almost periodically forced circle flow defined by Λ∗t (φ, y) = (φ + 0 a(y · s) ds mod 1, y · t), is not almost periodic. Hence Λ∗t is distal (in particular, point-distal) but not almost automorphic, simply because a distal almost automorphic minimal flow must be almost periodic. (5) In [29], concerning the 2D almost periodic Floquet problem, Johnson studied the APCF (projective bundle flow) ΛA t generated from (2.1). Minimal sets of the flow were shown to be almost periodic or almost automorphic for most cases except that the flow has only one minimal set M in which no fiber over the base admits a distal pair. Using Theorem 2, we conclude that

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M is either an almost 1–1 extension of the base (hence almost automorphic, see Theorem 3.2) or residually Li–Yorke chaotic (see also [7]). Similar classifications can be made for the projective bundle flow of a general sl(2, R)-valued, almost periodic, continuous or discrete cocycle (see Section 9 for details). 2.2. Topological classification of minimal sets In the case that an APCF is either discrete and unforced or continuous but periodically forced (i.e., Y = S 1 ), it follows from the classical Poincaré–Birkhoff–Denjoy classification that if the rotation number is rational (the resonant case), then each minimal set is a finite to one extension of S 1 , and if the rotation number is irrational (the non-resonant case), then a minimal set is either the entire phase space or of Denjoy type (in the continuous case, each of its Poincaré section is a Denjoy Cantor set). Our next result shows that one can have a similar topological classification of minimal sets for a general APCF. Theorem 3. Let M be a minimal set of an APCF. Then precisely one of the following holds: (a) M is an almost N –1 extension of Y for some positive integer N ; (b) M = S 1 × Y ; (c) M is a Cantorian. Theorem 3 will be proved in Section 6. A minimal set M of a SPCF (S 1 × Y, T) is said to be a Cantorian if there exists a residual subset Y0 of Y such that for each y ∈ Y0 , the fiber My = π −1 (y) ∩ M is a Cantor set. Remark 2. (1) Cantorians can arise in APCFs with or without mean motions. The Denjoy type of minimal set in a continuous, periodically forced circle flow is an example of Cantorian in (topologically non-transitive) APCFs with mean motion. An example of Cantorian in (topologically transitive) APCFs without mean motion is constructed in a recent work due to Béguin, Crovisier, Jäger, and Le Roux [4]. (2) It is clear that a minimal set in the case (a) of Theorem 3 cannot be residually Li– Yorke chaotic, hence by Theorem 2 it must be point-distal. Still, such a minimal set can well be Li–Yorke chaotic and topological complicated by being everywhere non-locally connected (see Remark 4(1) below). (3) We believe that a Cantorian minimal set of an APCF is more topologically complicated in the sense that it is not only a Cantorian but also everywhere non-locally connected. (4) According to the Poincaré–Birkhoff–Denjoy theory, dynamics of a minimal set M in a continuous, periodically forced circle flow can be completely classified according to its topological nature: in the resonant case M is periodic, while in the non-resonant case M is either 2-frequency almost periodic if it is the entire phase space or 2-frequency almost automorphic if it is of Denjoy type. To the contrary, dynamics of minimal sets in a general APCF is too complicated to be classified according to their topological natures. For instance, minimal sets in both cases (b) and (c) of Theorem 3 can be either point-distal or residually Li–Yorke chaotic which need not be almost automorphic. In fact, even in the case (a) of Theorem 3 we believe that minimal sets need not be almost automorphic in general. An almost N –1 extension of an almost periodic minimal flow need not be almost automorphic. A such example can be constructed in symbolic flows as follows. Let τ be the substitution on


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{0, 1} with τ (0) = 01 and τ (1) = 10. For any finite word w = w0 w1 · · · w −1 in {0, 1}, define τ (w) = τ (w0 )τ (w1 ) · · · τ (w −1 ) and τ i (w) = τ (τ i−1 (w)), i 2. Let X ⊆ {0, 1}Z be the set of all bi-infinite binary sequences x in X such that any finite word in x is a sub-word of τ i (0) for some i ∈ N, and let T be the left shift map on X. Then the discrete dynamical system (X, T ) is a so-called Morse system which is known to be minimal and an almost 2–1 extension of its maximal almost periodic factor [38]. As an almost automorphic minimal flow is necessary an almost 1–1 extension of its maximal almost periodic factor (see Theorem 3.2), (X, T ) is not almost automorphic. A suspension of (X, T ) also leads to an example of continuous flows. (5) Unlike the periodically forced continuous case, the topological classification for general APCFs given in Theorem 3 need not depend on the resonance type. For instance, case (a) of Theorem 3 can happen when the rotation number is rationally independent of forcing frequencies, as shown by an example of Johnson [33] in which an APCF admits a unique minimal set that is an almost 1–1 extension of the base, but the rotation number is not rationally dependent on the forcing frequencies. 2.3. Almost finite to one extensions We would like to exam cases of APCFs in which an almost N –1 extension of the base becomes almost automorphic. We first give a structural characterization of a minimal set if there are more than one minimal sets presented in an APCF. Theorem 4. Suppose that an APCF (S 1 × Y, T) has more than one minimal sets. Then the following holds. (1) There exists a positive integer N such that each minimal set of (S 1 × Y, T) is an almost N –1 extension of Y . (2) If one minimal set of (S 1 × Y, T) is almost automorphic, then so are others. (3) If Y is locally connected, then all minimal sets of (S 1 × Y, T) are almost automorphic. In the case that an APCF (S 1 × Y, T) admits more than one minimal sets, we feel that the local connectivity of Y is essential for the minimal sets to be almost automorphic. In the case that it only admits one minimal set which is an almost N –1 extension of Y for some N > 1, we believe that even local connectivity of Y would not be sufficient for the minimal set to be almost automorphic. However, we do have the following result. Theorem 5. Let M be a minimal set of an APCF (S 1 × Y, T) which is an almost N –1 extension of Y for some N 1. If M is not everywhere non-locally connected, then it is almost automorphic, and moreover, for any y ∈ Y , each fiber π −1 (y) ∩ M consists of exactly N connected components which are either singletons or closed intervals, if it is not homeomorphic to S 1 . Theorems 4, 5 will be proved in Section 7. Remark 3. (1) A Denjoy type of minimal set in a continuous, periodically forced circle flow is an almost 1–1 extension of a 2-torus with two points on each non-residual fiber. Hence by Theorem 5 it is everywhere non-locally connected. (2) In [32], Johnson constructed an example of continuous, quasi-periodically forced circle flow which has a unique minimal set M with the following properties: (i) M is an almost 1–1

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extension of the base torus (hence it is almost automorphic); (ii) M is locally connected at every points on singleton fibers; (iii) M is not locally connected at all points; (vi) there is a full (Haar) measure set in the base torus over which all fibers are non-degenerate intervals. This gives an example of Theorem 5. In fact, by Theorem 5, all non-singleton fibers in M are non-degenerate intervals. 2.4. Mean motion and dynamics In the next two results, we describe the behavior and structure of a minimal set of an APCF in both the cases with and without mean motion. Theorem 6 below is more or less known in the continuous case [54,62] but unknown in the discrete case. Theorem 6. Suppose that an APCF (S 1 × Y, T) admits mean motion. Then the following holds. (1) Each minimal set of (S 1 × Y, T) is almost automorphic whose frequency module is generated by the rotation number and the forcing frequencies. (2) If a minimal set of (S 1 × Y, T) is an almost N –1 extension of Y for some positive integer N , then N is the smallest positive integer whose multiplication to the rotation number is contained in the frequency module of the forcing. Theorem 7. Suppose that an APCF (S 1 × Y, T) admits no mean motion. Then the following holds. (1) Each minimal set of (S 1 × Y, T) is either the entire phase space S 1 × Y or is everywhere non-locally connected. (2) If Y is locally connected, then (S 1 × Y, T) is positively transitive and has only one minimal set. Theorems 6, 7 will be proved in Section 8 based on some general results on the connections between the lacking of mean motion, positive transitivity, and the uniqueness of minimal set. Theorem 7(2) is partially known for a quasi-periodically forced circle map with one forcing frequency [27,28]. But arguments in [27,28], being crucially depending on the one-dimensional forcing space, does not extend to the general situation completely. Corollary. Consider an APCF (S 1 × Y, T) with Y being locally connected. Then the following holds. (1) If (S 1 × Y, T) has more than one minimal set, then it admits mean motion. (2) If the entire phase space S 1 × Y is not minimal, then each minimal set of (S 1 × Y, T) is either everywhere non-locally connected or almost automorphic. (3) If the rotation number is rationally independent of the forcing frequencies, then (S 1 × Y, T) has a unique minimal set. In the above Corollary, (1) follows immediately from Theorem 7(2), (2) follows immediately from Theorem 6(1) and Theorem 7(1), and (3) follows immediately from Theorem 7(2), Theorem 4(1) and Theorem 6(2).


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Remark 4. (1) Consider the example of Johnson [33] in which an APCF admits a unique minimal set that is an almost 1–1 extension of the base, but the rotation number is not rationally dependent on the forcing frequencies. By Theorem 6, this example admits no mean motion, and, by Theorem 7, the unique almost automorphic minimal set is everywhere non-locally connected. Also consider the quasi-periodically forced circle flow constructed by Johnson [32] in which the unique minimal set is not everywhere non-locally connected. Theorem 7 implies that this flow does admit mean motion. (2) If an APCF has a minimal set which is residually Li–Yorke chaotic, then the minimal set cannot be almost automorphic and hence by Theorem 6 the APCF admits no mean motion. (3) An almost automorphic minimal set often occurs as intermediate dynamics in a parameter family of quasi-periodic forced circle flows. Consider a smooth family of quasi-periodically forced equations φ = λ + εf (φ, y · t),

φ ∈ R1 ,

(2.2)

where f : R 1 × T k → R 1 is sufficiently smooth, f (φ + 1, y) ≡ f (φ, y), y · t = y + ωt, ω ∈ R k is Diophantine, and λ, ε are bounded parameters. We let Σ be the set of (λ, ε) whose corresponding equation (2.2) is smoothly reducible to a pure rotation according to Arnold–Moser theorem [2,43], i.e., there is a smooth, near identity transformation φ = ψ + hλ,ε (ψ, y) with hλ,ε ∞ < 1, such that the transformed equation becomes ψ = λ (hence the corresponding quasi-periodically forced circle flow is quasi-periodic and Diophantine). Now consider a boundary point (λ0 , ε0 ) of Σ , i.e., there is a sequence (λn , εn ) ∈ Σ → (λ0 , ε0 ). Let y0 ∈ T k , ψ0 ∈ [0, 1) be given, and hn (t) = hλn ,εn (ψ0 , y0 + ωt) converges uniformly on compact sets to some h∞ (t) according to the Ascoli theorem. Then φn (t) = ψ0 + λn t + hn (t) converges uniformly on compact sets to φ∞ = ψ0 + λ∞ t + h∞ (t) which is a solution of (2.2) corresponding to (λ0 , ε0 ). Since h∞ ∞ 1, it follows that the quasi-periodically forced circle flow (2.2) corresponding to (λ0 , ε0 ) admits mean motion (see Theorem 8.2 in Section 8) and hence by Theorem 6 all its minimal sets are almost automorphic. The rotation number associated with (λ0 , ε0 ) may well depend on ω in a joint Louisville way so that the frequencies of the almost automorphic minimal sets need not be Diophantine. Dynamics of the flow associated with (λ, ε) lying in the complement of Σ¯ are expected to be more complicated due to the possible loss of mean motion property. Similar intermittency phenomenon can be observed in the spectral problem of an almost periodic Schrödinger operator (see Section 9 for detail). For a general parameter family of APCFs, we believe that almost automorphic intermittency (or bifurcation) may occur at the critical value when either almost periodicity is lost, or mean motion property becomes invalid, or when Li–Yorke chaos tends to appear (order to chaos). 2.5. An extended Denjoy theorem From Theorems 4, 7 and the Corollary above, we see that local connectivity of Y plays an important role in the dynamics and topological structures of minimal sets in the corresponding APCF (S 1 × Y, T). The case when Y is locally connected of course includes that of a quasi-periodically forced circle flow. In fact, when Y is a torus, one can have a more complete characterization on the topological structure of a minimal set. The following result can be regarded as a quasi-periodic extension of the classical Denjoy theorem with respect to the topological structure of minimal sets.

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Theorem 8. Consider an APCF (S 1 × Y, T) with Y being a torus (e.g., (Y, T) is quasi-periodic) and suppose that the rotation number is rationally independent of the forcing frequencies. Then (S 1 × Y, T) has a unique minimal set M and M is either the entire phase space S 1 × Y or is everywhere non-locally connected. If, in addition, the APCF admits mean motion, then M is almost automorphic, and moreover, M is either the entire phase space S 1 × Y or an everywhere non-locally connected Cantorian. Theorems 8 will also be proved in Section 8. Remark 5. (1) In light of Theorem 2, under the condition of Theorem 8, an everywhere nonlocally connected minimal set can be either a finite to one extension of the base or a Cantorian. (2) Our results give some information on possible topological and dynamical complexity of a SNA in a quasi-periodically forced, damped nonlinear oscillator and Mather sets in a quasiperiodically forced, damping-free nonlinear oscillator. Consider a quasi-periodically forced, damped, nonlinear oscillator (1.6) in which a SNA exists. If the damping is not too weak, then the attractor lies in a quasi-periodically forced circle flow through an integral manifolds reduction. The complexity of a SNA is often reflected by that on its minimal sets, because, using arguments in [55], such an attractor is made up by minimal sets and their “connecting orbits.” Due to the geometric strangeness, the SNA is however not the entire phase space if it is globally attracting. It follows from Theorem 3 that each minimal set in the SNA is either an almost finite cover of the forcing space (a torus) or a Cantorian, which, by part (2) of the Corollary, is almost automorphic and/or everywhere non-locally connected. In particular, if the rotation number is rationally independent of the forcing frequencies, then it follows from Theorem 8 that the SNA contains a unique minimal set which is everywhere nonlocally connected (which can be a Cantorian carrying almost automorphic dynamics). Of course, minimal dynamics in a SNA can well be Li–Yorke chaotic or even residually Li–Yorke chaotic according to Theorem 2 (as shown in [21], a SNAs can exhibit certain mild chaotic behavior). All these actually suggest that topologically a minimal set in a SNA should typically be everywhere non-local connected and be either an almost finite cover of the forcing space or a Cantorian; and dynamically a minimal set in a SNA should essentially be either almost automorphic or residually Li–Yorke chaotic (see Section 9 for more discussions in this regard on almost periodically forced projective bundle flows). We remark that the kind of complexity of SNAs described above for a quasi-periodically forced, damped, nonlinear oscillator is particularly significant when the damping is weak, in which case the reduced flow on the integral manifold becomes a less/non-smooth, quasiperiodically forced circle flow. To the contrary, when the damping is strong, one can well have cases in which topological complexity of minimal sets plays a less role to the geometric complexity of a SNA in comparing with its measure-theoretic complexity. For a quasi-periodically forced, damping-free nonlinear oscillator (1.8), we note that the flow on each projected Mather set is a (not necessarily smooth) skew-product flow lying in S 1 × Y for which all our results above are applicable. Hence if dynamics on a projected Mather set is not quasi-periodic, then similar topological and dynamical structures are expected for its minimal sets. 3. Preliminary For simplicity, we assume that all T-flows, for T = R or Z, to be considered in the rest of the paper are defined on complete separable metric spaces. We will use the same symbol | · |


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to denote the cardinality of a set, the absolute value of a number, and the norm of a matrix or a function. Also, for a compact metric space X, we let the set 2X of compact subsets of X be endowed with the Hausdorff metric. We say that a flow (X, T) is compact if the phase space X is a compact metric space. Recall that a nonempty compact invariant subset M of a flow (X, T) is minimal if it contains no nonempty, proper, closed invariant subset. A compact flow (X, T) is said to be minimal if X itself is a minimal set, to be strictly ergodic if it is both minimal and uniquely ergodic (i.e., it admits an unique invariant probability measure), and to be positively transitive if for each pair of nonempty open subsets U, V of X there exists t 0 such that U · t ∩ V = ∅. If (X, T) is positively transitive, then the set Tran+ (X) of positive transitive points of X is a residual subset of X, and moreover, Tran+ (X) =

∞

Un · t,

n=1 t0

where {Un }∞ n=1 is a countable basis of X. 3.1. Proximality, distality, and almost automorphy Let (X, T) be a flow and d be the metric on X. Two points x, y ∈ X are said to be positively proximal if lim inft→+∞ d(x · t, x · t) = 0; proximal if lim inft→∞ d(x · t, x · t) = 0. For any x ∈ X, we define PR+ (x) = {x ∈ X: x, x are positively proximal}; PR(x) = {x ∈ X: x, x are proximal}. Now assume that (X, T) is a compact flow and consider the flow maps Πt : X → X: Πt (x) = x ·t, t ∈ T. Then {Πt : t ∈ T} ⊂ X X —the compact Hausdorff space of self-maps of X endowed with the topology of pointwise convergence. The space X X is also a semigroup under the composition of maps on which the right multiplication p → pp0 is continuous for all p0 ∈ X X and the left multiplication p → p0 p is continuous only if p0 is a continuous map. The Ellis semigroup E(X, T) of X is simply defined as E(X, T) = {Πt : t ∈ T}, where the closure is taken under the topology of pointwise convergence. Hence E(X, T) is compact and a sub-semigroup of X X with identity e—the identity map, on which the right multiplication is continuous. We note that the flow (X, T) also induces a natural compact flow (E(X, T), T) on E(X, T): γ · t ≡ Πt γ , γ ∈ E(X, T), t ∈ T. Let ω(e) be the ω-limit set of the identity e in (E(X, T), T). It is clear that two points x, y ∈ X are proximal (resp. positively proximal) iff there exists p ∈ E(X, T) (resp. p ∈ ω(e)) such that p(x) = p(y). x ∈ X is called a positive distal point (resp. distal point) if PR+ (x) = {x} (resp. PR(x) = {x}). A minimal flow (X, T) is called point-distal if it contains a distal point. It is well known that if (X, T) is point-distal, then the set Xd of distal points of X is a residual subset [59]. It is clear that a distal point in a flow must be a positive distal point. In the following, we show that the converse is also true. Proposition 3.1. Let (X, T) be a minimal flow. Then a positive distal point in X is also a distal point. In particular, if (X, T) is not point-distal, then for any x ∈ X, PR+ (x) \ {x} = ∅.

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Proof. Let ω(e) be the ω-limit set of e in the flow (E(X, T), T). Using continuity of the right multiplication and the fact that e is the identity of E(X, T), we have that E(X, T)ω(e) = ω(e) and ω(e) is a sub-semigroup of E(X, T). Let x be a positive distal point. Since (X, T) is minimal, it is easy to see that for any y ∈ X, ω(e)y = {p(y): p ∈ ω(e)} = X. Let y ∈ X \ {x} and consider ωy = {p ∈ ω(e): p(y) = y}. Since ω(e)y = X, ωy is nonempty. Since ωy is a closed sub-semigroup of ω(e) on which the right multiplication is continuous, it follows from a general result due to Namakura [45] (see also [11, Lemma 1]) that ωy contains an idempotent point u, i.e., u2 = u. Clearly u(y) = y. Since u(x) = u(u(x)) and x is a positive distal point, x = u(x). Hence for any p ∈ E(X, T), p(x) = pu(x)

and p(y) = pu(y).

(3.1)

Since pu ∈ E(X, T)ω(e) = ω(e) and x, y are not positively proximal, pu(x) = pu(y). It follows from (3.1) that p(x) = p(y). Since p is arbitrary, x, y are not proximal. This shows that PR(x) = {x}, i.e., x is a distal point. 2 A function f ∈ C(T, X), where X is a complete separable metric space, is said to be almost automorphic if whenever {tn } is a sequence such that f (tn + t) → g(t) ∈ C(T, X) uniformly on compact sets, then also g(t − tn ) → f (t) uniformly on compact sets, as n → ∞. An almost automorphic function valued in a separable Banach space admits well-defined Fourier series which are however not necessarily unique and only converge point-wise in general in term of Bochner–Fejer summation [57,58]. But one can uniquely define the frequency module of an almost automorphic function in the usual way as the smallest additive sub-group of R containing a Fourier spectrum [57]. In this sense, both almost periodic and almost automorphic functions can be viewed as natural generalizations to the periodic ones in the strongest and the weakest sense respectively. A point x in a flow (X, T) is said to be almost automorphic if the orbit {x · t} is an almost automorphic function in t. A flow (X, T) is called almost automorphic minimal if X is the closure of an almost automorphic orbit. An almost automorphic minimal flow is compact, minimal, point-distal, and contains residually many almost automorphic points which are precisely the distal points [59]. Unlike an almost periodic minimal flow, an almost automorphic one can be non-uniquely ergodic, topologically complicated, can admit positive topological entropy, and its general measure-theoretic characterization can be completely random (see [5,15,17,57,62] and references therein). Hence, though an almost automorphic minimal flow resembles an almost periodic one harmonically, it can have certain dynamical, topological, and measure-theoretic complexities which significantly differ from an almost periodic one. Let (X, T) be a flow. A Δ-set S of T is the set of all increasing differences in a sequence ∗ {sn }∞ n=1 , i.e., S = {sn −sm : n > m}, and a Δ -set is a subset of T which has nonempty intersection with each Δ-set. A point x ∈ X is called Δ∗ -recurrent if for every neighborhood V of x, the recurrent time set N (x, V ) = {t ∈ T: x · t ∈ V } is a Δ∗ -set. Almost automorphic points can be characterized by Δ∗ -recurrency as follows. Theorem 3.1. A point x in a compact flow (X, T) is almost automorphic iff it is Δ∗ -recurrent.


847

Proof. The theorem is a classical result of Furstenberg [16] when T = Z. For T = R, the proof is completely similar. 2 3.2. Flow extensions A flow extension (or a flow homomorphism or a factor map) π : (X, T) → (Y, T) is a continuous onto map π : X → Y which preserves the flows. If such a flow extension exists, then (X, T) (or X) is called an extension of (Y, T) (or Y ) and (Y, T) (or Y ) is called a factor of (X, T) (X). Let π : (X, T) → (Y, T) be an extension between compact flows. π is called proximal (resp. distal) if for each y ∈ Y and any two points on π −1 (y) are proximal (resp. distal), called almost N –1 (resp. almost finite to one), if there exists a residual subset X0 ⊂ X such that for any x ∈ X0 π −1 π(x) consists of N points (resp. π −1 π(x) is a finite set), called N –1 (resp. finite to one) if π −1 π(x) consists of N points (resp. π −1 π(x) is a finite set) for all x ∈ X, called open if π : X → Y is an open map, and called semi-open if π : X → Y is a semi-open map, i.e., for any nonempty open subset U of X the image π(U ) has nonempty interior in Y . A 1–1 flow extension is also called a flow isomorphism. Let Rπ = {(x1 , x2 ) ∈ X × X: π(x1 ) = π(x2 )}. The flow (X, T) induces a natural flow (Rπ , T) on Rπ . π is called positively weakly mixing if the flow (Rπ , T) is positively transitive. Using the ω-limit sets of (Rπ , T), it is easy to see that if π is a proximal extension, then it must be a positive proximal extension, i.e., for any x ∈ X, any two points in π −1 π(x) are positively proximal. The general structure of an almost automorphic minimal flow is characterized by the following structure theorem due to Veech [58]. Theorem 3.2. A compact flow is almost automorphic minimal iff it is an almost 1–1 extension of an almost periodic minimal flow. By the above structure theorem, almost automorphic points in an almost automorphic minimal flow are precisely those lying in singleton fibers of the corresponding almost 1–1 extension of an almost periodic minimal flow. Hence an almost automorphic minimal set becomes almost periodic iff every point in the set is an almost automorphic point. Proposition 3.2. Let π : (X, T) → (Y, T) be an extension between minimal flows. Then the following holds. (1) If π is almost finite to one, then there is a positive integer N such that π is an almost N –1 extension. (2) If π is open and finite to one, then π is a distal and N –1 extension for some positive integer N . If, in addition, (Y, T) is point-distal, then so is (X, T). Proof. (1) We let Y∗ = {y ∈ Y : |π −1 (y)| < +∞}. Since π −1 : Y → 2X : y → π −1 (y) is upper semi-continuous, the set Y0 of all continuity points of π −1 is a residual subset of Y . For any given y∗ ∈ Y∗ and y0 ∈ Y0 , we let {tn } ⊂ T be a sequence such that y∗ · tn → y0 . It follows from the continuity of π −1 at y0 that |π −1 (y0 )| |π −1 (y∗ · tn )| = |π −1 (y∗ )| < +∞. Hence Y0 ⊂ Y∗ . Now, for any y1 , y2 ∈ Y0 , the above argument yields that |π −1 (y1 )| |π −1 (y2 )| and |π −1 (y2 )| |π −1 (y1 )|, i.e., the map Y0 → N: y → |π −1 (y)| is a constant, say N .

848


(2) Since π is open, π −1 is continuous, i.e., Y0 = Y . It follows from the proof of (1) above that the map Y → N: y → |π −1 (y)| is a constant N . The continuity of π −1 also implies that there cannot be any proximal pair on each fiber, for otherwise the number of points on some fiber would be smaller than N . A distal extension of a point-distal flow is easily seen to be point-distal. 2 Proposition 3.3. If π : (X, T) → (Y, T) is an open and proximal extension between minimal flows, then it is a positively weakly mixing extension. Proof. We follow the arguments of the proof of Theorem 6.3, in [19]. We first show the following Claim. If x1 , x2 , . . . , xn ∈ X are such that π(x1 ) = π(x2 ) = · · · = π(xn ), then for any x ∈ X there is a positive increasing sequence {tm } ⊂ T such that lim xi · tm = x,

m→∞

for all 1 i n.

(3.2)

Since (X, T) is minimal, the Claim clearly holds for n = 1. By induction, we assume that the Claim is true for some n = k. Then for any x ∈ X there exists a positive increasing sequence {tm } such that limm→∞ xi · tm = x for all 1 i k. Without loss of generality, we let xk+1 · tm be convergent, say to some x ∈ X. Then π(x) = π(x ), and hence lim inft→+∞ d(x · t, x · t) = 0. Using minimality of (X, T) we let {sj } be a positive increasing sequence such that limj →∞ x · sj = x and limj →∞ x · sj = x, i.e., lim lim xi · (tm + sj ) = x,

j →∞ m→∞

i = 1, 2, . . . , k + 1.

It follows that we can take a positive increasing sequence {rj = sj + tm(j ) } for sufficiently large {m(j )} such that lim xi · rj = x,

j →∞

i = 1, 2, . . . , k + 1.

This proves the Claim. Let W, W be two nonempty open subsets of Rπ = {(x1 , x2 ) ∈ X × X: π(x1 ) = π(x2 )}. Since π is an open map, there exist nonempty open sets U, V ; U , V of X such that π(U ) = π(V ), π(U ) = π(V ), W ⊇ (U × V ) ∩ Rπ = ∅, and W ⊇ (U × V ) ∩ Rπ = ∅. For a fixed z0 ∈ π(U ), we have by minimality of (X, T) that there are t1 , t2 , . . . , tn ∈ T such that ni=1 V · ti = X. By relabeling the ti ’s if necessary, we assume without loss of generality an integer 1 m n such that V · ti ∩ π −1 (z0 ) = ∅ for all i = 1, 2, . . . , m and mthat there is −1 −1 i=1 V · ti ⊃ π (z0 ). For each i = 1, 2, . . . , m, we let vi ∈ V be such that vi · ti = yi ∈ π (z0 ). Since π(U ) = π(V ), there is a point ui ∈ U with π(ui ) = π(vi ). Denote xi = ui · ti , i = 1, 2, . . . , m. Then it is clear that π(xi ) = π(ui · ti ) = π(vi · ti ) = π(yi ) = z0 , i.e., xi ∈ π −1 (z0 ), i = 1, 2, . . . , m. By the Claim, there exists t > 0 such that xi · t ∈ U and t + ti > 0 for all i = 1, 2, . . . , m. Since z0 · t = π(xi · t) ∈ π(U ) = π(V ),i = 1, 2, . . . , m, we can take a point b ∈ V such that π(b) = z0 · t. Then b · (−t) ∈ π −1 (z0 ) ⊆ m i=1 V · ti . Hence b · (−t) ∈ V · ti , i.e., b ∈ V · (t + ti ) ∩ V for some 1 i m. Now let a = xi · t. Then π(a) = z0 · t and a ∈ U · (t + ti ) ∩ U . It follows that (a, b) ∈ ((U × V ) · (t + ti )) ∩ (U × V ) ∩ Rπ ⊂ (W · (t + ti )) ∩ W . 2


849

The following is a classical result of Auslander [59]. Proposition 3.4. If π : (X, T) → (Y, T) is an extension between minimal flows, then π is semiopen. It is well known that every extension of minimal flows can be lifted to an open extension by almost 1–1 modifications. To be precise, let π : (X, T) → (Y, T) be an extension between minimal flows, and let Y0 be the set of continuity points of π −1 : Y → 2X : y → π −1 (y). Recall that Y0 is an invariant residual subset of Y . Let Y ∗ = cl({π −1 (y): y ∈ Y0 }) and (2X , T) be the flow on 2X induced from (X, T). Then Y ∗ is an invariant closed subset of 2X . It is easy to see that for any y ∗ ∈ Y ∗ , π(y ∗ ) is a singleton. Define τ : Y ∗ → Y as such that τ (y ∗ ) = π(x), x ∈ y ∗ . Then τ : (Y ∗ , T) → (Y, T) is a flow extension. Also let X ∗ = {(x, y ∗ ) ∈ X × Y ∗ : x ∈ Y ∗ }. Then X ∗ is a closed invariant subset of (X × Y ∗ , T). Denote τ : X ∗ → X and π : X ∗ → Y ∗ as the natural projections. Proposition 3.5. The following holds. (1) (X ∗ , T) is a minimal flow and the following diagram (X, T)

τ

(X ∗ , T) π

π

(Y, T)

τ

(Y ∗ , T)

commutes. (2) τ, τ are almost 1–1 extensions. (3) π is an open extension. Proof. See Theorem 3.1 in [59] or Lemma 14.41 in [3].

2

3.3. Entropies Let (X, T) be a compact flow and consider the time-1 map T : X → X: x → x · 1. We denote the discrete flow induced by T simply by (X, T ). Let BX denote the collection of all Borel subsets of X. A cover of X is a finite family of Borel subsets of X whose union is X. A partition of X is a cover of X whose elements are pairwise disjoint. We denote the set of partitions of X by PX and the set of open covers of X by CX . An open cover U is said to be finer than V (denoted by U V) if each element of U is contained in some element of V. Let U ∨ V

= {U ∩ V : U ∈ U, V ∈ V}. Given non-negative integers M, N N N = −n U . Also, given U ∈ C , we let N (U) be the minimal and U ∈ CX or PX , we let UM X n=M T cardinality among all cardinalities of sub-open-covers of U and let H (U) = log N (U). Clearly, if there is another open cover V U , then H (V) H (U). In fact, for any two covers U , V ∈ CX we have H (U ∨ V) H (U) + H (V). Consequently, for any open cover U ∈ CX ,

850


an = H (U0n−1 ) is a bounded sub-additive sequence, i.e., an+m an + am , n, m ∈ N. Hence limn→∞ ann exists and equals infn1 ann . This limit, denoted by htop (T , U), is called the entropy of U . The topological entropy htop (X, T ) of (X, T ) is simply defined as htop (X, T ) = sup htop (T , U), U ∈ CX

and, the topological entropy htop (X, T) of (X, T) is simply defined as htop (X, T ). Let M(X), M(X, T ), and Me (X, T ), respectively, be the set of Borel probability measures on X, the set of invariant Borel probability measures on X, and the set of invariant ergodic measures on X, respectively. For given α ∈ PX and μ ∈ M(X), define Hμ (α) = −μ(A) log μ(A). A∈α

Now let μ ∈ M(X, T ). Then for a given α ∈ PX , Hμ (α0n−1 ) is a non-negative sub-additive sequence. The measure-theoretic entropy of μ relative to α is defined by 1 n−1 1 Hμ α0 = inf Hμ α0n−1 , n→∞ n n1 n

hμ (T , α) = lim

and the measure-theoretic entropy of μ is defined by hμ (X, T ) = sup hμ (T , α). α∈PX

The classical variational principle of entropy says that htop (X, T ) =

sup

μ∈M(X,T )

hμ (X, T )

and the supremum can be attained by an invariant ergodic measure. We refer the readers to [10,46,60] for more information on the classical theory of measure-theoretic and topological entropies. Given partitions α, β ∈ PX , μ ∈ M(X) and σ -algebra A ⊆ BX , define Hμ (α | β) = Hμ (α ∨ β) − Hμ (β), Hμ (α | A) = −E(1A | A) log E(1A | A) dμ, A∈α X

Hμ (α | β ∨ A) = Hμ (α ∨ β | A) − Hμ (β | A), where E(1A | A) is the expectation of 1A with respect to A. Then Hμ (α | β) (resp. Hμ (α | A)) increases with respect to α and decreases with respect to β (resp. A). Let μ ∈ M(X, T ) and A be an invariant measurable σ -algebra of X. It is not hard to see that for a given α ∈ PX , Hμ (α0n−1 | A) is a bounded sub-additive sequence. The measure-theoretic conditional entropy of α with respect to A is defined by 1 n−1 1 A = inf Hμ α0n−1 A , Hμ α0 n→∞ n n1 n

hμ (T , α | A) = lim


851

and the measure-theoretic conditional entropy of (X, T , μ) with respect to A is defined by hμ (X, T | A) = sup hμ (T , α | A). α∈PX

∞

It is easy to see that hμ (T , α | A) = Hμ (α | i=1 (T −i α) ∨ A). Let π : (X, T ) → (Y, S) be an extension between compact discrete flows. For each α ∈ PX , the measure-theoretic conditional entropy of α with respect to (Y, S) is defined by 1 hμ (T , α | Y ) = hμ T , α π −1 (BY ) = lim Hμ α0n−1 π −1 (BY ) , n→∞ n and the measure-theoretic conditional entropy of (X, T , μ) with respect to (Y, S) is defined by hμ (X, T | Y ) = sup hμ (T , α | Y ). α∈PX

When (Y, S) is the trivial flow, the above coincides with the measure-theoretic entropy of (X, T ) with respect to μ. Let μ ∈ M(X, T ) and A be an invariant sub-σ -algebra of BX . The relative Pinsker σ -algebra Pμ (A) is defined as the smallest σ -algebra containing {ξ ∈ PX : hμ (T , ξ | A) = 0}. When A = {∅, X}, Pμ (A) coincides with Pμ —the classical Pinsker σ -algebra of the system. It is easy k) = Pμ (A) for any k ∈ Z \ {0}, to see that Pμ (A) is invariant, Pμ (A) ⊇ Pμ ∨ A, and Pμ (A, smallest σ -algebra containing {ξ ∈ PX : hμ (T k , ξ | A) = 0}. where Pμ (A, k) denotes the

+∞ ∞ − −i T −i Given α ∈ PX , we let α = n=1 T α and α = n=−∞ T α. Then a relative version of the classical Pinsker formula (see [18,20,46]) says that if α, β ∈ PX , then hμ (T , α ∨ β | A) = hμ (T , β | A) + Hμ α β T ∨ α − ∨ A . (3.3) In particular, when A is trivial, hμ (T , α ∨ β) = hμ (T , β) + Hμ (α | β T ∨ α − ). Proposition 3.6. Let μ and A be given as above. Then for each ξ ∈ PX , hμ T , ξ Pμ (A) = Hμ ξ ξ − ∨ Pμ (A) = Hμ (ξ | ξ − ∨ A) = hμ (T , ξ | A). Proof. For any α ∈ PX and any invariant sub-σ -algebra C of BX , it is easy to see that hμ (α, T | C) = Hμ (α | α − ∨ C). Hence for each ξ ∈ PX , we have hμ T , ξ Pμ (A) = Hμ ξ ξ − ∨ Pμ (A) and Hμ (ξ | ξ − ∨ A) = hμ (T , ξ | A). Now we fix ξ ∈ PX and let η ⊂ Pμ (A) be any finite measurable partition. It follows from (3.3) that Hμ (ξ | ξ − ∨ A) + Hμ η η− ∨ ξ T ∨ A = hμ (T , ξ | A) + Hμ η ξ T ∨ η− ∨ A = hμ (T , ξ ∨ η | A) = Hμ (ξ ∨ η | ξ − ∨ η− ∨ A) = Hμ (η | ξ − ∨ η− ∨ A ∨ ξ ) + Hμ (ξ | ξ − ∨ η− ∨ A).

(3.4)

852


Since η ⊂ Pμ (A), we have Hμ (η | η− ∨ A) = hμ (T , η | A) = 0, Hμ η η− ∨ ξ T ∨ A = 0,

andHμ (η | η− ∨ ξ − ∨ A ∨ ξ ) = 0.

(3.5)

Combining (3.4) and (3.5), we have Hμ (ξ | ξ − ∨ η− ∨ A) = Hμ (ξ | ξ − ∨ A).

(3.6)

Let ηn ⊂ Pμ (A) be an increasing sequence of finite measurable partitions of X such that

∞ − ∨ P (A)) = μ n=1 ηn = Pμ (A) (mod μ). It follows from (3.6) that Hμ (ξ | ξ − Hμ (ξ | ξ ∨ A). 2 3.4. Measure-theoretic extensions Let (X, B, μ) be a standard Borel space, μ be a regular probability measure on X, and T : X → X be a measurable transformation. The quadruple (X, B, μ, T ) is said to be a metric dynamical system (MDS for short) if T is measure preserving, i.e., μ(B) = μ(T −1 B) for all B ∈ B. If, in addition, T is bijective and T −1 is also measure-preserving, then (X, B, μ, T ) is said to be invertible. In the following, a MDS is always assumed to be invertible. A MDS (X, B, μ) is said to be ergodic if whenever A ∈ B is such that μ(AΔT −1 A) = 0 then either μ(A) = 0 or μ(A) = 1. Let (X, B, μ, T ) be a MDS and (Y, C, ν, S) be a measure-theoretic factor of (X, B, μ, T ), i.e., there exists a measure-preserving map π : X → Y , called a measure-theoretic factor map or extension, such that π ◦ T = S ◦ π μ-a.e. It is well known that μ admits a ν-a.s. unique disintegration μ = Y μy dν(y) over (Y, C, ν, S) [16, Proposition 5.9], where μy , y ∈ Y , are Borel probability measures on X satisfying μSy = T μy ,

ν-a.e. y ∈ Y.

(3.7)

For each i = 1, 2, . . . , n, let πi : (Xi , Bi , μi , Ti ) → (Y, C, ν, S) be a factor map between MDSs and μi = Y μi,y dν(y) be the disintegration of μi over (Y, C, ν, S). Define μ 1 × Y μ 2 × Y · · · ×Y μ n =

μ1,y × μ2,y × · · · × μn,y dν(y).

(3.8)

Y

Then by (3.7), T1 × T2 × · · · × Tn preserves the measure μ1 ×Y μ2 ×Y · · · ×Y μn . The MDS (X1 × X2 × · · · × Xn , B1 × B2 × · · · × Bn , μ1 ×Y μ2 ×Y · · · ×Y μn , T1 × T2 × · · · × Tn ) is called the product of (Xi , Bi , μi , Ti ), i = 1, 2, . . . , n, relative to (Y, C, ν, S). Let π : (X, B, μ, T ) → (Y, C, ν, S) be a factor map between ergodic MDSs and let μ = Y μy dν(y) be the disintegration of μ over (Y, C, ν, S). π is said to be relatively weakly mixing if the MDS (X × X, B × B, μ ×Y μ, T × T ) is ergodic, and is said to be compact if there exists a dense set F of functions in L2 (X, B, μ) with the following properties: for any f ∈ F and δ > 0, there exists a finite set of functions g1 , g2 , . . . , gk ∈ L2 (X, B, μ) such that min1ik T n f − gj L2 (μy ) < δ, ν-a.e. y ∈ Y , for all n ∈ Z. Now consider a compact discrete flow (X, T ). For μ ∈ M(X, T ), we let Pμ (A) be the relative Pinsker σ -algebra of invariant σ -algebra A of BX and denote the completion of Borel σ -algebra BX of X under μ by Bμ . Then (X, Bμ , μ, T ) is a Lebesgue system. Let (Z, Z, η, R) be the


853

Pinsker factor of (X, Bμ , μ, T ) and π : (X, Bμ , μ, T ) → (Z, Z, η, R) be the measure-theoretic Pinsker factor map with respect to A, i.e., π : X → Z is measure-preserving, π ◦ T = R ◦ π −1 μ-a.e., and π Z = Pμ (A) (mod μ). Let μ = Z μz dη(z) be the disintegration of μ over (Z, Z, η, R). Then for each integer n 2, (n) (n) -invariant measure on X (n) , where λA n (μ) = Z μz dη(z) is a T μ(n) μz × μz × · · · × μz , z =

X (n) = X ×X×

· · · × X,

T (n) = T ×T ×

· · · × T .

n

n

n

Moreover, it follows from basic properties of disintegration [16,50] that for any A1 , A2 , . . . , An ∈ Bμ and α ∈ PX , λA n (μ)

n

Ai =

i=1

n E 1A Pμ (A) (x) dμ(x)

(3.9)

X i=1

and Hμ α Pμ (A) =

−E 1A Pμ (A) (x) log E 1A Pμ (A) (x) dμ(x)

X A∈α

=

−μπ(x) (A) log μπ(x) (A) dμ(x)

X A∈α

=

Z

=

−μπ(x) (A) log μπ(x) (A) dμz (x) dη(z)

X A∈α

Hμz (α) dη(z).

(3.10)

Z

The following result should be well-known. As we are not aware of a suitable reference for it, a proof of the result is given for the sake of completeness. Theorem 3.3. Let (X, T ) be a compact discrete flow, μ ∈ Me (X, T ), and A be a T -invariant σ -algebra of BX . Let π : (X, Bμ , μ, T ) → (Z, Z, η, R) be the measure-theoretic Pinsker factor map with respect to A and μ = Z μz dη(z) be the disintegration of μ over (Z, Z, η, R). If hμ (X, T | A) > 0, then (1) μz is non-atomic for η-a.e. z ∈ Z; (n) -invariant ergodic measure on X (n) . (2) λA n (μ) is a T Proof. (1) If (1) is not true, then the ergodicity of μ implies that there exists a positive integer k such that μz is purely atomic with exactly k points in its support for η-a.e. z ∈ Z. Hence for each β ∈ PX and η-a.e. z ∈ Z, Hμz (β) log k.

(3.11)

854


Now for any α ∈ PX , we have by Proposition 3.6, (3.10), and (3.11) that n−1 1 −i T α Pμ (A) hμ (T , α | A) = hμ T , α Pμ (A) = lim Hμ n→∞ n i=0 n−1 1 1 −i = lim Hμz T α dη(z) lim log k dη(z) = 0. n→∞ n n→∞ n Z

i=0

Z

Since α is arbitrary, hμ (X, T | A) = 0, a contradiction. (2) We first claim that π : (X, Bμ , μ, T ) → (Z, Z, η, T ) is a relatively weakly mixing extension. If not, then by a classical result of Furstenberg and Zimmer [16,63,64] there exist (Y, C, ν, S) and factor maps π1 : X → Y , π2 : Y → Z such that π = π2 π1 and π2 is a nontrivial compact extension. We note that π1−1 π2−1 Z = Pμ (A). Now, for any A ∈ C, we have by Proposition 3.6 that hμ T , π1−1 A, π1−1 (Y \ A) A = hμ T , π1−1 A, π1−1 (Y \ A) Pμ (A) = hμ T , π1−1 A, π1−1 (Y \ A) π1−1 π2−1 Z = hν S, {A, Y \ A} π2−1 Z . Since π2 is a compact extension, the conditional sequential entropy characterization of compact extensions [24] implies that hμ (T , {π1−1 A, π1−1 (Y \ A)} | A) = 0. This shows that π1−1 A ∈ Pμ (A). Since A is arbitrary, π1−1 C ⊆ Pμ (A) (mod μ), and moreover, Pμ (A) = π1−1 π2−1 Z ⊆ π1−1 C ⊆ Pμ (A) (mod μ). Hence π1−1 π2−1 Z = π1−1 C = Pμ (A) (mod μ), i.e., π2−1 Z = C (mod ν). This shows that π2 is an isomorphism, a contradiction to the fact that π2 is a nontrivial compact extension. Hence π : (X, Bμ , μ, T ) → (Z, Z, η, T ) is a relatively weakly mixing extension. × μ ×Z · · · ×Z μ and μ is ergodic, we have by Proposition 6.3 in [16] Since λA n (μ) = μ Z

n

that (X (n) , B (n) , μ(n) , T (n) ) is ergodic, where B (n) = B ×B×

· · · × B.

2

n

4. Entropy and ordering The main purpose of this section is to prove Theorem 1. While Theorem 1 should more or less follow from an entropy inequality in [9], we will prove a general entropy preservation result (Theorem 4.1 below) under an ordering condition which not only implies Theorem 1 but also has the advantage of treating other zero entropy problems. For instance, applying our entropy preservation result, one can similarly show that all almost automorphic minimal sets obtained in [56,57] for an almost periodically forced monotone system admit zero topological entropy. 4.1. Entropy preservation Given a compact discrete flow (X, T ), a finite subset A of X is called a full scrambled set if for each map f : A → A there exists an infinite sequence {ni } ⊂ N such that limi→∞ T ni x = f (x) for any x ∈ A.


855

Lemma 4.1. Let π : (X, T ) → (Y, S) be an extension between compact discrete flows and μ be an ergodic measure of (X, T ). If hμ (X, T | Y ) > 0, then for each integer n 2 there exist y ∈ Y and x1 , x2 , . . . , xn ∈ π −1 (y) such that x1 , x2 , . . . , xn are pairwise different and {x1 , x2 , . . . , xn } is a full scrambled set of (X, T ). Proof. We follow the arguments in the proof of Theorem 2.1 in [8]. −1 Let (Z, Z, η, R) be the Pinsker factor of (X, Bμ , μ, T ) with respect to π BY and μ = μ dη(z) be the disintegration of μ over (Z, Z, η, R). For each integer n 2, let λn (μ) = z Z (n) n Z μz dη(z) and W = supp(λn (μ)). Since μ is ergodic and hμ (X, T | Y ) > 0, we have by Proposition 3.3 that μz is non-atomic for η-a.e. z ∈ Z and λn (μ) is an ergodic measure. Hence n (W n , T (n) ) is transitive, i.e., it contains a transitive point—point whose orbit is dense. Let Wtrans n (n) denote the set of all transitive points of (W , T ) and Gn be the set of generic points in W n −1 with respect to λn (μ), i.e., Gn = {w ∈ W n : N1 N i=0 δT i ×T i ×···×T i (w) → λn (μ) as N → ∞}. n Since λn (μ) is ergodic, we have Gn ⊂ WTrans . Then by Birkhoff ergodic theorem, 1 = λn (μ)(Gn ) =

μ(n) z (Gn ) dη(z). Z (n)

It follows that there exists a subset Zn of Z with η(Zn ) = 1 such that μz (Gn ) = 1 and μz is non-atomic for all z ∈ Zn . For each z ∈ Zn , let Sz = supp(μz ). Then Sz is a closed subset of X without isolated points and n ∩ Sz(n) =: Lnz , Gn ∩ Sz(n) ⊂ Wtrans

where Sz(n) = Sz × Sz × · · · × Sz .

n

(n)

(n)

(n)

Since μz (Gn ∩ Sz ) = 1, Gn ∩ Sz

(n)

is a dense subset of Sz . This shows that for each z ∈ Zn ,

Sz(n) = cl Lnz ⊆ W n .

(4.1)

n . By (4.1) and the fact that S is not a Now, fix z ∈ Zn and take (x1 , x2 , . . . , xn ) ∈ Lnz ⊂ Wtrans z singleton, x1 , x2 , . . . , xn are pairwise different. Let A = {x1 , x2 , . . . , xn } and f : A → A be any (n) map. We have by (4.1) that (f (x1 ), f (x2 ), . . . , f (xn )) ∈ Sz ⊂ W n . Since (x1 , x2 , . . . , xn ) ∈ n Wtrans , there exists an infinite sequence {ni } ⊂ N such that

lim T ni x1 , T ni x2 , . . . , T ni xn = f (x1 ), f (x2 ), . . . , f (xn ) ,

i→∞

i.e., limi→∞ T ni x = f (x) for all x ∈ A. This shows that A is a full scrambled set of (X, T ). It remains to show that there exists y ∈ Y such that {x1 , x2 , . . . , xn } ⊆ π −1 (y). If this is not true, then there exist two disjoint nonempty open subsets U1 and U2 of Y such that {π(xi )}ni=1 ⊂ U1 ∪ U2 and {π(xi )}ni=1 ∩ Uj = ∅, j = 1, 2. For each i = 1, 2, . . . , n, take s(i) ∈ {1, 2} such that xi ∈ π −1 Us(i) . Since (x1 , x2 , . . . , xn ) ∈ supp(λn (μ)) and ni=1 π −1 Us(i) is an open neighborhood of (x1 , x2 , . . . , xn ), we have n −1 π Us(i) > 0. λn (μ) i=1

856


But by (3.9) and the facts that π −1 Us(i) ∈ Pμ (π −1 BY ), π −1 U1 ∩ π −1 U2 = ∅, and {s(1), . . . , s(n)} = {1, 2}, we also have λn (μ)

n

π

−1

Us(i)

n E 1π −1 Us(i) Pμ π −1 BY (x) dμ(x) =

i=1

X i=1

=

n

1π −1 Us(i) (x) dμ(x) = 0,

X i=1

which is a contradiction.

2

Lemma 4.2. Let π : (X, T ) → (Y, S) be an extension between compact discrete flows. If htop (X, T ) > htop (Y, S), then for each integer n 2 there exist y ∈ Y and x1 , x2 , . . . , xn ∈ π −1 (y) such that x1 , x2 , . . . , xn are pairwise different and {x1 , x2 , . . . , xn } is a full scrambled set of (X, T ). Proof. Since htop (X, T ) > htop (Y, S), htop (Y, S) < +∞. By the variational principle of entropy there exists an ergodic measure μ of (X, T ) with hμ (X, T ) > htop (Y, S). Hence ν = φμ ∈ Me (Y, S) and hμ (X, T ) > hν (Y, S).

∞ ∞ Let {αi }∞ X , {βj }j =1 ⊂ PY be such that α1 α2 · · ·, i=1 αi = BX (mod μ), i=1 ⊂ P

β1 β2 · · · , and ∞ β = B (mod ν). We have by (3.3) that j Y j =1 T hμ T , αi ∨ π −1 βj = hμ T , π −1 βj + Hμ αi π −1 βj ∨ (αi )− .

(4.2)

Since hμ (T , π −1 βj ) = hν (S, βj ), (4.2) yields that T Hμ αi π −1 βj ∨ (αi )− hμ (T , αi ) − hν (βj , S) hμ (T , αi ) − hν (Y, S).

(4.3)

T

Note that (π −1 βj ) ∨ (αi )− π −1 BY ∨ (αi )− as j → ∞. Taking j → ∞ in (4.3), we have by Matingale theorem that Hμ (αi | π −1 BY ∨ (αi )− ) hμ (T , αi ) − hν (Y, S). Hence hμ (X, T | Y ) hμ (αi , T | Y ) = Hμ αi π −1 BY ∨ (αi )− hμ (T , αi ) − hν (Y, S). (4.4) Taking i → ∞ in (4.4), we have by Kolmogorov–Sinai theorem that hμ (X, T | Y ) hμ (X, T ) − hν (Y, S) > 0. The lemma now follows from Lemma 4.1.

2

When hν (Y, S) < +∞, it can be shown that hμ (X, T | Y ) = hμ (X, T ) − hν (Y, S). When hν (Y, S) = +∞, we note that hμ (X, T ) = hν (Y, S) = +∞. But in this case, it can also happen that hμ (X, T | Y ) > 0. Therefore the condition hμ (X, T | Y ) > 0 in Lemma 4.1 is more general than the condition hμ (X, T ) > hν (Y, S) in Lemma 4.2. Given an integer n 2, we denote by Pern (X) the set of all coordinate permutations on X (n) . An n-partial order relation R on X is a subset of X (n) such that


857

(a) τ0 (R) ∩ R = ∅ for some τ0 ∈ Pern (X); (n) (b) R is essentially closed, i.e., for any {wk }∞ k=1 ⊂ R and w = (x1 , x2 , . . . , xn ) ∈ X , if limk→∞ wk = w and xi = xj for 1 i = j n, then w ∈ R. We say that a compact flow (X, T) = (X, {Πt }t∈T ) preserves an n-partial order R if Π × Πt × · · · × Πt (R) ⊆ R t

n

for all t > 0. Also, we refer the relation PRne (X, T) = {(x1 , x2 , . . . , xn ): xi = xj , 1 i < j n, and lim inft→+∞ diam({x1 · t, x2 · t, . . . , xn · t}) = 0} as the proper n-proximal relation of (X, T). Theorem 4.1. Let π : (X, T) → (Y, T) be an extension between compact flows. If for some integer n 2, the flow (X, T) preserves an n-partial order relation R on X and PRne (X, T) ∩ n Rπ ⊆ τ ∈Pern (X) τ (R), where Rπn = {(x1 , x2 , . . . , xn ) ∈ X (n) : π(x1 ) = · · · = π(xn )}, then htop (X, T) = htop (Y, T). Proof. Let T and S be the time-1 maps of (X, T) and (Y, T) respectively. If htop (X, T ) = htop (Y, S), then htop (X, T ) > htop (Y, S). It follows from Lemma 4.2 that there exist y ∈ Y and x1 , x2 , . . . , xn ∈ π −1 (y) such that xi = xj , 1 i = j n, and {x 1 , x2 , . . . , xn } is a full scrambled set of (X, T ). Clearly, (x1 , x2 , . . . , xn ) ∈ PRen (X, T) ∩ Rπ ⊆ τ ∈Pern (X) τ (R). Without loss of generality, we assume that (x1 , x2 , . . . , xn ) ∈ R. Since R is an n-partial relation, there exists τ0 ∈ Pern (X) such that τ0 (R) ∩ R = ∅. Denote τ0 (x1 , x2 , . . . , xn ) = (x1 , x2 , . . . , xn ). Then xi = xj for all 1 i = j n. Since {x1 , x2 , . . . , xn } is a full scrambled set of (X, T ), there exists a sequence {ni } ⊂ N such that lim T ni x1 , T ni x2 , . . . , T ni xn = x1 , x2 , . . . , xn . i→∞

Note that (T ni x1 , T ni x2 , . . . , T ni xn ) ∈ R and xi = xj for all 1 i = j n. We have by the essential closeness of R that (x1 , x2 , . . . , xn ) ∈ R. Now (x1 , x2 , . . . , xn ) ∈ τ0 (R), a contradiction to the fact that τ0 (R) ∩ R = ∅. Hence htop (X, T ) = htop (Y, S). 2 Corollary 4.1. Let (X, T) be a compact flow which preserves an n-partial order relation R on X for some integer n 2. If PRne (X, T) ⊆ τ ∈Pern (X) τ (R), then htop (X, T) = 0. Proof. It follows from Theorem 4.1 by taking (Y, T) as the trivial flow.

2

4.2. Zero entropy of APCFs The follows theorem immediately implies Theorem 1. Theorem 4.2. For a SPCF (S 1 × Y, T), htop (S 1 × Y, T) = htop (Y, T). Proof. Let π : S 1 × Y → Y be the natural projection. Clearly, π : (S 1 × Y, T) → (Y, T) is a flow extension. Consider R = e2πφ1 i , y , e2πφ2 i , y , e2πφ3 i , y : y ∈ Y and φ1 < φ2 < φ3 < 1 + φ1 .

858


It is clear that R is a 3-partial relation on X = S 1 × Y which is preserved by (X, T) and PR3e (X, T) ∩ Rπ3 ⊆ τ ∈Per3 (X) τ (R). It follows from Theorem 4.1 that htop (S 1 × Y, T) = htop (Y, T). 2 5. Li–Yorke chaos and proximality 5.1. General conditions on the existence of Li–Yorke chaos The following lemmas will be needed in the proof of our general result on Li–Yorke chaos of a proximal extension. Lemma 5.1. Let X and Y be compact metric spaces, π : X → Y be a semi-open, surjective, continuous map, and K be a residual subset of X. Then A = AK = y ∈ Y : K ∩ π −1 (y) is a residual subset of π −1 (y) is a residual subset of Y . Proof. See Proposition 3.1 in [59].

2

Let X be a compact metric space. A subset K ⊆ X is called a Mycielski set if it is a countable union of Cantor sets. The following result is a special case of Mycielski theorem [44]. Lemma 5.2. Let X be a compact metric space with no isolated point. If R is a residual subset of X × X, then there exists a Mycielski set K of X which is dense in X such that for any two distinct points x, y in K, (x, y) ∈ R. Proof. See Theorem 1 in [44] or Lemma 2.6 in [8].

2

The following result is more or less known for maps [1] but the proof does not automatically carry over to the case of R-flows. Theorem 5.1. Let π : (X, T) → (Y, T) be a proximal extension of minimal flows which is not almost 1–1. Then there exists a residual subset Yc of Y such that each fiber π −1 (y), y ∈ Yc , admits an uncountable scrambled set. In particular, (X, T) is Li–Yorke chaotic. Proof. Let Y0 , Y ∗ , X ∗ and π , τ, τ be as in Proposition 3.5. Recall that π is an open extension. Let y ∗ ∈ Y ∗ and x, x ∈ y ∗ . Since π(x) = π(x ) = τ (y ∗ ) and π is a proximal extension, x, x are proximal, so are (x, y ∗ ), (x , y ∗ ). This shows that π is a proximal extension as well. It follows from Proposition 3.3 that (Rπ , T) is positively transitive. Thus Tran+ (Rπ ) is a residual subset of Rπ . Since ρ : Rπ → Y ∗ : ((x, y ∗ ), (x , y ∗ )) → y ∗ is an open map, it follows from Lemma 5.1 that there is a residual subset Yc∗ ⊆ Y ∗ such that for every point y ∗ ∈ Yc∗ , Tran+ (Rπ ) ∩ ((π )−1 (y ∗ ) × (π )−1 (y ∗ )) is a residual subset of (π )−1 (y ∗ ) × (π )−1 (y ∗ ). Let Yc = τ (Yc∗ ). Since by Proposition 3.4 τ is semi-open, Lemma 5.1 implies that Yc is a residual set. Fix y0 ∈ Yc and let y0∗ ∈ Yc∗ be such that τ (y0∗ ) = y0 , i.e., y0∗ ⊂ π −1 (y0 ). We first claim that the closed subset (π )−1 (y0∗ ) = {(x, y0∗ ): x ∈ y0∗ } has no isolated point. Suppose for


859

contradiction that there exists x ∈ y0∗ such that (x, y0∗ ) is an isolated point of (π )−1 (y0∗ ). Obviously, {((x, y0∗ ), (x, y0∗ ))} is a relatively open subset of (π )−1 (y0∗ ) × (π )−1 (y0∗ ), hence ((x, y0∗ ), (x, y0∗ )) ∈ Trans+ (Rπ ) as Trans+ (Rπ )∩((π )−1 (y0∗ )×(π )−1 (y0∗ )) is a residual subset of (π )−1 (y0∗ )×(π )−1 (y0∗ ). It follows that Rπ = {(x ∗ , x ∗ ): x ∗ ∈ X ∗ }, i.e., π is an isomorphism, and hence π −1 (y ∗ ) is a singleton for each y ∗ ∈ Y ∗ . In particular, π −1 (y) is a singleton for each y ∈ Y0 , i.e., π is an almost 1–1 extension, a contradiction. Next, by applying Lemma 5.2 with R = Tran+ (Rπ ) ∩ ((π )−1 (y0∗ ) × (π )−1 (y0∗ )), we obtain a Mycielski set Ky∗0 ⊂ (π )−1 (y0∗ ) such that for any two distinct points x ∗ , x1∗ in Ky∗0 , (x ∗ , x1∗ ) ∈ Tran+ (Rπ ). Now let Ky0 = {x ∈ X : (x, y0∗ ) ∈ Ky∗0 }. Since Ky0 and Ky∗0 are homeomorphic, Ky0 ⊆ π −1 (y0 ) is also a Mycielski set hence it is uncountable. It remains to show that Ky0 is a scrambled subset of (X, T). Let x, x1 be any two distinct points in Ky0 . We note that ((x, y0∗ ), (x1 , y0∗ )) ∈ Tran+ (Rπ ). Since both ((x, y0∗ ), (x, y0∗ )) and ((x, y0∗ ), (x1 , y0∗ )) are in Rπ , there are positive increasing sequences ti → +∞, sj → +∞ such that x, y0∗ , x1 , y0∗ · ti = x, y0∗ , x1 , y0∗ , i→∞ lim x, y0∗ , x1 , y0∗ · sj = x, y0∗ , x, y0∗ . lim

j →∞

This implies that limi→∞ (x, x1 ) · ti = (x, x1 ) and limj →∞ (x, x1 ) · sj = (x, x), i.e., {x, x1 } is a Li–Yorke pair of (X, T). Hence Ky0 is an uncountable scrambled set of X. This completes the proof. 2 An extension π : (X, T) → (Y, T) between compact flows is said to be positively asymptotic if for each y ∈ Y , any two points x, x ∈ π −1 (y) are positively asymptotic, i.e., limt→+∞ d(x · t, x · t) = 0, where d is a compatible metric on X. Corollary 5.1. Let π : (X, T) → (Y, T) be a positively asymptotic extension between minimal flows. Then π is an almost 1–1 extension. Proof. Obviously, π is a proximal extension. If π is not almost 1–1, then by Theorem 5.1 there exists an uncountable scrambled set Ky ⊂ π −1 (y) for some y ∈ Y . In particular, there exist x1 , x2 ∈ Ky ⊂ π −1 (y) which are not positively asymptotic, a contradiction. 2 5.2. A strict dynamical dichotomy of minimal sets We now consider a SPCF (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ) in the form (1.1), i.e., Λt (s0 , y0 ) = ψ(s0 , y0 , t), y0 · t , t ∈ T. We denote dY as a compatible metric on Y and π : S 1 × Y → Y as the natural projection. For s1 = s2 ∈ S 1 , we denote [s1 , s2 ] as the closed arc from s1 to s2 oriented counter-clockwise, and let (s1 , s2 ) = [s1 , s2 ] \ {s1 , s2 }. We also denote [s, s] = {s} for any s ∈ S 1 . For a fixed y0 ∈ Y , consider the family of maps ft : S 1 → S 1 : s → ψ(s, y0 , t), t ∈ T. Then each ft is an orientation preserving homeomorphism of S 1 .

860


Lemma 5.3. Let s1 = s2 ∈ S 1 and {tj } ⊂ T be such that limj →∞ ftj (si ) = si for some si ∈ S 1 , i = 1, 2. Then the following holds. (1) If s1 = s2 , then limj →∞ ftj ([s1 , s2 ]) = [s1 , s2 ] under the Hausdorff metric on 2S . (2) If s1 = s2 = s , then either limj →∞ ftj ([s1 , s2 ]) = {s } or limj →∞ ftj ([s2 , s1 ]) = {s } by taking subsequences if necessary. (3) If s1 = s2 = s and lim supj →∞ ftj ([s1 , s2 ]) = S 1 , then limi→∞ ftj ([s1 , s2 ]) = {s }. (4) If limt→+∞ |ft (s1 ) − ft (s2 )| = 0, then there exists t0 > 0 such that, as t t0 , |ft (s1∗ ) − ft (s2∗ )| |ft (s1 ) − ft (s2 )| for either all s1∗ , s2∗ ∈ [s1 , s2 ] or all s1∗ , s2∗ ∈ [s2 , s1 ]. In particular, limt→+∞ |ft (s1∗ ) − ft (s2∗ )| = 0 for either all s1∗ , s2∗ ∈ [s1 , s2 ] or all s1∗ , s2∗ ∈ [s2 , s1 ]. 1

Proof. (1)–(3) are obvious. (4) Denote A1 = [s1 , s2 ] and A2 = [s2 , s1 ]. We let 0 < 0 < diam(S 1 )

diam(S 1 ) 3

be such that if |s −s | 0

then |ψ(s, y, r) − ψ(s , y, r)| for all y ∈ Y and r ∈ [0, 1]. We also let t0 > 0 be such 3 that |ft (s1 ) − ft (s2 )| 0 for all t t0 . Then for any t t0 , there exists i(t) = 1 or 2 such that diam(ft (Ai(t) )) |ft (s1 ) − ft (s2 )| 0 . Since diam(A1 ∪ A2 ) = diam(S 1 ), we have 2 diam ft S 1 \ Ai(t) diam S 1 − 0 > diam S 1 3 for all t t0 . In the following, we show that i(t) equals a constant, say i0 = 1 or 2, on [t0 , +∞) ∩ T. If this is not true, then there exist t1 ∈ [t0 , +∞) ∩ T and r ∈ (0, 1] ∩ T such that i(t1 ) = i(t1 + r). On one hand, since i(t1 ) = i(t1 + r), we have 2 diam ft1 +r (Ai(t1 ) ) diam ft1 +r S 1 \ Ai(t1 +r) diam S 1 − 0 > diam S 1 . 3 But on the other hand, since diam(ft1 (Ai(t1 ) )) 0 , we have diam(S 1 ) diam ft1 +r (Ai(t1 ) ) = diam ψ(s, y · t1 , r): s ∈ ft1 (Ai(t1 ) ) . 3 This is a contradiction. Now for any s1∗ , s2∗ ∈ Ai0 and t t0 , we have |ft (s1∗ ) − ft (s2∗ )| diam(ft (Ai(t) )) |ft (s1 ) − ft (s2 )|. 2 We now assume that base flow (Y, T) is minimal in the SPCF (S 1 × Y, T). Let X be a minimal set of (S 1 × Y, T), Y0 be the set of continuity points of π −1 : Y → 2X : y → π −1 (y), and (X ∗ , T), (Y ∗ , T), τ, τ , π be defined as in Proposition 3.5 with respect to the extension π : (X, T) → (Y, T). Recall that Y0 is an invariant residual subset of Y , Y ∗ = cl{π −1 (y): y ∈ Y0 }, X ∗ = {(x, y ∗ ) ∈ X × Y ∗ : x ∈ y ∗ }, (X ∗ , T) is a minimal flow, τ : (Y ∗ , T) → (Y, T) and τ : (X ∗ , T) → (X, T) are almost 1–1 extensions (hence (Y ∗ , T) is point-distal if (Y, T) is), and π : (X ∗ , T) → (Y ∗ , T) is an open extension. Let Z = S 1 × Y ∗ and define the skew-product flow (Z, T) = (Z, {ΠtZ }t∈T ) by ΠtZ (s, y ∗ ) = ψ s, τ (y ∗ ), t , y ∗ · t .


861

Consider ρ : X ∗ → Z: ((s, τ (y ∗ )), y ∗ ) → (s, y ∗ ) and let Z ∗ = ρ(Z) endow with the metric dZ ∗ s1 , y1∗ , s2 , y2∗ = |s1 − s2 | + dY ∗ y1∗ , y2∗ ,

s1 , y1∗ , s2 , y2∗ ∈ Z ∗ ,

where dY ∗ is a compatible metric on Y ∗ . Since for any ((s, τ (y ∗ )), y ∗ ) ∈ X ∗ and t ∈ T, ρ

s, τ (y ∗ ) , y ∗ · t = ρ Λt s, τ (y ∗ ) , y ∗ · t = ρ ψ s, τ (y ∗ ), t , τ (y ∗ ) · t , y ∗ · t = ψ s, τ (y ∗ ), t , y ∗ · t = ΠtZ (s, y ∗ ) = ΠtZ ρ s, τ (y ∗ ) , y ∗ ,

we see that ρ : (X ∗ , T) → (Z ∗ , T) is a flow isomorphism. Hence (Z ∗ , T) is a minimal flow. Let π ∗ : Z → Y ∗ be the natural projection and denote π1 = π ∗ |Z ∗ . Lemma 5.4. The following diagram (X, T)

τ

ρ

(Z ∗ , T)

π

π

(Y, T)

(X ∗ , T)

τ

(Y ∗ , T)

π1

(Y ∗ , T)

commutes, where τ, τ are almost 1–1, π , π1 are open, and ρ is 1–1. Proof. Since π is open, so is π1 = π ◦ ρ −1 . With Proposition 3.5, we only need to check the commutativity of the right-half of the diagram. Let ((s, τ (y ∗ )), y ∗ ) ∈ X ∗ . Then π1 ρ s, τ (y ∗ ) , y ∗ = π1 (s, y ∗ ) = y ∗ = π s, τ (y ∗ ) , y ∗ .

2

Proposition 5.1. Suppose that the base flow (Y, T) of the SPCF (S 1 ×Y, T) is point-distal. If there exists a second category subset Yu∗ of Y ∗ such that for each y ∗ ∈ Yu∗ there exists no uncountable scrambled set in π1−1 (y ∗ ), then (Z ∗ , T) is point-distal. Proof. Let A denote the collection of all proper, closed, sub-arcs of S 1 with end points being roots of unity and consider the set D = (I1 , I2 ): I1 , I2 ∈ A and I2 ⊂ int(I1 ) . It is clear that D is countable. Since (Y ∗ , T) is point-distal, the set Yd∗ of distal points of Y ∗ is a residual subset. Clearly, ∗ Yw = Yu∗ ∩ Yd∗ is a second category subset of Y ∗ . For each y ∗ ∈ Y ∗ , we let S(y ∗ ) = {s ∈ S 1 : (s, y ∗ ) ∈ Z ∗ }. Then S(y ∗ ) is a closed subset of S 1 . 1 Since π1 is open, the map θ : Y ∗ → 2S : y ∗ → S(y ∗ ) is continuous. Define Yi∗ = y ∗ ∈ Yw∗ : S(y ∗ ) contains an isolated point

and Yp∗ = Yw∗ \ Yi∗ .

Since Yw∗ is a second category subset of Y ∗ , either Yi∗ or Yp∗ is a second category subset of Y ∗ .

862


There are two cases to consider. Case 1. Yi∗ is a second category subset of Y ∗ . y∗

y∗

y∗

y∗

We note that for each y ∗ ∈ Yi∗ , there exists (I1 , I2 ) ∈ D such that S(y ∗ ) ∩ I1 = S(y ∗ ) ∩ I2 y∗

y∗

is a singleton. Thus the map Φ : Yi∗ → D: y ∗ → (I1 , I2 ) is well defined. Since Yi∗ = (I1 ,I2 )∈D Φ −1 (I1 , I2 ), D is countable, and Yi∗ is a second category subset of Y ∗ ,

there exist (I10 , I20 ) ∈ D and a nonempty open subset U of Y ∗ such that Φ −1 (I10 , I20 ) ⊇ U . Using

the continuity of the map θ : Y ∗ → 2S : y ∗ → S(y ∗ ), we have that for each y ∗ ∈ Φ −1 (I10 , I20 ), S(y ∗ ) ∩ int(I10 ) = S(y ∗ ) ∩ I20 is a singleton. In particular, for each y ∗ ∈ U , S(y ∗ ) ∩ int(I10 ) = S(y ∗ ) ∩ I20 is a singleton. Let W = int(I10 ). Then W ∩ S(y ∗ ) is a singleton for each y ∗ ∈ U . Fix points y ∗ ∈ Y ∗ and s ∈ S(y ∗ ). Then (s, y ∗ ) ∈ Z ∗ . Since (W × U ) ∩ Z ∗ is a nonempty open ∗ subset of Z ∗ and (Z ∗ , T) is a minimal flow, there exists t0 ∈ T such that ΠtZ0 (s, y ∗ ) ∈ W × U . ∗ Hence there exists an open neighborhood V of s in S 1 such that ΠtZ0 ((V × {y ∗ }) ∩ Z ∗ ) ⊂ ∗ (W × U ) ∩ Z ∗ . Since ΠtZ0 ((V × {y ∗ }) ∩ Z ∗ ) ⊂ S(y ∗ · t0 ) × {y ∗ · t0 } and y ∗ · t0 ∈ U , ∗ ΠtZ0 ((V × {y ∗ }) ∩ Z ∗ ) ⊆ (S(y ∗ · t0 ) ∩ W ) × {y ∗ · t0 } is a singleton, it follows that (V × {y ∗ }) ∩ Z ∗ is a singleton, i.e., s is an isolated point of S(y ∗ ). Thus, for each y ∗ ∈ Y ∗ , S(y ∗ ) is a discrete closed subset of S 1 , hence a finite subset of S 1 . This shows that π1 : (Z ∗ , T) → (Y ∗ , T) is a finite to one and open extension of the point-distal flow (Y ∗ , T). By Proposition 3.2(2), (Z ∗ , T) is point-distal. 1

Case 2. Yp∗ is a second category subset of Y ∗ . Let y0∗ ∈ Yp∗ be fixed such that S(y0∗ ) contains no isolated point. Then S(y0∗ ) is a perfect subset of S 1 . It is not hard to see that there exists s0 ∈ S(y0∗ ) such that for any > 0 sufficiently small, [s0 , s0 e2πi ] ∩ S(y0∗ ) and [s0 e−2πi , s0 ] ∩ S(y0∗ ) are two uncountable sets. Clearly, z0∗ = (s0 , y0∗ ) ∈ Z ∗ . Let y0 = τ (y0∗ ) and ft : S 1 → S 1 : s → ψ(s, y0 , t). Suppose for contradiction that (Z ∗ , T) is not point-distal. Claim 1. There exists 0 > 0 such that if (s1 , y ∗ ), (s2 , y ∗ ) ∈ Z ∗ are distal, then supt0 ψ s1 , τ (y ∗ ), t − ψ s2 , τ (y ∗ ), t > 0 . Since (Z ∗ , T) is not point-distal, we have by Proposition 3.1 that there exists z1∗ = (s , y1∗ ) ∈ \ {z0∗ } such that z1∗ , z0∗ are positively proximal. In particular, y1∗ , y0∗ are positively proximal. Since y0∗ ∈ Yd∗ , y1∗ = y0∗ . Hence s = s0 and there exists a sequence ti → +∞ and a point s ∈ S 1 such that limi→∞ fti (s ) = limi→∞ fti (s0 ) = s . It follows from Lemma 5.3(2) that there exists a subsequence {ik } ⊂ {i} such that either limk→∞ ftik ([s , s0 ]) = {s } or limk→∞ ftik ([s0 , s ]) = {s } under Hausdorff metric. We let B = [s , s0 ] or [s0 , s ] be such that for any s1 , s2 ∈ B ∩ S(y0∗ ), Z∗

lim inf dZ ∗ (s1 , y0∗ ) · t, (s2 , y0∗ ) · t = lim ftik (s1 ) − ftik (s2 ) = 0. t→+∞

k→∞

By the choice of s0 , B ∩ S(y0∗ ) is an uncountable set. For each (s, y ∗ ) ∈ Z ∗ , we consider the set AR+ (s, y ∗ ) = s1 ∈ S(y ∗ ): lim dZ ∗ (s1 , y ∗ ) · t, (s, y ∗ ) · t = 0 . t→+∞


863

Obviously, if s ∈ S(y0∗ ), then AR+ (s, y0∗ ) = {s1 ∈ S(y0∗ ): limt→+∞ |ft (s1 ) − ft (s)| = 0}. Hence for any s1 , s2 ∈ S(y0∗ ), either AR+ (s1 , y0∗ ) = AR+ (s2 , y0∗ ) or AR+ (s1 , y0∗ ) ∩ AR+ (s2 , y0∗ ) = ∅. It follows that there exists a set I ⊂ B ∩ S(y0∗ ) such that (1) for any s ∈ I , B ∩ AR+ (s, y0∗ ) = ∅; ∗ ∗ (2) for any s1 = s 2 ∈ I , AR+ (s1 , y0 ) ∩ AR+ (s2 , y0 ) = ∅; (3) B ∩ S(y0∗ ) = s∈I (B ∩ AR+ (s, y0∗ )). By (2) above, if s1 = s2 ∈ I , then lim supt→+∞ dZ ∗ ((s1 , y0∗ ) · t, (s2 , y0∗ ) · t) > 0, hence (s1 , y0∗ ) and (s2 , y0∗ ) form a Li–Yorke pair. This shows that I × {y0∗ } ⊆ π1−1 (y0∗ ) is a scrambled set of (Z ∗ , T). Since π1−1 (y0∗ ) contains no uncountable scrambled set, I must be countable. Note that B ∩S(y0∗ ) is uncountable. There must be a point s 0 ∈ I such that B ∩AR+ (s 0 , y0∗ ) is uncountable, in particular, AR+ (s 0 , y0∗ ) is uncountable. Let > 0 be given. Since AR+ (s 0 , y0∗ ) is uncountable, there exist s11 , s21 ∈ AR+ (s 0 , y0∗ ) such that [s11 , s21 ] ∩ AR+ (s 0 , y0∗ ) and [s21 , s11 ] ∩ AR+ (s 0 , y0∗ ) are both uncountable. Using Lemma 5.3(4) and the fact that limt→+∞ dZ ∗ ((s11 , y0∗ ) · t, (s21 , y0∗ ) · t) = 0, we have that there exists t0 0 such that, as t t0 , dZ ∗ s1 , y0∗ · t, s2 , y0∗ · t dZ ∗ s11 , y0∗ · t, s21 , y0∗ · t for all s1 , s2 ∈ [s11 , s21 ] or s1 , s2 ∈ [s21 , s11 ]. Let [s10 , s20 ] ⊆ [s11 , s21 ] or [s21 , s11 ] be such that [s10 , s20 ]∩S(y0∗ ) is an uncountable set, [s10 , s20 ]∩S(y0∗ ) ⊆ AR+ (s0 , y ∗ ), and supt0 dZ ∗ ((s1 , y ∗ )·t, (s2 , y ∗ ) · t) for all s1 , s2 ∈ [s10 , s20 ] ∩ S(y0∗ ). Also let s30 ∈ (s10 , s20 ) ∩ S(y0∗ ) and 0 > 0 be such that {s ∈ S 1 : |s − s30 | 20 } ⊂ (s10 , s30 ). Let (s1 , y ∗ ), (s2 , y ∗ ) ∈ Z ∗ be distal. Since (Z ∗ , T) is minimal, there exists a positive sequence ∗ ∗ tn → +∞ such that ΠtZn (s1 , y ∗ ) → (s30 , y0∗ ) and ΠtZn (s2 , y ∗ ) → (s40 , y0∗ ) for some s40 ∈ S(y0∗ ). / (s10 , s20 ). Hence |s30 − s40 | 20 . It follows that Since (s30 , y0∗ ), (s40 , y0∗ ) ∈ Z ∗ are also distal, s40 ∈ supt0 |ψ(s1 , τ (y ∗ ), t) − ψ(s2 , τ (y ∗ ), t)| |s30 − s40 | > 0 . This proves Claim 1. Claim 2. For each > 0, there exist a nonempty open set U ⊂ Y ∗ and a point (I1 , I2 ) ∈ D such that if y ∗ ∈ U , then S(y ∗ ) ∩ I2 = ∅ and supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t) for all s1 , s2 ∈ int(I1 ) ∩ S(y ∗ ). Let y ∗ ∈ Yp∗ . Then S(y ∗ ) contains no isolated point and no uncountable scrambled set. Similar to the proof of Claim 1 there are s1 = s2 ∈ S(y ∗ ) such that [s1 , s2 ] ∩ S(y ∗ ) is an uncountable set, [s1 , s2 ] ∩ S(y ∗ ) ⊆ AR+ (s, y ∗ ) for some s ∈ S(y ∗ ), and supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t) y∗

y∗

y∗

for all s1 , s2 ∈ [s1 , s2 ] ∩ S(y ∗ ). Hence there exists (I1 , I2 ) ∈ D such that I2 ∩ S(y ∗ ) = ∅ y∗

and supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t) for any s1 , s2 ∈ I1 ∩ S(y ∗ ). We define the map y∗

y∗

Φ : Yp∗ → D as Φ (y ∗ ) = (I1 , I2 ), y ∗ ∈ Yp∗ . Since Yp∗ = (I1 ,I2 )∈D Φ−1 (I1 , I2 ), D is countable, and Yp∗ is a second category subset of Y ∗ , we have that there exists (I1 , I2 ) ∈ D and a nonempty open subset U of Y ∗ such that

Φ−1 (I1 , I2 ) ⊇ U . We note that for any y ∗ ∈ Φ−1 (I1 , I2 ), I2 ∩ S(y ∗ ) = ∅, supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t)

for any s1 , s2 ∈ I1 ∩ S(y ∗ ). It follows from the continuity of the map θ : Y ∗ → 2S : y ∗ → S(y ∗ ) 1

that for each y ∗ ∈ Φ−1 (I1 , I2 ), S(y ∗ ) ∩ I2 = ∅, and supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t)

864


for any s1 , s2 ∈ int(I1 ) ∩ S(y ∗ ). In particular, for each y ∗ ∈ U , we have S(y ∗ ) ∩ I2 = ∅ and supt0 dZ ∗ ((s1 , y ∗ ) · t, (s2 , y ∗ ) · t) for any s1 , s2 ∈ int(I1 ) ∩ S(y ∗ ). This proves Claim 2. Claim 3. If (s1 , y ∗ ), (s2 , y ∗ ) ∈ Z ∗ are proximal, then s1 ∈ AR+ (s2 , y ∗ ). Moreover, for each (s, y ∗ ) ∈ Z ∗ , AR+ (s, y ∗ ) is an open subset of S(y ∗ ). Let (s1 , y ∗ ), (s2 , y ∗ ) ∈ Z ∗ be proximal. For any > 0, we let U ⊂ Y ∗ and (I1 , I2 ) ∈ D be as in Claim 2. Then for each y1∗ ∈ U , we have S(y1∗ ) ∩ I2 = ∅ and supt0 dZ ∗ ((s1 , y1∗ ) · t, (s2 , y1∗ ) · t) for all s1 , s2 ∈ int(I1 ) ∩ S(y1∗ ). Since (Z ∗ , T) is minimal and (int(I1 ) × U ) ∩ Z ∗ is an open subset of Z ∗ , there exists t1 ∈ T such that (s1 , y ∗ )·t1 , (s2 , y ∗ )·t1 ∈ (int(I1 )×U )∩Z ∗ . Hence sup dZ ∗ (s1 , y ∗ ) · (t1 + t), (s2 , y ∗ ) · (t1 + t) . t0

Since > 0 is arbitrary, s1 ∈ AR+ (s2 , y ∗ ). For a fixed ∈ (0, 0 ), we let y1∗ ∈ Y ∗ and s1 , s2 ∈ int(I1 ) ∩ S(y1∗ ). We have by Claims 1 and 2 that (s1 , y1∗ ), (s2 , y1∗ ) are proximal. Repeat the above arguments for (s1 , y1∗ ), (s2 , y1∗ ) in place of (s1 , y ∗ ), (s2 , y ∗ ) respectively, we conclude that s1 ∈ AR+ (s2 , y1∗ ). Hence int(I1 ) ∩ S(y1∗ ) ⊂ AR+ (s, y1∗ ) for each s ∈ int(I1 ) ∩ S(y1∗ ). Let (s, y ∗ ) ∈ Z ∗ and t ∈ T be such that (s, y ∗ ) · t ∈ (int(I1 ) × U ) ∩ Z ∗ , i.e., ψ(s, τ (y ∗ ), t) ∈ int(I1 ) and y ∗ · t ∈ U . Clearly, there exists an open neighborhood V of s in S 1 such that ψ s , τ (y ∗ ), t : s ∈ V ∩ S(y ∗ ) ⊂ int I1 ∩ S(y ∗ · t) ⊂ AR+ (s, y ∗ ) · t . This implies that V ∩ S(y ∗ ) ⊂ AR+ (s, y ∗ ), i.e., AR+ (s, y ∗ ) is an open subset of S(y ∗ ). Claim 4. π1 : (Z ∗ , T) → (Y ∗ , T) is a finite to one extension. Since for any s1 , s2 ∈ S(y0∗ ), either AR+ (s1 , y0∗ ) = AR+ (s2 , y0∗ ) or ∗ ∗ ∗ AR+ (s1 , y0 ) ∩ AR+ (s2 , y0 ) = ∅, there exists J ⊂ S(y0 ) such that {AR+ (s, y0∗ )}s∈J is a partition of S(y0∗ ). By Claim 3, for each s ∈ J , AR+ (s, y0∗ ) is an open subset of S(y0∗ ). So {AR+ (s, y0 )}s∈J is an open cover and also a partition of S(y0∗ ). Hence J mustbe a finite set, say, J = {s1 , s2 , . . . , sn }. For each i = 1, 2, . . . , n, since AR+ (si , y0∗ ) = S(y0∗ ) \ j =i AR+ (sj , y0∗ ), we see that AR+ (si , y0∗ ) is also a closed subset of S(y0∗ ). If n = 1, then for any s1 , s2 ∈ S(y0∗ ), (s1 , y0∗ ), (s2 , y0∗ ) are proximal. This implies that π1 is a proximal extension. Since for each y ∗ ∈ Yu∗ there exists no uncountable scrambled set in π1−1 (y ∗ ), we have by Theorem 5.1 that π1 is an almost 1–1 extension. Hence there exists y1∗ such that |S(y1∗ )| = |π1−1 (y1∗ )| = 1. Moreover, since θ : y ∗ → S(y ∗ ) is continuous, |π1−1 (y ∗ )| = |S(y ∗ )| = 1 for any y ∗ ∈ Y ∗ , i.e., π1 is a flow isomorphism. If n 2, then there exists s ∈ S(y0∗ ) \ AR+ (s1 , y0∗ ). Since AR+ (s1 , y0∗ ) is closed, we can find points a1 , b1 ∈ AR+ (s1 , y0∗ ) such that AR+ (s1 , y0∗ ) ⊆ [a1 , b1 ] (if a1 = b1 , then [a1 , b1 ] = {a1 }) and s ∈ S 1 \ [a1 , b1 ]. By Lemma 5.3(4) and the fact that s ∈ / AR+ (s1 , y0∗ ), we have that ∗ ∗ [a1 , b1 ] ∩ S(y0 ) ⊆ AR+ (s1 , y0 ) and diam(ft ([a1 , b1 ])) |ft (a1 ) − ft (b1 )| as t sufficiently large, where ft (s) = ψ(s, τ (y0∗ ), t), t ∈ T. Hence AR+ (s1 , y0∗ ) = [a1 , b1 ] ∩ S(y0∗ ). Similarly, for each i = 2, 3, . . . , n, there exist points ai , bi ∈ AR+ (si , y0∗ ) such that AR+ (si , y0∗ ) = [ai , bi ] ∩ S(y0∗ ) and diam(ft ([ai , bi ])) |ft (ai ) − ft (bi )| as t sufficiently large.


865

Let tk → +∞ be such that y0∗ · tk → y0∗ and limk→∞ ftk ([ai , bi ]) = {ci } for some ci ∈ S(y0∗ ). Using the continuity of θ : y ∗ → S(y ∗ ) and the fact that n n S y0∗ · t = ft S y0∗ = ft AR+ si , y0∗ = ft [ai , bi ] ∩ S y0∗ , i=1

t ∈ T,

i=1

we have n S y0∗ = lim S y0∗ · tk = lim ftk [ai , bi ] ∩ S y0∗ = {c1 , c2 , . . . , cn }. k→∞

k→∞

i=1

By the continuity of θ again, we have |π1−1 (y ∗ )| = |S(y ∗ )| = |S(y0∗ )| for any y ∗ ∈ Y ∗ . This shows that π1 is a finite to one extension. The proof of Claim 4 is now complete. Now, since (Y ∗ , T) is point-distal and π1 is finite to one and open, we have by Proposition 3.2(2) that (Z ∗ , T) is point-distal, a contradiction. 2 Theorem 5.2. Let X be a minimal set of a SPCF (S 1 × Y, T) with point-distal base flow (Y, T). Then X is either point-distal or residually Li–Yorke chaotic. Proof. Let X ∗ , Y ∗ , Z ∗ , τ, τ , ρ, π, π1 , π be as in Lemma 5.4 for the present minimal set X. We first consider the case that there exists a second category subset Yu∗ of Y ∗ such that for each y ∗ ∈ Yu∗ there exists no uncountable scrambled set in π1−1 (y ∗ ). By Proposition 5.1, (Z ∗ , T) is point-distal. Since ρ : (X ∗ , T) → (Z ∗ , T) is a flow isomorphism, (X ∗ , T) is also point-distal. Note that τ : (X ∗ , T) → (X, T) is almost 1–1. We conclude that (X, T) is point-distal in this case. Next, we consider the case that there exists a residual subset Yc∗ of Y ∗ such that for each ∗ y ∈ Yc∗ there exists an uncountable scrambled set in π1−1 (y ∗ ). Let y0∗ ∈ Yc∗ and F be an uncountable scrambled set in π1−1 (y0∗ ). Then there exists an uncountable subset S of S 1 such that F = {(s, y0∗ ): s ∈ S}. Let y0 = τ (y0∗ ) and F = {(s, y0 ): s ∈ S}. Clearly, F is an uncountable set. Since π2 = τ ◦ ρ −1 : (Z ∗ , T) → (X, T) is a flow extension and π2 (F ) = F , we have F ⊂ π −1 (y0 ). Let s1 = s2 ∈ S. Then (s1 , y0∗ ), (s2 , y0∗ ) form a Li–Yorke pair. Hence lim infψ(s1 , y0 , t) − ψ(s2 , y0 , t) = 0 and t→+∞

lim supψ(s1 , y0 , t) − ψ(s2 , y0 , t) > 0, t→+∞

i.e., {(s1 , y0 ), (s2 , y0 )} is also a Li–Yorke pair. Therefore, F ⊆ π −1 (y0 ) is an uncountable scrambled set of (X, T). Let Yc = τ (Yc∗ ). Since Yc∗ is a residual subset of Y ∗ and τ is an almost 1–1 extension, Yc is a residual subset of Y and there exists an uncountable scrambled set in π −1 (y) for each y ∈ Yc . Hence (X, T) is residually Li–Yorke chaotic. 2 A point-distal or even an almost automorphic minimal set can also be Li–Yorke chaotic. But our next theorem says that it cannot be residually Li–Yorke chaotic.

866


Lemma 5.5. Let X and Y be compact metric spaces, and π : X → Y be a semi-open, surjective, continuous map. Then X0 = x ∈ X: for any open neighborhood U of x, π(U ) is a neighborhood of π(x) is a residual subset of X. Proof. It follows from arguments of Lemma 3.1 in [59].

2

Theorem 5.3. Consider a SPCF (S 1 × Y, T) with point-distal base flow (Y, T). If a minimal set is point-distal, then it is not residually Li–Yorke chaotic. Proof. We use the explicit expression (1.1) for a SPCF (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ). Suppose for contradiction that the SPCF has a point-distal minimal set M which is also residually Li–Yorke chaotic. Then the set Md of distal points of M is a residual subset. It follows from Lemma 5.5 that Yd = y ∈ Y : Md ∩ π −1 (y) is a residual subset of π −1 (y) ∩ M is a residual subset of Y . Since M is residually Li–Yorke chaotic, there exists a residual subset Yc of Y such that each fiber π −1 (y), y ∈ Yc , admits an uncountable scrambled set Wy . Fix y ∈ Yd ∩ Yc and let Ey = {s ∈ S 1 : (s, y) ∈ Wy }. Then Ey is an uncountable subset of 1 S and it is not hard to see that there exists s0 ∈ Ey such that for any > 0 sufficiently small, [s0 , s0 e2πi ] ∩ Ey and [s0 e−2πi , s0 ] ∩ Ey are two uncountable sets. Consider the family of maps ft : S 1 → S 1 : s → ψ(s, y, t), t ∈ T. Take s1 ∈ Ey \ {s0 }. Then (s0 , y), (s1 , y) are positively proximal. Hence there exists a sequence ti → +∞ and a point s ∈ S 1 such that limi→∞ fti (s1 ) = limi→∞ fti (s0 ) = s . It follows from Lemma 5.3(2) that there exists a subsequence {ik } ⊂ {i} such that either limk→∞ ftik ([s1 , s0 ]) = {s } or limk→∞ ftik ([s0 , s1 ]) = {s } under Hausdorff metric. We let B = [s1 , s0 ] or [s0 , s1 ] be such that for any s2 ∈ B, lim inf d (s0 , y) · t, (s2 , y) · t = lim ftik (s0 ) − ftik (s2 ) = 0. t→+∞

k→∞

According to the choice of s0 , B ∩ Ey is an uncountable set. Now since π −1 (y) ∩ Md is a residual subset of π −1 (y) ∩ M, there exists s2∗ ∈ int(B) ∩ Ey such that (s2∗ , y) ∈ Md . This is impossible since lim inft→+∞ d((s0 , y) · t, (s2∗ , y) · t) = 0. 2 Now Theorem 2 immediately follows from Theorems 5.2, 5.3 above. 6. A general topological classification of minimal sets In this section, we consider a SPCF (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ) with minimal base flow (Y, T). We adopt the explicit form (1.1), i.e., Λt (s0 , y0 ) = ψ(s0 , y0 , t), y0 · t , t ∈ T, and denote dY as a compatible metric on Y and π : S 1 × Y → Y as the natural projection.


867

Let M be a minimal set of the SPCF (S 1 × Y, T). For each y ∈ Y , we denote My = {s ∈ S 1 : (s, y) ∈ M}. Since My is a closed subset of S 1 , each connected component of My is either the whole circle or a closed interval in the circle (which can be degenerate). Consider the function ζM : Y → R 1 : ζM (y) = sup |B|: B is a connected component of My , where |B| denotes the length of B. For each y ∈ Y , it is clear that there exists a component B in My such that ζM (y) = |B|, i.e., ζM (y) = max |B|: B is a connected component of My . Lemma 6.1. The function ζM is non-negative and upper semi-continuous, i.e., ζM (y) 0 and lim supyn →y ζM (yn ) ζM (y) for each y ∈ Y . Proof. The lemma is clear because M is compact.

2

Let Y0 (ζM ) = {y ∈ Y : ζM (y) = 0}. Lemma 6.2. Either infy∈Y ζM (y) > 0 or Y0 (ζM ) is a residual subset of Y . Proof. By Lemma 6.1, Y0 (ζM ) is a Gδ -set. Hence it is sufficient to show that if infy∈Y ζM (y) = 0, then Y0 (ζM ) is a dense subset of Y . Assume that infy∈Y ζM (y) = 0 and let Yc (ζM ) be the set of points of continuity of ζM . Since ζM is upper semi-continuous, Yc (ζM ) is a residual set. If ζM (y0 ) > 0 for some y0 ∈ Yc (ζM ), then there exist open neighborhood U of y0 and c > 0 such n that ζM (y) c for all y ∈ U . By the minimality of Y , we let t1 < t2 < · · · < tn be such that i=1 U · ti = Y . Since infy∈U ζM (y) > 0, we have that ci =: infy∈U ζM (y · ti ) > 0, for all i = 1, 2, . . . , n. Hence infy∈Y ζM (y) min{ci : i = 1, 2, . . . , n} > 0, a contradiction to the fact that infy∈Y ζM (y) = 0. This shows that for any y0 ∈ Yc (ζ (M)), ζM (y0 ) = 0, i.e., Yc (ζM ) ⊆ Y0 (ζM ). Hence Y0 (ζM ) is a residual subset of Y . 2 Recall that a minimal set M of a SPCF (S 1 × Y, T) is a Cantorian if there exists a residual subset Y0 of Y such that for each y ∈ Y0 , My is a Cantor set. The following theorem immediately yields Theorem 3. Theorem 6.1. Let M be a minimal set of a SPCF (S 1 × Y, T) with minimal base flow (Y, T). Then precisely one of the following holds: (a) M is an almost N –1 extension of Y for some positive integer N ; (b) M = S 1 × Y ; (c) M is a Cantorian. Proof. By Lemma 6.2, there are two cases: (a) infy∈Y ζM (y) > 0; (b) Y0 (ζM ) is a residual subset of Y .

868


We first consider the case (a). Let A = e2πir , e2πil : r < l < 1 + r, r, l ∈ Q . Since A is countable, we can rewrite it as A = {[ai , bi ]}i∈N . Since infy∈Y ζM (y) > 0, we have that for each y ∈ Y , there exists i(y) ∈ N such that [ai(y) , bi(y) ] ⊆ My . In particular, [ai(y) , bi(y) ] × {y} ⊆ M. Denote Yi = {y ∈ Y : i(y) = i} for i ∈ N. Then i∈N Yi = Y . Hence there exists i∗ ∈ N such that W =: int(Y i∗ ) = ∅. Since [ai∗ , bi∗ ] × Yi∗ ⊆ M and M is closed, we have that [ai∗ , bi∗ ] × Y i∗ ⊆ M, which implies that (ai∗ , bi∗ ) × W ⊆ M. Let y ∈ Y and denote S(y) = {s ∈ S 1 : (s, y) ∈ My }. For any s ∈ S(y), since M is minimal, there exists t ∈ T such that (s, y) · t ∈ (ai∗ , bi∗ ) × W , which implies that s ∈ intS 1 (S(y)). It 1 1 follows that S(y) is an open subset1of S . Since S(y) is also closed, S(y) = S . Since y is arbitrary, M = y∈Y S(y) × {y} = S × Y . We now consider the case (b). Let Y c (M) = {y ∈ Y : My is a Cantor set}, Y i (M) = {y ∈ Y : My has an isolated point}. Since it is clear that My is a Cantor set for any y ∈ Y0 (ζM ) \ Y i (M), we have that Y c (M) ⊇ Y0 (ζM ) \ Y i (M). If Y c (M) is a residual subset of Y , then by definition M is a Cantorian. If Y c (M) is not a residual subset of Y , then Y i (M) is of second category, or Y c (M) ⊇ Y0 (ζM ) \ Y i (M) is a residual subset of Y since Y0 (ζM ) is a residual subset of Y . Consider the countable set D = (I1 , I2 ): I1 , I2 ∈ A and I2 ⊂ int(I1 ) . y

y

y

y

We note that for each y ∈ Y i (M), there exists (I1 , I2 ) ∈ D such that S(y) ∩ I1 = S(y) ∩ I2 is a y y singleton. Thus the map Φ : Y i (M) → D: y → (I1 , I2 ) is well defined. i −1 Since Y (M) = (I1 ,I2 )∈D Φ (I1 , I2 ), D is countable, and Y i (M) is of second category, there exist (I10 , I20 ) ∈ D and a nonempty open subset U of Y such that Φ −1 (I10 , I20 ) ⊇ U . Since θ : y → S(y) is upper semi-continuous, the set Y0 of all continuity points of θ is an invariant residual subset of Y . Let W = int(I10 ). For each y ∈ U ∩ Y0 ⊂ Φ −1 (I10 , I20 ) ∩ Y0 , since S(y) ∩ int(I10 ) = S(y) ∩ I20 is a singleton, W ∩ S(y) is also a singleton. Fix y ∈ Y0 and s ∈ S(y). Then (s, y) ∈ M. Since (W × U ) ∩ M is a nonempty open subset of M and M is a minimal set, there exists t0 ∈ T such that (s, y) · t0 ∈ W × U . Hence there exists an open neighborhood V of s in S 1 such that (V × {y}) · t0 ∩ M ⊂ (W × U ) ∩ M. Since (V × {y}) · t0 ∩ M ⊂ S(y · t0 ) × {y · t0 } and y · t0 ∈ U ∩ Y0 , we have that (V × {y}) · t0 ∩ M ⊆ (S(y · t0 ) ∩ W ) × {y · t0 } is a singleton. It follows that (V × {y}) ∩ M is a singleton, i.e., s is an isolated point of S(y). Thus, for each y ∈ Y0 , S(y) is a discrete closed subset of S 1 , hence it is a finite subset of S 1 . This shows that π : M → Y is an almost finite to one extension. It follows from Proposition 3.2(1) that π : M → Y is an almost N –1 extension for some positive integer N . Finally, the above is a strict trichotomy because in both cases (a) and (b), M cannot be a Cantorian. 2


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7. Finite to one extensions and almost automorphic dynamics 7.1. Local connectivity and almost automorphy Let X be a complete metric space. Recall that x ∈ X is a locally connected point if for any open neighborhood U of x there is a connected closed neighborhood V of x such that V ⊆ U . We denote by Xlc the set of locally connected points in X. Lemma 7.1. Suppose that Xlc = ∅ and (X, T) is minimal. Then Xlc is an invariant residual subset of X. Proof. The invariance of Xlc is clear. For each k ∈ N, we consider the open set 1 k Xlc = x ∈ X: there exists a connected closed neighborhood V of x such that V ⊆ B x, , k k where B(x, k1 ) = {z ∈ X: d(x, z) < k1 }. Then Xlc = ∞ k=1 Xlc , i.e., Xlc is a Gδ subset of X. It follows from the minimality of X that Xlc is also dense. 2 The following result is known as the Ramsey theorem [49]. Lemma 7.2. If the set C = {(i, j ) ∈ N × N: 1 i < j < ∞} is divided into finite sets C1 , C2 , . . . , C , then there is a sequence {in } of natural numbers for which all pairs (im , in ), m < n, are in Cj for some j ∈ {1, 2, . . . , }. Our main result Theorem 5 is a direct consequence of the following theorem. Theorem 7.1. Consider an almost n–1 extension π : (X, T) → (Y, T) between minimal flows in which (Y, T) is almost periodic minimal. If Xlc = ∅, then the following holds. (1) (X, T) is almost automorphic; (2) For each y ∈ Y , the fiber π −1 (y) has precisely n connected components. Proof. (1) Let X0 = x ∈ X: for any open neighborhood U of x, π(U ) is a neighborhood of π(x) . By Lemma 5.5, X0 is a residual subset of X. Since by Lemma 7.1 Xlc is residual, X0 ∩ Xlc is also a residual subset of X. Hence by Lemma 5.1, Y1 = y ∈ Y : π −1 (y) ∩ X0 ∩ Xlc is a residual subset of π −1 (y) is a residual subset of Y . Let Y0 be the set of continuity points of the map Φ : Y → 2X : y → π −1 (y). Since Y0 is a residual subset of Y , so is Y0 ∩ Y1 . Let y0 ∈ Y0 ∩ Y1 . Then |π −1 (y0 )| = n, say, π −1 (y0 ) = {x1 , x2 , . . . , xn }. Since π −1 (y0 ) ∩ X0 is a residual subset of π −1 (y0 ), we have that xi ∈ X0 for all i = 1, 2, . . . , n. By Theorem 3.1, we want to show that x1 is a Δ∗ -recurrent point, i.e., for any open neighborhood U of x1 , the

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recurrent time set N (x1 , U ) = {t ∈ T: x1 · t ∈ U } is a Δ∗ -set. More precisely, let {si }∞ i=1 be a sequence in T. We need to show that N (x1 , U ) ∩ {sk − sk : k > k } = ∅. Let Ui be open neighborhoods of xi , for i = 1, 2, . . . , n respectively, such that U1 ⊆ U and cl(Ui ) ∩ cl(Uj ) = ∅ for any 1 i = j n. Since y0 ∈ Y0 , there exists an open neighborhood W of y0 such that for each y ∈ W , π −1 (y) ∩ Ui = ∅, i = 1, 2, . . . , n, and π −1 (y) ⊂ ni=1 Ui . Since (Y, T) is almost periodic, there exists an invariant compatible metric dY on Y , i.e., dY (y1 · t, y2 · t) = dY (y1 , y2 ) for any y1 , y2 ∈ Y and t ∈ T. Let δ > 0 be such that the open ball Bδ (y0 ) centered at y0 with radius δ is contained in W . Let m = n!. Then there exist r ∈ si −r {0, 1, . . . , m − 1}, a subsequence {ik } ⊂ N, and a homeomorphism g : Y → Y such that km ∈ T and sik − r δ , g(y) k sup dY y · m 2 y∈Y

(7.1)

for all k = 1, 2, . . . . Then for any u, v ∈ N with u = v, it follows from (7.1) that siu − siv si − r si − r , y = sup dY y · u ,y · v sup dY y · m m m y∈Y y∈Y si − r si − r , g(y) + sup dY y · v , g(y) sup dY y · u m m y∈Y y∈Y

δ δ + . 2u 2v

si −r

Denote rk = km , k = 1, 2, . . ., R = {ri − rj : i > j }, and Per(n) as the set of all permutations of {1, 2, . . . , n}. Let t ∈ R. Since y0 · t ∈ W , π −1 (y0 · t) ∩ Ui = ∅ for all i = 1, 2, . . . , n. Hence there exists a unique Pt ∈ Per(n) such that xj · t ∈ UPt (j ) for all j = 1, 2, . . . , n. For eachP ∈ Per(n), we let RP = {t ∈ R : Pt = P } and CP = {(i, j ) ∈ N × N: rj − ri ∈ P }. Since R = P ∈Per(n) RP = {ri − rj : i > j }, we have C = P ∈Per(n) CP . Applying Lemma 7.2, one finds a subsequence {lj } ⊂ N and Q ∈ Per(n) such that RQ ⊇ {rli − rlj : i > j }. Let ui = rli , i ∈ N. It is clear that (a) {ui − uj : i > j } ⊆ RQ ; (b) {m(ui − uj ): i > j } ⊆ {sk − sk : k > k }; (c) supy∈Y d(y · (ui − uj ), y) < 2δi + 2δj for any i > j . Since Q ∈ Per(n), Qm (j ) = j for each j = 1, 2, . . . , n. In particular, Qm (1) = 1. Let Wm = U1 . Since limi>j →∞ supy∈Y d(y · (ui − uj ), y) = 0, there exists a positive integer Nm and an open neighborhood Vm ⊆ W of y0 such that Vm · (ui − uj ) ⊆ W for all i > j Nm . Since X is local connected, there exists a connected closed neighborhood Wm−1 of xQm−1 (1) such that Wm−1 ⊆ UQm−1 (1) ∩ π −1 (Vm ). Now for any i > j Nm , since π −1 (Vm · (ui − uj )) ⊆ π −1 (W ) ⊆ nk=1 Uk , we have Wm−1 · (ui − uj ) ⊆ nk=1 Uk . Note that Wm−1 · (ui − uj ) is both connected and closed, xQm−1 (1) · (ui − uj ) ∈ Wm = U1 , and cl(Uk ) ∩ cl(Uk ) = ∅ for 1 k < k n. It follows that Wm−1 · (ui − uj ) ⊆ Wm , i.e., we find a connected closed neighborhood Wm−1 of xQm−1 (1) and a positive integer Nm such that Wm−1 · (ui − uj ) ⊆ Wm for all i > j Nm .


871

By repeating the above process and using induction, we find that, for each v = m − 1, m − 2, . . . , 1, 0, there is a connected closed neighborhood Wv of xQv (1) and a positive integer Nv+1 such that Wv · (ui − uj ) ⊆ Wv+1 for all i > j Nv+1 . Let N = max{Nv+1 : v = m − 1, m − 2, . . . , 1, 0}. Then for any i > j N , we have W0 · m(ui − uj ) = W0 · (ui − uj ) · (m − 1)(ui − uj ) ⊆ W1 · (m − 1)(ui − uj ) ⊆ · · · ⊆ Wv · (m − v)(ui − uj ) ⊆ · · · ⊆ Wm = U1 ⊆ U. Since x1 ∈ W0 , x1 · (m(ui − uj )) ∈ U . This together with (b) above implies that m(ui − uj ) ∈ ∗ N(x1 , U ) ∩ {sk − sk : k > k } = ∅. Since {si }∞ i=1 is arbitrary, N (x, U ) is a Δ -set for any neigh∗ borhood U of x1 . Hence x1 is a Δ -recurrent point, i.e., (X, T) is almost automorphic. (2) Let dY be an invariant compatible metric on Y , d be the metric on X, and Y0 , Y1 be the residual sets defined in (1). First, we show that for each y ∈ Y , the fiber π −1 (y) has at least n connected components. Suppose this is not true. Then there exists y1 ∈ Y such that π −1 (y1 ) has m-connected components −1 {Br }m r=1 for some m n − 1. For a fixed y0 ∈ Y0 ∩ Y1 , we let π (y0 ) = {x1 , x2 , . . . , xn }. Also let Ui be open neighborhoods of xi , for i = 1, 2, . . . , n respectively, such that U1 ⊆ U and cl(Ui ) ∩ cl(Uj ) = ∅ for any 1 i = j n. Since y0 ∈ Y0 , there exists an open neighborhood W of n −1 (y) ⊂ π U . Let y0 such that for each y ∈ W , π −1 (y) ∩ Ui = ∅, i =1, 2, . . . , n, and i i=1 n −1 t ∈ T be such that y1 · t ∈ W . Then π −1 (y1 · t) = m r=1 Br · t ⊆ i=1 Ui and π (y1 · t) ∩ Ui = ∅, i = 1, 2, . . . , n. For each r = 1, 2, . . . , m, since Br · t is a closed connected subset of π −1 (y1 · t) and cl(Ui ) ∩ cl(Uj ) = ∅, there exists i(r) ∈ {1, 2, . . . , n} such that Br · t ⊆ Ui(r) . Hence {1, 2, . . . , n} \ {i(1), i(2), . . . , i(m)} = ∅. Let i0 ∈ {1, 2, . . . , n} \ {i(1), i(2), . . . , i(m)}. Then Ui0 ∩ π −1 (y1 · t) ⊆ m r=1 Ui0 ∩ Ui(r) = ∅, a contradiction. Next, suppose for contradiction that there exists y ∈ Y such that π −1 (y) has at least (n + 1)−1 connected components {Aj }n+1 j =1 . For a fixed y0 ∈ Y0 ∩ Y1 , we let π (y0 ) = {x1 , x2 , . . . , xn } and

= min{d(xi , xj ) : 1 i < j n}. Also let N be a natural number such that > N2 . For a given integer m N , we consider the sets Uim = π −1 (B(y0 , m1 )) ∩ B(xi , m1 ), i = 1, 2, . . . , n, where B(y0 , r) = {z ∈ Y : dY (z, y0 ) < r} and B(xi , r) = {x ∈ X: d(xi , x) < r}, i = 1, 2, . . . , n. For each i = 1, 2, . . . , n, since xi ∈ Xlc , there exists a connected closed neighborhood Vim of xi such that Vim ⊆ Uim . Let V m = ni=1 Vim . Then V m is a closed neighborhood of π −1 (y0 ). Since π −1 is continuous at y0 , there exists a neighborhood Wm of y0 such that π −1 (Wm ) ⊂ V m . Let tm ∈ T be such that y · tm ∈ Wm . Then Aj · tm ⊆ V m for all j = 1, 2, . . . , n + 1. Since Aj · tm is connected and Vim ∩ Vjm = ∅ for all 1 i < j n, there exists a unique integer m . Hence there have to be integers j1 (m), j2 (m) n(j, m) ∈ {1, 2, . . . , n} such that Aj · tm ⊆ Vn(j,m) with 1 j1 (m) < j2 (m) n + 1 such that n(j1 (m), m) = n(j2 (m), m), which we denote as n(m). m · (−t ). It is clear that E is a connected closed set, A Let Em = Vn(m) m m j1 (m) ∪ Aj2 (m) ⊆ Em , m ) · (−t ) ⊆ B(y , 1 ) · (−t ) = B(y · (−t ), 1 ). Since y ∈ π(E ), we and π(Em ) = π(Vn(m) m 0 m m 0 m m m have π(Em ) ⊆ B(y, m2 ). Now we take a sequence N m1 < m2 < · · · such that (i) j1 (m1 ) = j1 (m2 ) = · · ·, denoted by j1 ; (ii) j2 (m1 ) = j2 (m2 ) = · · ·, denoted by j2 ; (iii) lim →∞ Em = E for some E ∈ 2X under the Hausdorff metric on 2X .

872


It is clear that 1 j1 < j2 n + 1, Aj1 ∪ Aj2 ⊆ E, and E is a connected closed set of X. Since π(Em ) ⊆ B(y, m2 ), E ⊆ π −1 (y). Note that E is connected, Aj1 ∪ Aj2 ⊆ E, and Aj1 , Aj2 are two connected components of π −1 (y). We must have E = Aj1 = Aj2 , which is a contradiction to the fact that Aj1 ∩ Aj2 = ∅. 2 7.2. SPCF with at least two minimal sets We consider a SPCF (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ) in the form (1.1), i.e., Λt (s0 , y0 ) = ψ(s0 , y0 , t), y0 · t ,

t ∈ T.

We denote dY as a compatible metric on Y and π : S 1 × Y → Y as the natural projection. The following result may be regarded as a topological counterpart to the Furstenberg measuretheoretic characterization [15] for SPCFs. Theorem 7.2. Consider a SPCF (S 1 × Y, T) with minimal base flow (Y, T) and assume that it has at least two minimal sets. Then the following holds. (a) There is a positive integer n such that each minimal set is an almost n–1 extension of Y . (b) If the SPCF becomes APCF and one of its minimal set is almost automorphic, then so are others. Proof. (a) Let M, M0 be two minimal sets of (S 1 × Y, T). For each y ∈ Y , we consider the sets S(y) = {s ∈ S 1 : (s, y) ∈ M} and S0 (y) = {s ∈ S 1 : (s, y) ∈ M0 }. Clearly, for each y ∈ Y , S(y) 1 and S0 (y) are closed subsets of S 1 , S(y) ∩ S0 (y) = ∅, and the maps ρ, ρ0 : Y → 2S ×Y defined by ρ(y) = S(y), ρ0 (y) = S0 (y), y ∈ Y , are upper semi-continuous. We denote by Y c and Y0c , respectively, as the sets of continuity points of ρ, ρ0 , respectively. Then both Y c and Y0c are residual subsets of Y . Fix a point y0 ∈ Y c ∩ Y0c . Since S 1 \ S0 (y0 ) is an open subset of S 1 , S 1 \ S0 (y) is a countable union of proper, open sub-arcs of S 1 , i.e., S 1 \ S0 (y0 ) = Ii=1 Ai , where 1 I +∞ and each Ai is a proper, open sub-arc of S 1 . Since Ii=1 Ai ⊇ S(y0 ), there exists a positive integer N (y0 ) N (y ) such that i=1 0 Ai ⊃ S(y0 ). Without loss of generality, we assume that Ai ∩ S(y0 ) = ∅ for all i = 1, 2, . . . , N(y0 ). Claim. For each i = 1, 2, . . . , N (y0 ), Ai ∩ S(y0 ) is a singleton. In particular, |S(y0 )| = N(y0 ) < ∞. Suppose for contradiction that the Claim is not true. Then there exists some 1 i N (y0 ) such that Ai ∩ S(y0 ) is not a singleton. We denote Ai = (c, d) and let Bi = [a, b] be a closed subarc of Ai such that Ai ∩ S(y0 ) = Bi ∩ S(y0 ). It is clear that a, b ∈ S(y0 ) and c, d ∈ S0 (y0 ). Using minimality of M, we let tj → +∞ be a sequence such that limj →∞ (b, y0 ) · tj = (a, y0 ). By taking subsequences if necessary, we assume that limj →∞ (a, y0 ) · tj = (a , y0 ), limj →∞ (c, y0 ) · tj = (c , y0 ), and limj →∞ (d, y0 ) · tj = (d , y0 ), for some a ∈ S(y0 ) and c , d ∈ S0 (y0 ). Let ft be as in Lemma 5.3. Then ftj (b) → a, ftj (a) → a , ftj (c) → c , and ftj (d) → d , as j → ∞. We first show that c = c, d = d, and a = a. Suppose c = c. Since c = a, we have by Lemma 5.3(1) that limj →∞ ftj ([c, b]) = [c , a] [c, a]. Hence there is a sufficiently small open neighborhood V of c in S 1 such that V ⊂ (c , a). Since y0 ∈ Y0c and c ∈ S0 (y0 ), there


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exists an open neighborhood U of y0 in Y such that S0 (y) ∩ V = ∅ for all y ∈ U . Note that ftj ((c, b)) = (ftj (c), ftj (b)), V ⊂ (c , a), and y0 · tj → y0 . There exists a positive integer J such that ftj ((c, b)) ⊃ V and y · tj ∈ U as j J . Moreover, for fixed j J , there exists s ∈ (c, b) such that ftj (s) ∈ S0 (y0 · tj ) ∩ V . This implies that (ftj (s), y0 · tj ) ∈ M0 , i.e., (s, y0 ) · tj ∈ M0 . Hence (s, y0 ) ∈ M0 , i.e., s ∈ S0 (y0 ), which contradicts to the fact that (c, b) ∩ S0 (y0 ) = ∅. Therefore, c = c. Similarly, d = d. Since a ∈ [c, b], a = limj →∞ ftj (a) ∈ [c, a]. Hence a ∈ [c, a] ∩ S(y0 ) = {a}, i.e., a = a. Next, we show that S(y0 · tj ) S(y0 ).

(7.2)

Since ftj ([a, b]) ⊆ ftj ([c, b]), lim supj →∞ ftj ([a, b]) ⊆ [c, a] = S 1 . Also note that ftj (a) → a and ftj (b) → a. We have by Lemma 5.3(3) that limj →∞ ftj ([a, b]) = {a}. If c = d, then S(y0 ) ⊆ [a, b]. Hence S(y0 · tj ) = ftj (S(y0 )) = ftj (S(y0 ) ∩ [a, b]) → {a} = S(y0 ). Now suppose that c = d. Since limj →∞ ftj (c) = c and limj →∞ ftj (d) = d, we have by Lemma 5.3(1) that limj →∞ ftj ([d, c]) = [d, c]. Note that S(y0 · tj ) = ftj (S(y0 )) = ftj (S(y0 ) ∩ [a, b]) ∪ ftj (S(y0 ) ∩ [d, c]). It follows that lim supj →∞ S(y0 · tj ) ⊆ ({a} ∪ [d, c]). Since b ∈ / {a} ∪ [d, c], (7.2) holds. Now, since y0 ∈ Y c and y0 · tj → y0 , we have that S(y0 · tj ) → S(y0 ), which is a contradiction (7.2). This proves the Claim. It follows from the Claim that S(y) is a finite set for any y ∈ Y c ∩ Y0c . Hence M is an almost finite to one extension of Y . It follows from Proposition 3.2(1) that M is an almost n–1 extension of Y for some positive integer n = n(M). Similarly, M0 is an almost n0 –1 extension for some positive integer n0 = n(M0 ). In fact, from the proof of Proposition 3.2(1), we also see that |S(y)| = n for any y ∈ Y c and |S(y)| = n0 for any y ∈ Y0c . For a fixed y ∈ Y c ∩ Y0c , since S 1 \ S0 (y) has precisely n0 connected components, N (y) n0 . Using the Claim, we also have n = |S(y)| = N(y). Hence n n0 . Similarly, n0 n. This shows that n = n0 . (b) Let M0 , M be two minimal sets of (S 1 × Y, T) among which M0 is almost automorphic. We consider the set A0 of almost automorphic points of M0 . Since A0 is a residual subset of M0 , it follows from Lemma 5.5 that Y0∗ = y ∈ Y : A0 ∩ π −1 (y) is a residual subset of π −1 (y) ∩ M0 is a residual subset of Y . Let Y0c , Y c , S0 (y), S(y) be as in (a) for the minimal sets M0 , M and take any point y0 ∈ Y c ∩ Y0c ∩ Y0∗ . By (a), n = |S0 (y0 )| = |S(y0 )|. Since A0 ∩ π −1 (y0 ) = π −1 (y0 ) ∩ M0 , we see that for each s ∈ S0 (y0 ), (s, y0 ) ∈ A0 . If n = 1, then M is an almost 1–1 extension of Y , hence it is almost automorphic by Theorem 3.2. We now assume n 2. Since S 1 \ S0 (y0 ) has precisely n connected components and each connected component contains precisely one point in S(y0 ), there exist a1 = a2 ∈ S0 (y0 ) and b ∈ S(y0 ) such that {b} = [a1 , a2 ] ∩ S(y0 ). We want to show that (b, y0 ) is an almost automorphic point of M. Let {ti } be any sequence in T. Since (a1 , y0 ) and (a2 , y0 ) are almost automorphic points of M0 , there exist a subsequence {ti } ⊆ {ti } and a1 , a2 ∈ S0 (y) for some y ∈ Y such that limi→∞ (aj , y0 )· ti = (aj , y) and limi→∞ (aj , y) · (−ti ) = (aj , y0 ), for j = 1, 2. Taking a subsequence of {ti } if necessary, we may assume that there exist b ∈ S(y) and b ∈ S(y0 ) such that limi→∞ (b, y0 )·ti = (b , y) and limi→∞ (b , y) · (−ti ) = (b , y0 ).

874


Since limi→∞ limm→∞ (aj , y0 ) · (tm − ti ) = (aj , y0 ), j = 1, 2, and limi→∞ limm→∞ (b, y0 ) · (tm − ti ) = (b , y0 ), there exist sequences {mk } and {ik } such that if rk = tmk − tik for all k, then limk→∞ (aj , y0 ) · rk = (aj , y0 ), j = 1, 2, and limk→∞ (b, y0 ) · rk = (b , y0 ). Again, let ft be as in Lemma 5.3. Since limk→∞ frk (aj ) = aj , we have by Lemma 5.3(1) that limk→∞ frk ([a1 , a2 ]) = [a1 , a2 ]. Now, b = limk→∞ frk (b) ∈ limk→∞ frk ([a1 , a2 ]) = [a1 , a2 ]. Hence b ∈ [a1 , a2 ] ∩ S(y0 ) = {b}, i.e., b = b. This shows that (b, y0 ) is an almost automorphic point of M, implying that M is almost automorphic. 2 Theorem 7.3. Consider an APCF (S 1 × Y, T) in which Y is locally connected. If there are at least two minimal sets, then each minimal set is almost automorphic. Proof. Let M, M0 be two minimal sets of (S 1 × Y, T) and let S(y), S0 (y), ρ, ρ0 , Y c , Y0c be defined as in the proof of Theorem 7.2 for the present M, M0 . Fix y0 ∈ Y c ∩ Y0c . It follows from the proof of Theorem 7.2 that n =: |S0 (y0 )| = |S(y0 )| < +∞. Again, if n = 1, then M is an almost 1–1 extension of Y , hence by Theorem 3.2 it is almost automorphic. We now assume that n 2. Since S 1 \ S0 (y0 ) has precisely n connected components and each of them has precisely one point in S(y0 ), there exist points 0 t1 < r1 < t2 < r2 < · · · < tn < rn < 1 + t1 such that S(y0 ) = {a1 , a2 , . . . , an } and S0 (y0 ) = {b1 , b2 , . . . , bn }, where aj = e2πitj , bj = e2πirj , j = 1, 2, . . . , n. We want to show that (a1 , y0 ) is an almost automorphic point of M. By Theorem 3.1, it is sufficient to show that for any open neighborhood U of (a1 , y0 ) in S 1 × Y , the recurrent time set N((a1 , y0 ), U ) is a Δ∗ -set. Let U be an open neighborhood of (a1 , y0 ) in S 1 × Y . It is clear that there exist open neighborhoods W1 of y0 in Y and E of a1 in S 1 such that E × W1 ⊆ U . Let δ > 0 be sufficiently small and aj+ =: e2πi(tj +δ) ,

aj− =: e2πi(tj −δ) ,

bj+ =: e2πi(rj +δ) ,

bj− =: e2πi(rj −δ) ,

Aj = aj− , aj+ , Bj = bj− , bj+ ,

j = 1, 2, . . . , n, be such that A1 ⊂ E and A1 , B1 , A2 , B2 , . . . , An , Bn are pairwise disjoint, i.e., t1 − δ < t1 + δ < r1 − δ < r1 + δ < t2 − δ < · · · < tn + δ < rn − δ < rn + δ < 1 + t1 − δ. Since y0 ∈ Y c ∩Y0c , there exists an open neighborhood W of y0 with W ⊆ W1 such that for all y ∈ W , S(y) ⊆ nj=1 int(Aj ), S0 (y) ⊆ nj=1 int(Bj ), S(y) ∩ int(Aj ) = ∅, and S0 (y) ∩ int(Bj ) = ± ± , rj,y with ∅ for all j = 1, 2, . . . , n. Thus for each y ∈ W and j = 1, 2, . . . , n there exist tj,y − + − + tj − δ tj,y tj,y tj + δ and rj − δ rj,y rj,y rj + δ such that if +

aj− =: e2πitj,y ∈ S(y),

+

bj− =: e2πirj,y ∈ S0 (y),

+ =: e2πitj,y , aj,y + bj,y =: e2πirj,y ,

−

−

then S(y) ∩ Aj = S(y) ∩ Aj,y ⊆ int(Aj ) and S0 (y) ∩ Bj = S0 (y) ∩ Bj,y ⊆ int(Bj ), where Aj,y = − + − + , aj,y ] and Bj,y = [bj,y , bj,y ]. It is clear that A1,y , B1,y , A2,y , B2,y , . . . , An,y , Bn,y are pair[aj,y wise disjoint for each y ∈ W , and, for each j = 1, 2, . . . , n, the map Ej− : W → [tj − δ, tj + δ]:


875

− + y → tj,y is lower semi-continuous and the map Ej+ : W → [tj − δ, tj + δ]: y → tj,y is up[t −δ,t j j +δ] : per semi-continuous. It follows that for each j = 1, 2, . . . , n, both maps Ej : W → 2 1 − + − + − + y → [tj,y .tj,y ] and φj : W → 2S : y → Aj,y = [aj,y , aj,y ] = {e2πt : t ∈ [tj,y , tj,y ]} are upper semi-continuous.

Claim 1. Given j ∈ {1, 2, . . . , n}, y ∈ W , and t ∈ T such that y · t ∈ W , there exists a unique Ltj (y) ∈ {1, 2, . . . , n} such that (Aj,y × {y}) · t = ALt (y),y·t × {y · t}. Moreover, for fixed y, t as j

above, the map Lt{·} (y) : {1, 2, . . . , n} → {1, 2, . . . , n} is a permutation of {1, 2, . . . , n}. Let j, y, t be given as above. We consider the orientation preserving homeomorphism h : S 1 → S 1 : s → ψ(s, y, t). It is clear that (Aj,y × {y}) · t = h(Aj,y ) × {y · t}, h(Aj,y ) = − + − + [h(aj,y ), h(aj,y )], and h(aj,y ), h(aj,y ) ∈ S(y · t). If there exists b ∈ h(Aj,y ) ∩ S0 (y · t), i.e., (b, y · t) ∈ (Aj,y × {y}) · t and (b, y · t) ∈ M0 , then there exists b ∈ Aj,y such n that (b , y) · t = (b, y · t) ∈ M0 . It follows that (b , y) ∈ M0 , i.e., b ∈ Aj,y ∩ S0 (y) ⊆ Aj ∩ ( i=1 Bi ) = ∅, which is impossible. Hence h(Aj,y ) ∩ S0 (y · t) = ∅.

(7.3)

− + Since y · t ∈ W and h(aj,y ), h(aj,y ) ∈ S(y · t), there exist i1 , i2 ∈ {1, 2, . . . , n} such that − + − + h(aj,y ) ∈ Ai1 ,y·t ⊆ Ai1 and h(aj,y ) ∈ Ai2 ,y·t ⊆ Ai2 . Since [h(aj,y ), h(aj,y )] ∩ S0 (y · t) = ∅, − + we must have i1 = i2 . For otherwise, i1 = i2 , and the arc [h(aj,y ), h(aj,y )] intersects both − + − Ai1 and Ai2 . It follows that Bi1 ⊆ [h(aj,y ), h(aj,y )], and hence bi1 ,y·t ∈ Bi1 ∩ S0 (y · t) ⊆ − + [h(aj,y ), h(aj,y )] ∩ S0 (y · t), which is impossible by (7.3). − + t Now let Lj (y) = i1 . Then h(aj,y ), h(aj,y ) ∈ ALt (y),y·t . Since ALt (y),y·t is a sub-arc of S 1 , j

j

− + − + either (i) [h(aj,y ), h(aj,y )] ⊆ ALt (y),y·t or (ii) [h(aj,y ), h(aj,y )] ⊇ S 1 \ ALt (y),y·t . But the case j

j

− + (ii) is impossible because S0 (y · t) ⊆ S 1 \ ALt (y),y·t and [h(aj,y ), h(aj,y )] ∩ S0 (y · t) = ∅. It now j follows from (i) that

Aj,y × {y} · t ⊆ ALt (y),y·t × {y · t}.

(7.4)

j

Such Ltj (y) is unique because A1,y·t , A2,y·t , . . . , An,y·t are pairwise disjoint. Let y = y · t and t = −t. Then y , y · t ∈ W . From the above, for each i = 1, 2, . . . , n, there exists a unique Lti (y ) ∈ {1, 2, . . . , n} such that Ai,y × {y } · t ⊆ ALt (y ),y ·t × {y · t }.

(7.5)

i

For each j = 1, 2, . . . , n, we let i(j ) = Ltj (y). By (7.4) and (7.5), we have Aj,y × {y} ⊆ Ai(j ),y × {y } · t ⊆ ALt

i(j ) (y

),y

× {y}.

Since A1,y·t , A2,y·t , . . . , An,y·t are pairwise disjoint, Lti(j ) (y ) = j and Aj,y × {y} = (Ai(j ),y × {y }) · t . This implies that

876


Aj,y × {y} · t = Ai(j ),y × {y } · t · t = Ai(j ),y × {y } = ALt (y),y·t × {y · t}, j

i.e., (Aj,y × {y}) · t = ALt (y),y·t × {y · t}. j

Note that Ltj (y) ∈ {1, 2, . . . , n}, (Aj,y × {y}) · t = ALt (y),y·t × {y · t} for each j = j 1, 2, . . . , n, and A1,y , A2,y , . . . , An,y are pairwise disjoint. We have that Lt{·} (y) : {1, 2, . . . , n} → {1, 2, . . . , n} is one to one, i.e., a permutation of {1, 2, . . . , n}. This proves Claim 1. Claim 2. Let V be a nonempty, connected, closed subset of W and t ∈ T be such that V · t ⊂ W . Then there exists a permutation P of {1, 2, . . . , n} such that for each j = 1, 2, . . . , n and y ∈ V , (Aj,y × {y}) · t = AP (j ),y·t × {y · t}. Let j ∈ {1, 2, . . . , n} and y ∈ V be given. By Claim 1, there exists a unique Ltj (y) ∈ {1, 2, . . . , n} such that (Aj,y × {y}) · t = ALt (y),y·t × {y · t}. j

We first show that the map Ltj (·) : V → {1, 2, . . . , n} is continuous. Let {yk }∞ k=1 ⊂ V cont t t verges to some y ∈ V and denote i = Lj (y). If Lj (yk ) Lj (y), then by Claim 1 there exists a subsequence {k1 < k2 < · · ·} of {k} and r ∈ {1, 2, . . . , n} \ {i} such that Ltj (yk ) = r for each ∈ N. Take a sequence of points z ∈ Aj,yk , ∈ N. We assume without loss of generality 1

that lim →∞ z = z for some z ∈ S 1 . By the upper semi-continuity of the map φj : W → 2S : y → Aj,y , we have that z ∈ Aj,y . Since (z , yk ) · t ∈ Ar,yk × {yk · t}, it again follows from the upper semi-continuity of φj that (z, y) · t = lim →∞ (z , yk ) · t ∈ Ar,y·t × {y · t}. But since (z, y) · t ∈ (Aj,y × {y}) · t ∈ Ai,y·t × {y · t}, (z, y) · t ∈ Ai,y·t × {y · t} ∩ Ar,y·t × {y · t}. Hence Ai,y·t ∩ Ar,y·t = ∅, a contradiction to the fact that i = r. This shows the continuity of Ltj (·) : V → {1, 2, . . . , n}. Now, for each j = 1, 2, . . . , n, Ltj (·) : V → {1, 2, . . . , n} must be a constant map since its domain is connected and its range is discrete. Let y ∈ V and P (j ) = Ltj (y), j = 1, 2, . . . , n. Then Aj,y × {y} · t = AP (j ),y·t × {y · t}. It follows from Claim 1 that P : {1, 2 · · · , n} → {1, 2, . . . , n} is a permutation of {1, 2, . . . , n}. Claim 3. For any sequence {si } in T, N ((a1 , y0 ), U ) ∩ {sk − sk : k > k } = ∅, i.e., N ((a1 , y0 ), U ) is a Δ∗ -set. Let m = n! and d be an invariant compatible metric on Y , i.e., d(y1 · t, y2 · t) = d(y1 , y2 ), y1 , y2 ∈ Y , t ∈ T. Since (Y, T) is almost periodic, there exists an increasing subsequence si −r {ik } ⊂ N, an integer r ∈ {0, 1, . . . , m − 1}, and a homeomorphism g : Y → Y such that km ∈ T and 1 si − r sup d y · k , g(y) k m 2 y∈Y

(7.6)

for all k = 1, 2, . . . . Since Y is locally connected, there exists a connected closed neighborhood V of y0 such that Bδ (V ) =: {y ∈ Y : d(y, V ) < δ} ⊆ W and B(y0 , δ) ⊆ V for some δ > 0. Let K δ be a natural number such that 21K < 2m . Then for any u, v ∈ N with u > v K, we have by (7.6) that


877

si − r siu − siv si − r , y = sup d y · u ,y · v sup d y · m m m y∈Y y∈Y si − r si − r sup d y · u , g(y) + sup d y · v , g(y) m m y∈Y y∈Y It follows that d(V · (

siu −siv m

), V )
k }, i.e., N ((a1 , y0 ), U ) ∩ {sk − sk : k > k } = ∅. Since {si } is arbitrary, N ((a1 , y0 ), U ) is a Δ∗ -set. This completes the proof. 2 Now, parts (1) and (2) of Theorem 4 are the respective parts in Theorem 7.2 above, and, part (3) of Theorem 4 is just Theorem 7.3 above. 8. Mean motion, transitivity, and connectivity In this section, we consider a SPCF (S 1 × Y, T) = (S 1 × Y, {Λˆ t }t∈T ) in the angular form (1.3), i.e., Λˆ t (φ0 , y0 ) = φ(φ0 , y0 , t), y0 · t , t ∈ T, (8.1) ˜ φ˜ 0 , y0 , t) be the lift of φ(φ0 , y0 , t) in R 1 satisfying where φ, φ0 ∈ R 1 (mod 1), y0 ∈ Y . Let φ( ˜ φ˜ 0 + 1, y0 , t) ≡ φ( ˜ φ˜ 0 , y0 , t) + 1. Then it is clear that Λˆ t is generated from the flow Λ˜ t : R 1 × φ( Y → R1 × Y : ˜ φ˜ 0 , y0 , t), y0 · t , t ∈ T Λ˜ t (φ˜ 0 , y0 ) = φ( when φ˜ 0 , φ˜ are identified modulo 1.

878


Through the section, for simplicity, we will often use the same symbol φ0 to denote a point φ0 ∈ S 1 and its lift φ˜ 0 ∈ R 1 . 8.1. Rotation number and mean motion It is more or less known that a SPCF (8.1) with uniquely ergodic base flow (Y, T) admits a well-defined rotation number (see [23] for the discrete case and [34] for certain almost periodic continuous case). Below, for the sake of completeness, we give a unified proof of this result for both discrete and continuous cases. The following result is known as the Oxtoby ergodic theorem (see [15]). Lemma 8.1. Let (X, T) be uniquely ergodic and f ∈ C(X, R 1 ). Then for any x ∈ X, 1 lim T →+∞ λT ([0, T ) ∩ T)

f (x · t) dλT (t) = [0,T )∩T

1 T →+∞ λT ((−T , 0] ∩ T)

f (z) dμ(z), X

f (x · t) dλT (t) =

lim

f (z) dμ(z), X

(−T ,0]∩T

where λT is the Haar measure on T with λT ([0, 1) ∩ T) = 1 and μ is the unique T-invariant probability measure on (X, T). Theorem 8.1. Consider the SPCF (8.1) with uniquely ergodic base flow (Y, T). Then for any φ0 ∈ R 1 , y0 ∈ Y , the limit ˜ 0 , y0 , t) φ(φ t→∞ t

ρ = lim

exists and is independent of choice of (φ0 , y0 ) ∈ R 1 × Y . Proof. For simplicity, we only consider the limit as t → +∞. First, we observe from the periodicity of φ˜ in the first argument that for any t ∈ T, y ∈ Y , if ∗ φ1 , φ2∗ ∈ R 1 are such that |φ1∗ − φ2∗ | < l for some positive integer l, then also ∗ φ˜ φ , y, t − φ˜ φ ∗ , y, t < l. 1 2 Next, for any φ0 ∈ R 1 , y ∈ Y , t, s ∈ T, we let 0 φ1 , φ2 < 1 be such that φ1 ≡ φ0 ,

˜ 0 , y, t) φ2 ≡ φ(φ

(mod 1).

Then ˜ 0 , y · t, s) − φ0 = φ(φ ˜ 1 , y · t, s) − φ1 , φ(φ ˜ 0 , y, t + s) − φ(φ ˜ 0 , y, t) = φ(φ ˜ 2 , y · t, s) − φ2 . φ(φ It follows from (8.2) that

(8.2)


879

φ(φ ˜ 0 , y, t) − φ(φ ˜ 0 , y · t, s) + φ0 ˜ 0 , y, t + s) − φ(φ ˜ 2 , y · t, s) − φ(φ ˜ 1 , y · t, s) + φ2 − φ1 = φ(φ ˜ 2 , y · t, s) − φ(φ ˜ 1 , y · t, s)| + |φ2 − φ1 4, φ(φ i.e., ˜ 0 , y, t + s) − φ(φ ˜ 0 , y, t) − φ(φ ˜ 0 , y · t, s) + φ0 4. −4 φ(φ

(8.3)

Integrating (8.3) with respect to t from 0 to a positive number T ∈ T yields that −4 ł e

1 T

˜ 0 , y, t) dλT (t) ˜ 0 , y, t + s) − φ(φ φ(φ

[0,T )∩T

−

˜ φ(φ0 , y · t, s) − φ0 dλT (t) 4.

(8.4)

[0,T )∩T

˜ 0 , z, τ ) − φ0 |: φ0 ∈ R 1 , z ∈ Y, |τ | s}. It For any positive number s ∈ T, we let Ms = sup{|φ(φ is clear that 0 Ms < +∞ and ˜ ˜ ˜ φ(φ0 , y, t + s) − φ(φ0 , y, t) dλT (t) − s φ(φ0 , y, T ) [0,T )∩T

=

˜ 0 , y, T + t) − φ(φ ˜ 0 , y, t) dλT (t) − s φ(φ ˜ 0 , y, T ) φ(φ

[0,s)∩T

=

˜ 0 , y, T ) dλT (t) ˜ 0 , y, T ), y · T , t − φ(φ φ˜ φ(φ

[0,s)∩T

−

˜ φ(φ0 , y, t) − φ0 dλT (t) − sφ0

[0,s)∩T

s |φ0 | + 2Ms =: sΦs . Combining this with (8.4) yields that 4T + sΦs 1 1 ˜ 0 , y, T ) − − φ(φ sT T sT

˜ 0 , y · t, s) − φ0 dλT (t) φ(φ

[0,T )∩T

4T + sΦs . sT

(8.5) ˜

˜

Now consider functions ρ ∗ (y) = lim supt→+∞ φ(φ0t,y,t) and ρ∗ (y) = lim inft→+∞ φ(φ0t,y,t) . We note by (8.2) that both ρ ∗ (y) and ρ∗ (y) are independent of φ0 . Since (Y, T) is uniquely ergodic, we have by Lemma 8.1 that

880


1 lim T →+∞ T

˜ 0 , y · t, s) − φ0 dλT (t) = φ(φ

[0,T )∩T

˜ 0 , z, s) − φ0 dμ(z), φ(φ

(8.6)

Y

where μ is the unique T-invariant Borel probability measure on (Y, T). By letting T → +∞ in (8.5) and applying (8.5) and (8.6), we have that 4 1 4 1 ∗ ˜ ˜ 0 , z, s) − φ0 dμ(z). φ(φ0 , z, s) − φ0 dμ(z) ρ∗ (y) ρ (y) + φ(φ − + s s s s Y

Y

Now, taking limit s → +∞ in the above, we see that ρ∗ (y) = ρ ∗ (y) and equals 1 ˜ 0 , z, s) − φ0 dμ(z), lim φ(φ s→+∞ s Y

which is a constant, denoted by ρ.

2

The limit in the theorem is referred to as the rotation number associated with the SPCF (8.1). Recall that the SPCF (8.1) is said to admit mean motion if ˜ 0 , y0 , t) − φ0 − ρt < ∞ supφ(φ t∈T

for all (φ0 , y0 ) ∈ R 1 × Y . Theorem 8.2. Consider the SPCF (8.1) with strictly ergodic base flow (Y, T). Then the followings are equivalent: (a) (b) (c) (d) (e) (f) (g)

(8.1) admits mean motion; ˜ 0 , y0 , t) − φ0 − ρt| < +∞ for some (φ0 , y0 ) ∈ R 1 × Y ; supt0 |φ(φ ˜ 0 , y0 , t) − φ0 − ρt| < +∞ for some (φ0 , y0 ) ∈ R 1 × Y ; supt0 |φ(φ ˜ 0 , y0 , t) − φ0 − ρt) < +∞ for all (φ0 , y0 ) ∈ R 1 × Y ; supt0 (φ(φ ˜ 0 , y0 , t) − φ0 − ρt) < +∞ for all (φ0 , y0 ) ∈ R 1 × Y ; supt0 (φ(φ ˜ 0 , y0 , t) − φ0 − ρt) > −∞ for all (φ0 , y0 ) ∈ R 1 × Y ; inft0 (φ(φ ˜ 0 , y0 , t) − φ0 − ρt) > −∞ for all (φ0 , y0 ) ∈ R 1 × Y . inft0 (φ(φ

Proof. It is clear that (a) implies (b)–(g). Suppose (b) holds and let ˜ 0 , y0 , t) − φ0 − ρt . M = supφ(φ t0

Then by the flow property, φ˜ φ(φ ˜ 0 , y0 , s), y0 · s, t − φ(φ ˜ 0 , y0 , s) − ρt + φ(φ ˜ 0 , y0 , s) − φ0 − ρs M

(8.7)

for all s + t 0. Consider the ω-limit set ω(φ0 , y0 ) of (φ0 , y0 ) with respect to the flow (8.1). For simplicity, we view ω(φ0 , y0 ) as a subset of [0, 1] × Y . Let (φ∗ , y∗ ) ∈ ω(φ0 , y0 ) and sn → +∞


881

be a sequence such that Λ˜ sn (φ0 , y0 ) → (φ∗ , y∗ ). By taking a subsequence if necessary, we let ˜ 0 , y0 , sn ) − φ0 − ρsn |. It follows from (8.7) that r(φ∗ , y∗ ) = limn→∞ |φ(φ φ(φ ˜ ∗ , y∗ , t) − φ∗ − ρt M − r(φ∗ , y∗ ) for all t ∈ T. Hence by (8.2), supφ˜ φ 0 , y 0 , t − φ 0 − ρt < ∞ t∈T

for all (φ 0 , y 0 ) ∈ R 1 × Y , i.e., (a) holds. This shows that (b) implies (a). Similarly, (c) implies (a). Now let (d) hold. Suppose for contradiction that (a) fails. It follows from the flow property and the equivalence between (a) and (b) that ˜ ∗ , y ∗ , n) − φ ∗ − ρn = +∞ (8.8) supφ(φ n∈N

for all (φ ∗ , y ∗ ) ∈ R 1 × Y . Let E be a minimal set of the time-1 map Λˆ 1 and consider the function ˜ y, 1) − φ − ρ. Since u(φ + 1, y) ≡ u(φ, y), u can be viewed u : R 1 × Y → R 1 : u(φ, y) = φ(φ, 1 as a continuous function on S × Y . Using flow property and induction, it is easy to see that n−1 ˜ u Λ˜ i (φ, y) = φ(φ, y, n) − φ − ρn

(8.9)

i=0

˜ ∗ ∗ for all n ∈ N and (φ, y) ∈ R 1 × Y . Hence by (8.8), | n−1 i=0 u(Λi (φ , y ))| is unbounded on n−1 1 ∗ ∗ ∗ ∗ N for any (φ , y ) ∈ E. Since limn→∞ n i=0 u(Λ˜ i (φ , y )) = 0 for any (φ ∗ , y ∗ ) ∈ E, there ˜ is a residual subset E∗ of E such that for any (φ∗ , y∗ ) ∈ E∗ the function n−1 i=0 u(Λi (φ∗ , y∗ )) oscillates from −∞ to +∞ as n → +∞ (see e.g., [30,35]). In particular, n−1 ˜ ∗ , y∗ , n) − φ∗ − ρn = sup u Λ˜ i (φ∗ , y∗ ) = +∞ sup φ(φ

n∈N

(8.10)

n∈N i=0

for all (φ∗ , y∗ ) ∈ E∗ . This is a contradiction to the condition in (d). Hence (d) implies (a). Similarly, either (e) or (f) or (g) implies (a). 2 8.2. APCF with mean motion The aim of this subsection is to prove Theorem 6. We will need the following Lemma 8.2. Let M be a minimal set of an almost periodically forced skew-product flow (R 1 × Y, T) = (R 1 × Y, {Πt }t∈T ): Πt (x0 , y0 ) = x(x0 , y0 , t), y · t , t ∈ T, where (Y, T) is an almost periodic minimal flow. Then M is an almost 1–1 extension of Y hence is almost automorphic.

882


Proof. In the case T = R, the lemma is a special case of the main result in [56] concerning totally monotone skew-product semiflows. The proof for the discrete case follows from that of the continuous case (see also [62]) almost word by word. 2 The following theorem is just our main result Theorem 6. Theorem 8.3. Suppose that (8.1) is an APCF which admits mean motion. Then the following holds. (1) Each minimal set of (8.1) is almost automorphic whose frequency module is generated by the rotation number and the forcing frequencies. (2) If a minimal set of (8.1) is an almost N –1 extension of Y for some positive integer N , then N is the smallest positive integer whose multiplication to the rotation number is contained in the frequency module of the forcing. Proof. (1) Let E be a minimal set of (8.1). It is sufficient to only consider the case T = Z, because, if T = R, then a point of E is almost automorphic iff it is almost automorphic for the time-1 map Λ˜ 1 [5]. Let Y ∗ = S 1 × Y be given the flow (φ0 , y0 ) · t = (φ0 + ρt (mod 1), y0 · t), t ∈ Z. Then (Y ∗ , Z) is almost periodic (need not be minimal). Consider the skew-product flow Λ∗t : R 1 × Y ∗ → R1 × Y ∗ : ˜ 0 , y0 , t) − φ0 − ρt + x0 , φ0 + ρt (mod 1), y0 · t , Λ∗t (x0 , φ0 , y0 ) = φ(φ

t ∈ Z.

It follows from Lemma 8.2 that each minimal set of Λ∗t is an almost 1–1 extension of a minimal set in (Y ∗ , Z). Using this fact and (8.9), the rest of the proof follows from that of Theorem 3.2(1) in [62] almost word by word. (2) Let ρ be the rotation number of (8.1) and M be a minimal set of (8.1) which is an almost N –1 extension of Y . We denote by M(M), M(Y ) as the frequency modules of M, Y , respectively. Then by the general module containment property for almost automorphic minimal sets (e.g., [57]), N is the smallest positive integer such that N M(M) ⊂ M(Y ). But by (1), M(M) is generated by ρ and M(Y ). The theorem follows. 2 8.3. APCF without mean motion We first prove a general result on positive transitivity and the uniqueness of minimal set in a SPCF (S 1 × Y, T). Theorem 8.4. If a SPCF (S 1 × Y, T) with minimal base flow (Y, T) is positively transitive, then it has a unique minimal set. Proof. Suppose for contradiction that (S 1 × Y, T) has two distinct minimal sets M, M0 . We let S(y), S0 (y), ρ, ρ0 , Y c and Y0c be defined as in the proof of Theorem 7.2 for the present M, M0 . Since (S 1 × Y, T) is positively transitive, the set Tran+ (S 1 × Y ) of positively transitive points in S 1 × Y is a residual subset of S 1 × Y . Let YT = y ∈ Y : Tran+ S 1 × Y ∩ S 1 × {y} is a residual subset of S 1 × {y} .


883

Then by Lemma 5.1, YT is a residual subset of Y . For a given y0 ∈ Y c ∩ Y0c ∩ YT , we have by the proof of Theorem 7.2 that n =: |S0 (y0 )| = |S(y0 )| < +∞ and each of the n connected components of S 1 \ S0 (y0 ) contains precisely one point in S(y0 ). Thus there exists points 0 t1 < r1 < t2 < r2 < · · · < tn < rn < 1 + t1 such that S(y0 ) = {a1 , a2 , . . . , an } and S0 (y0 ) = {b1 , b2 , . . . , bn }, where aj = e2πitj and bj = e2πirj for j = 1, 2, . . . , n. Since Tran+ (S 1 × Y ) ∩ (S 1 × {y0 }) is dense in S 1 × {y0 }, there exists w1 ∈ (t1 , r1 ) such that (c1 , y0 ) ∈ Tran+ (S 1 × Y ), where c1 = e2πiw1 . Take a number w2 such that w2 ∈ (r1 , t2 ) when n 2 and w2 ∈ (r1 , 1 + t1 ) when n = 1, and let c2 = e2πiw2 . Then c2 ∈ (b1 , a2 ) when n 2 and c2 ∈ (b1 , a1 ) when n = 1. In any case, c2 ∈ / S(y0 ) ∪ S0 (y0 ). Since (c1 , y0 ) ∈ Tran+ (S 1 × Y ), there exists a monotonically increasing, positive sequence si → +∞ such that limi→∞ (c1 , y0 ) · si = (c2 , y0 ). Without loss of generality, we assume that limi→∞ (a1 , y0 ) · si = (aj , y0 ) for some 1 j n. Consider the family of functions ft : S 1 → S 1 : u → ψ(u, y0 , t), t ∈ T. Then each ft is an orientation preserving homeomorphism. Since limi→∞ fsi (c1 ) = c2 , limi→∞ fsi (a1 ) = aj , and aj = c2 , we have by Lemma 5.3(1) that limi→∞ fsi ([a1 , c1 ]) = [aj , c2 ]. Using the fact that b1 ∈ (a1 , c2 ) ⊆ (aj , c2 ), we can find a sufficiently small open neighborhood V of b1 in S 1 such that V ⊂ (aj , c2 ). Since y0 ∈ Y0c and b1 ∈ S0 (y0 ), there exists an open neighborhood U of y0 in Y such that S0 (y)∩V = ∅ for each y ∈ U . Note that fsi ((a1 , c1 )) = (fsi (a1 ), fsi (c1 )), V ⊂ (aj , c2 ), and y0 · sj → y0 . It follows that there exists a positive integer N such that fsi ((a1 , c1 )) ⊃ V and y0 · si ∈ U as i N . Moreover, for a fixed i N , there exists b ∈ (a1 , c1 ) such that fsi (b) ∈ S0 (y0 · si ) ∩ V . This implies that (fsj (b), y0 · sj ) ∈ M0 , i.e., (b, y0 ) · si ∈ M0 . Hence (b, y0 ) ∈ M0 , i.e., b ∈ S0 (y0 ). This is a contradiction to the fact that (a1 , c1 ) ∩ S0 (y0 ) = ∅. 2 To prove Theorem 7, we need the following lemmas. Lemma 8.3. Consider the SPCF (8.1) with strictly ergodic base flow (Y, T). Then for any φ1 , φ2 ∈ R 1 , y ∈ Y , and t ∈ T, ˜ 1 , y, t) − φ1 φ(φ ˜ 2 , y, t) − φ2 + 2. φ(φ ˜ y, t) : R 1 → R 1 Proof. Let φ1 , φ2 ∈ R 1 , y ∈ Y , and t ∈ T be given. Since the function φ(·, ˜ 1 , y, t) φ(φ ˜ 2 + [φ1 − φ2 ] + 1, y, t) = φ(φ ˜ 2 , y, t) + [φ1 − φ2 ] + 1, is strictly increasing, φ(φ where for each r ∈ R 1 , [r] denotes the largest integer which is less than or equal to r. Thus ˜ 1 , y, t) − φ1 φ(φ ˜ 2 , y, t) + [φ1 − φ2 ] − φ1 + 1 φ(φ ˜ 2 , y, t) + (φ1 − φ2 + 1) − φ1 + 1, i.e., φ(φ ˜ 1 , y, t) − φ1 φ(φ ˜ 2 , y, t) − φ2 + 2. 2 φ(φ Lemma 8.4. Consider the SPCF (8.1) with strictly ergodic base flow (Y, T). If there exists (φ∗ , y∗ ) ∈ R 1 × Y such that ˜ ∗ , y∗ , t) − φ∗ − ρt = +∞ lim sup φ(φ t→+∞

˜ ∗ , y∗ , t) − φ∗ − ρt = −∞ , resp. lim inf φ(φ t→+∞

(8.11)

˜ 0 , y0 , t) − φ0 − ρt) = +∞ then there exists a residual subset Y∗ of Y such that lim supt→+∞ (φ(φ ˜ (resp. lim inft→+∞ (φ(φ0 , y0 , t) − φ0 − ρt) = −∞) for all (φ0 , y0 ) ∈ R 1 × Y∗ .

884


Proof. We only consider the case that ˜ ∗ , y∗ , t) − φ∗ − ρt = +∞. lim sup φ(φ t→+∞

˜ ∗ , y∗ , s), s ∈ R 1 . Let M ∈ N and s ∈ T be given. It follows from (8.11) that Denote φs = φ(φ there exists t (s) > M such that φ˜ φ∗ , y∗ , s + t (s) − φ∗ − ρ s + t (s) > M + 2 + φs − φ∗ − ρs, ˜ s , y∗ · s, t (s)) − φs − ρt (s) > M + 2. By continuity, we let UsM be an open neighborhood i.e., φ(φ ˜ s , y, t (s)) − φs − ρt (s) > M + 2 for all y ∈ UsM . Then by Lemma 8.3, of y∗ · s such that φ(φ φ˜ φ, y, t (s) − φ − ρt (s) > M for all (φ, y) ∈ R 1 × UsM . Let UM = s∈T UsM . Since {y∗ · s}s∈T is dense in Y , UM is a dense ˜ y, t) − φ − ρt > open subset of Y . Moreover, for each y ∈ UM there exists t > M such that φ(φ, M for all φ ∈R 1 . Let Y∗ = M∈N UM . Then Y∗ is a residual subset of Y , and, ˜ 0 , y0 , t) − φ0 − ρt = +∞ lim sup φ(φ t→+∞

for any (φ0 , y0 ) ∈ R 1 × Y∗ .

2

Lemma 8.5. Consider the SPCF (8.1) with strictly ergodic base flow (Y, T). Then there exist y1 , y2 ∈ Y such that ˜ sup φ(φ, y1 , t) − φ − ρt 4,

(8.12)

t1

˜ y2 , t) − φ − ρt −4 inf φ(φ,

t1

(8.13)

for all φ ∈ R 1 . Proof. Suppose for contradiction that (8.12) is not true. Then for a given y ∈ Y there exist ˜ y , y, ty ) − φy − ρty > 4. By continuity, there exists an open φy ∈ R 1 and ty 1 such that φ(φ ˜ y , y , ty ) − φy − ρty > 4 for all y ∈ Uy . Since by Lemma 8.3 neighborhood Uy of y such that φ(φ ˜ y , z, ty ) − φy + 2 for all φ ∈ R 1 and z ∈ Y , we have ˜ φ(φ, z, ty ) − φ φ(φ ˜ φ(φ, y , ty ) − φ − ρty > 2 for all (φ, y ) ∈ R 1 × Uy . Since {Uy }y∈Y is an open cover of Y and Y is compact, there exists a finite set {y1 , y2 , . . . , yk } ⊂ Y such that ki=1 Uyi = Y . We denote Ui = Uyi , ti = tyi for short and let T = max{t1 , t2 , . . . , tk }.


885

Given (φ0 , y0 ) ∈ R 1 × Y , we inductively define sequences {Ti } ⊂ T, {ki } ⊂ {1, 2, . . . k}, and ˜ {mi } by letting T0 = 0, y0 · Ti ∈ Uki , mi = tki , and Ti+1 = Ti + mi . Then φ(φ, y0 · Ti , mi ) − φ − ρmi > 2 for all φ ∈ R 1 . It follows that ˜ 0 , y0 , Ti+1 ) − φ0 − ρTi+1 φ(φ ˜ 0 , y0 , Ti ) − ρmi = φ˜ φ(φ0 , y0 , Ti ), y0 · Ti , mi − φ(φ ˜ 0 , y0 , Ti ) − φ0 − ρTi . ˜ 0 , y0 , Ti ) − φ0 − ρTi 2 + φ(φ + φ(φ By induction, ˜ 0 , y0 , Ti ) − φ0 − ρTi 2i φ(φ for all i ∈ N. Since i Ti T i for all i ∈ N, we have lim sup i→∞

1 1 ˜ 0 , y0 , Ti ) lim sup (φ0 + ρTi + 2i) φ(φ Ti T i i→∞ = ρ + lim sup i→∞

2i 2 ρ + > ρ, Ti T

which contradicts to the definition of the rotation number. This proves (8.12). The proof of (8.13) is similar. 2 Theorem 8.5. Suppose that the SPCF (8.1) is an APCF which admits no mean motion. Then each of its minimal set is either the entire phase space S 1 × Y or is everywhere non-locally connected. Proof. We let d be an invariant compatible metric on Y , i.e., d(y1 · t, y2 · t) = d(y1 , y2 ) for all y1 , y2 ∈ Y and t ∈ T. Since (8.1) admits no mean motion, the condition (d) of Theorem 8.2 fails, ˜ ∗ , y∗ , t) − φ∗ − ρt) = +∞. It i.e., there exists (φ∗ , y∗ ) ∈ R 1 × Y such that lim supt→+∞ (φ(φ follows from Lemma 8.4 that there exists a residual subset Y∗ of Y such that ˜ 0 , y0 , t) − φ0 − ρt = +∞ lim sup φ(φ

(8.14)

t→+∞

for any (φ0 , y0 ) ∈ R 1 × Y∗ . By Lemma 8.5(1), there exists y1 ∈ Y such that ˜ sup φ(φ, y1 , t) − φ − ρt 4

(8.15)

t1

for all φ ∈ R 1 . Suppose that the entire phase space S 1 × Y is not minimal and let X be a minimal set of (8.1). Then there exist nonempty open subsets U2 ⊂ S 1 and V2 ⊂ Y such that X ∩ (U2 × V2 ) = ∅. Since (Y, T) is minimal, there exist t1 , t2 , . . . , t ∈ T such that V = {V2 · ti } i=1 is an open cover of Y . Let δ > 0 be the Lebesgue number of V with respect to the metric d.

886


If X is not everywhere non-locally connected, then by Lemma 7.1 the set Xlc of locally connected points in X is an invariant residual subset of X. Since by Proposition 3.4 the projection π : X → Y is semi-open, it follows from Lemma 5.5 that X0 = x ∈ X: for any open neighborhood U of x, π(U ) is a neighborhood of π(x) is also a residual subset of X. Take x∗ ∈ Xlc ∩ X0 and denote y ∗ = π(x∗ ). Let φ1 ∈ [0, 1] be such that x∗ = (a, y ∗ ), where a = e2πiφ1 and denote A = [φ1 − 14 , φ1 + 14 ]. Also let V1 be an open neighborhood of y ∗ such that diam(V1 ) < δ and let U1 = {e2πiφ : φ ∈ A}. Since x∗ is a locally connected point of X, there exists a connected closed neighborhood W of x∗ in X such that W ⊆ U1 × V1 . Since x∗ ∈ X0 , π(W ) is also a closed neighborhood of y ∗ in Y . Using minimality of (Y, T), we let t ∗ 1 be such that y2 =: y1 · t ∗ ∈ π(W ) and denote ˜ y1 , t ∗ ) − φ − ρt ∗ |. Then C(t ∗ ) < ∞ and it follows from (8.15) that, for C(t ∗ ) = maxφ∈R 1 |φ(φ, any φ ∈ R 1 , ˜ y1 , t ∗ ), y2 , t − φ − ρ(t + t ∗ ) 4 sup φ˜ φ(φ, t0

˜ ˜ ˜ y1 , t ∗ ), y2 , t − φ(φ, y1 , t ∗ ) − φ − ρt ∗ y1 , t ∗ ) − ρt ∗ + φ(φ, = sup φ˜ φ(φ, t0

˜ ˜ y1 , t ∗ ) − ρt ∗ − C(t ∗ ), sup φ˜ φ(φ, y1 , t ∗ ), y2 , t − φ(φ, t0

i.e., ˜ sup φ(φ, y2 , t) − φ − ρt 4 + C(t ∗ ).

(8.16)

t0

Since y2 ∈ π(W ), there exists φ2 ∈ A such that (a2 , y2 ) ∈ W , where a2 = e2πiφ2 . For any (b, y) ∈ W ⊆ U1 ×V1 , there exists a unique φ(b) ∈ A such that e2πiφ(b) = b. Consider the map h : W → A × V1 : (b, y) → (φ(b), y). Clearly, h is continuous. Let F = h(W ). Then (φ2 , y2 ) ∈ F and it follows from the closeness and connectivity of W that F is a closed connected subset of A × V1 . Since π(W ) is a closed neighborhood of y ∗ , π(W ) ∩ Y∗ = ∅. Take y0 ∈ π(W ) ∩ Y∗ . Then ˜ y, ti ) − φ − ρti : φ ∈ R 1 , there exists φ0 ∈ A such that (φ0 , y0 ) ∈ F . Let L = max i=1 max{φ(φ, y ∈ Y }. Then L < ∞ and by (8.14), there exists t > max{t1 , t2 , . . . , t } such that ˜ 0 , y0 , t ) − φ0 − ρt 6 + L + C(t ∗ ). φ(φ Since diam(V1 · t ) = diam(V1 ) < δ , there exists i ∈ {1, 2, . . . , } such that V1 · t ⊆ V2 · ti , i.e., V1 · (t − ti ) ⊆ V2 . Denote t∗ = t − ti . Then t∗ > 0, V1 · t∗ ⊆ V2 , and ˜ 0 , y0 , t∗ ) − φ0 − ρt∗ φ(φ ˜ 0 , y0 , t∗ ), y0 · t∗ , ti − φ(φ ˜ 0 , y0 , t∗ ) − ρti ˜ 0 , y0 , t ) − φ0 − ρt − φ˜ φ(φ = φ(φ ˜ 0 , y0 , t∗ ) − ρti 6 + C(t ∗ ), ˜ 0 , y0 , t∗ ), y0 · t∗ , ti − φ(φ 6 + L + C(t ∗ ) − φ˜ φ(φ i.e., ˜ 0 , y0 , t∗ ) (φ0 + ρt∗ ) + 6 + C(t ∗ ). φ(φ

(8.17)


887

˜ Consider the function Φ : F → R 1 : (φ, y) → φ(φ, y, t∗ ). Obviously, Φ is continuous. Moreover, by noting that φ2 , φ0 ∈ A, we have by (8.16) that Φ(φ2 , y2 ) (φ2 + ρt∗ ) + 4 + C(t ∗ ) (φ0 + ρt∗ ) + 5 + C(t ∗ ), and by (8.17) that Φ(φ0 , y0 ) (φ0 + ρt∗ ) + 6 + C(t ∗ ). Since F is connected and (φ0 , y0 ), (φ2 , y2 ) ∈ F , we have that Φ(F ) ⊇ [(φ0 + ρt∗ ) + 5, (φ0 + ρt∗ ) + 6]. Hence there exists (φ3 , y3 ) ∈ F such that e2πiΦ(φ3 ,y3 ) ∈ U2 . It follows from the definition of F that (e2πiφ3 , y3 ) ∈ W ⊂ X. Now, on one hand, (e2πiφ3 , y3 ) · t∗ ∈ X, and on the other hand, (e2πiφ3 , y3 ) · t∗ = 2πiΦ(φ 3 ,y3 ) , y · t ) ∈ U × V as V · t ⊆ V . This implies that X ∩ (U × V ) = ∅, a con(e 3 ∗ 2 2 1 ∗ 2 2 2 tradiction. 2 Theorem 8.6. Suppose that the SPCF (8.1) is an APCF with locally connected base space Y . If it admits no mean motion, then it is positively transitive and has only one minimal set. Proof. Let d, Y∗ and y1 be defined in the proof of Theorem 8.5. Let U1 , U2 ⊂ S 1 and V1 , V2 ⊂ Y be any nonempty open subsets. Since (Y, T) is minimal, there exist t1 , t2 , . . . , tk ∈ T such that V = {V2 · ti }ki=1 is an open cover of Y . Let δ > 0 be the Lebesgue number of V with respect to the metric d. Since {y1 · r}r1 is dense in Y , there exists r1 1 such that y1 · r1 ∈ V1 , i.e., V1 is an open neighborhood of y∗ = y1 · r1 . Since Y is local connected, there exists a connected closed neighborhood V of y∗ such that V ⊆ V1 and diam(V ) < δ. By (8.15), we have ˜ ˜ ˜ y1 , r1 ) − φ − ρr1 y1 , r1 ), y∗ , t − φ(φ, y1 , r1 ) − ρt 4 + φ(φ, sup φ˜ φ(φ, t0

˜ ˜ for all φ ∈ R 1 . Let C = supφ∈R 1 |φ(φ, y1 , r1 ) − φ − ρr1 | = max0φ1 |φ(φ, y1 , r1 ) − φ − ρr1 |. Then C < ∞ and ˜ sup φ(φ, y∗ , t) − φ − ρt 4 + C

(8.18)

t0

for all φ ∈ R 1 . Take z1 ∈ Y∗ ∩ V and φ1 ∈ [0, 1] such that e2πiφ1 ∈ U1 . Then ˜ 1 , z1 , t) − φ1 − ρt = +∞. lim sup φ(φ t→+∞

˜ Let L = maxki=1 max{φ(φ, y, ti ) − φ − ρti : φ ∈ R 1 , y ∈ Y }. Then L < ∞. Take t0 > max{t1 , t2 , . . . , tk } such that ˜ 1 , z1 , t0 ) − φ1 − ρt0 5 + C + L. φ(φ Since diam(V · t0 ) = diam(V ) < δ, there exists i ∈ {1, 2, . . . , k} such that V · t0 ⊆ V2 · ti , i.e., V · (t0 − ti ) ⊆ V2 . Let t∗ = t0 − ti . Then t∗ > 0, V · t∗ ⊂ V2 , and ˜ 1 , z1 , t∗ ) − φ1 − ρt∗ φ(φ ˜ 1 , z1 , t∗ ) − ρti ˜ 1 , z1 , t0 ) − φ1 − ρt0 − φ˜ φ(φ1 , z1 , t∗ ), z1 · t∗ , ti − φ(φ = φ(φ ˜ 1 , z1 , t∗ ) − ρti 5 + C + L − φ˜ φ(φ1 , z1 , t∗ ), z1 · t∗ , ti − φ(φ 5 + C.

888


˜ 1 , y, t∗ ). Clearly, Ψ is continuous, Ψ (y∗ ) Consider the function Ψ : V → R 1 : y → φ(φ 4 + C + φ1 + ρt∗ , and Ψ (z1 ) 5 + C + φ1 + ρt∗ . Since V is connected, Ψ (V ) ⊇ [4 + C + φ1 + ρt∗ , 5 + C + φ1 + ρt∗ ]. Hence there exists y2 ∈ V such that e2πiΨ (y2 ) ∈ U2 . This shows that (e2πiφ1 , y2 ) ∈ U1 × V1 and (e2πiφ1 , y2 ) · t∗ = (e2πiΨ (y2 ) , y2 · t∗ ) ∈ U2 × V2 as V · t∗ ⊆ V2 . Therefore, (e2πiΨ (y2 ) , y2 · t∗ ) ∈ (U1 × V1 ) · t∗ ∩ (U2 × V2 ) = ∅. Since U1 , U2 , V1 , V2 are arbitrary, the flow (8.1) is positively transitive. It follows from Theorem 8.4 the flow (8.1) has a unique minimal set. 2 Now, parts (1), (2) of Theorem 7 are just the respective Theorems 8.5, 8.6 above. We note that using Theorem 6(1) and Theorem 7(2) we also obtain an alternative proof for Theorem 4(3) (Theorem 7.3). 8.4. Quasi-periodically forced circle flows Our aim of this subsection is to prove Theorem 8 in which the base space Y is further assumed to be a torus. We will use a classical result of E. Cartan that every closed subgroup of a Lie group is also a Lie group. Hence any closed subgroup of a Lie group is a Lie group which cannot be a Cantor set. The following theorem is just our main result Theorem 8. Theorem 8.7. Consider an APCF (S 1 × Y, T) = (S 1 × Y, {Λt }t∈T ) with Y being a torus (e.g., (Y, T) is quasi-periodic) and suppose that the rotation number is rationally independent of the forcing frequencies. Then (S 1 × Y, T) has a unique minimal set M and M is either the entire phase space S 1 × Y or is everywhere non-locally connected. If, in addition, the APCF admits mean motion, then M is almost automorphic, and moreover, either M = S 1 × Y or M is an everywhere non-locally connected Cantorian. Proof. Since Y is local connected, it follows from Theorem 4(1), Theorem 6(2) in the case with mean motion and from Theorem 7(2) in the case without mean motion that (S 1 × Y, T) has a unique minimal set M. Suppose that M = S 1 × Y . We want to show that M is everywhere non-locally connected. In the case that the APCF (S 1 × Y, T) admits no mean motion, we have by Theorem 7(1) that M is everywhere non-locally connected. We now consider the case that the APCF (S 1 ×Y, T) = (S 1 ×Y, {Λt }t∈T ) admits mean motion. By Theorem 3 and Theorem 6(1), M is both a Cantorian and an almost automorphic minimal set. Suppose for contradiction that M has a locally connected point. Then the set Mlc of locally connected points in M is an invariant residual subset of M. Let Y ∗ be a maximal almost periodic factor of M and p : (M, T) → (Y ∗ , T) be the almost 1–1 extension according to Theorem 3.2. Since the proximal relation P (M) = (e1 , e2 ) ∈ M × M: inf d Λt (e1 ), Λt (e2 ) = 0 , t∈T

where d denotes the standard metric on S 1 × Y , is a closed (in particular, an equivalence), equivariance relation, Y ∗ can be identified to M/P (M) with flow being induced by Λt . Let π : M → Y be the natural projection. Then it is clear that P (M) ⊂ Rπ =: (e1 , e2 ) ∈ M × M: π(e1 ) = π(e2 ) .


889

Thus there exists an extension η : (Y ∗ , T) → (Y, T) such that π = η ◦ p. Since p is almost 1–1 and Mlc = ∅, we have by Theorem 7.1(2) that each fiber p −1 (y ∗ ), y ∗ ∈ Y ∗ , is connected. For each y ∗ ∈ Y ∗ , we note that p −1 (y ∗ ) ⊆ π −1 η(y ∗ ) = s, η(y ∗ ) : s, η(y ∗ ) ∈ M . It follows that for each y ∗ ∈ Y ∗ there is a subinterval Iy ∗ (which can be degenerate) of S 1 such that p −1 (y ∗ ) = Iy ∗ × {η(y ∗ )}. Since M is a Cantorian, there exists a residual subset Y0 of Y such that for each y ∈ Y0 , π −1 (y) is a Cantor set. For any y ∈ Y0 and y ∗ ∈ η−1 (y), since p −1 (y ∗ ) = Iy ∗ × {y} ⊆ π −1 (y), π −1 (y) is a Cantor set and Iy ∗ is a subinterval of S 1 , it follows that Iy ∗ is a singleton. Thus for each y ∈ Y0 the map p : π −1 (y) → η−1 (y) is a homeomorphism, i.e., η−1 (y) is also a Cantor set. j Fix y0 ∈ Y0 and y0∗ ∈ π −1 (y0 ). For any y1∗ , y2∗ ∈ Y ∗ , there exist sequences {ti }∞ i=1 , j = 1, 2, j ∗ ∗ such that limi→∞ y0 · ti = yj , j = 1, 2. We define y1∗ ◦ y2∗ = lim y0∗ · ti1 + ti2 . i→∞

(8.19)

Since (Y ∗ , T) is almost periodic, (8.19) is well-defined and is independent of the choose of sej ∗ ∗ ∗ quences {ti }∞ i=1 , j = 1, 2. With the operation y1 ◦ y2 , Y becomes a compact Abelian topological ∗ group with unity y0 (see Theorem 3.2.1 in [58]). Using y0 , we can define an operation “◦” on Y similarly so that it becomes a compact topological group with unity y0 . j j ∗ ∗ For any y1∗ , y2∗ ∈ Y ∗ , we take sequences {ti }∞ i=1 , j = 1, 2, such that limi→∞ y0 · ti = yj , j = 1, 2. Then limi→∞ y0 · ti = limi→∞ η(y0∗ · ti ) = η(yj∗ ), j = 1, 2, and j

j

η y1∗ ◦ y2∗ = lim η y0∗ · ti1 + ti2 = lim η y0∗ · ti1 + ti2 i→∞

i→∞

= lim y0 · ti1 + ti2 = η y1∗ ◦ η y2∗ . i→∞

Hence η is a group homomorphism from (Y ∗ , ◦) to (Y, ◦) and η−1 (y0 ) = ker(η) is a closed subgroup of (Y ∗ , ◦). Since p is semi-open, the set M0 = m ∈ M: for any open neighborhood U of m, p(U ) is a neighborhood of p(m) is a residual subset of M by Lemma 5.5. Hence Mlc ∩ M0 is also a residual subset of M. Since ∅ = p(Mlc ∩ M0 ) ⊆ Ylc∗ , Y ∗ admits a locally connected point y∗ . For each y ∗ ∈ Y ∗ , we consider the map Hy ∗ ,y∗ : Y ∗ → Y ∗ : y → y ◦ (y ∗ −1 ◦ y∗ ), where y ∗ −1 is the inverse of y ∗ . Since Hy ∗ ,y∗ is a homeomorphism and Hy ∗ ,y∗ (y ∗ ) = y∗ , y ∗ is also a locally connected point of Y ∗ . This shows that Y ∗ is locally connected. For any y ∈ Y and y ∗ ∈ η−1 (y), since Hy0∗ ,y ∗ : η−1 (y0 ) → η−1 (y) is a homeomorphism, η−1 (y) is a Cantor set and hence dim(η−1 (y)) = 0, where dim(·) denotes the covering dimension. It follows from Theorem VI.7 in [25] that dim(Y ∗ ) dim(Y ) + sup dim η−1 (y) = dim(Y ) < ∞. y∈Y

890


Summarizing up, we have shown that (Y ∗ , ◦) is a locally connected, finite dimensional, compact Abelian topological group. It follows from a classical result of Pontrjagin (see Theorem 56 in [48]) that Y ∗ is a Lie group. Now, since η−1 (y0 ) is a closed subgroup of the Lie group (Y ∗ , ◦), η−1 (y0 ) is a Lie group, which is in particular not a Cantor set. This contradicts to the fact that η−1 (y) is a Cantor set for all y ∈ Y . 2 R)-valued cocycles 9. Projective bundle flows of sl(2,R Let T = R or Z and (Y, T) be an almost periodic minimal flow. We consider an almost periodic, sl(2, R)-valued cocycle {Φ(y, t)}y∈Y,t∈T , i.e., the map (y, t) → Φ(y, t) ∈ sl(2, R) is continuous, Φ(y, 0) ≡ I —the identity matrix, and {Φ(y, t)}y∈Y,t∈T satisfies the following cocycle property: Φ(y, t + s) = Φ(y · s, t)Φ(y, s),

y ∈ Y, t, s ∈ T.

We refer the cocycle as continuous cocycle if T = R and as discrete cocycle if T = Z. In the discrete case, we always assume that the cocycle is homotopic to identity. An important example of continuous, almost periodic, sl(2, R)-valued cocycles is the one generated from an almost periodic, 2-dimensional, linear system of ordinary differential equations: x = A(y · t)x,

x ∈ R 2 , y ∈ Y, t ∈ R,

(9.1)

where tr A(y, t) ≡ 0. In this case, the principal matrix solution of the linear system clearly forms a continuous cocycle. The average exponential growth of the norm of {Φ(y, t)} is measured by the (maximal) Lyapunov exponent log Φ(y, t) dμ(y) 0, λ = lim t→+∞ t Y

where μ denotes the Haar measure on Y . We note that the limit exists by subadditivity, is independent of the matrix norm, and is non-negative because Φ(y, t) ∈ sl(2, R). By Kigman sub-additive ergodic theorem [36], lim

t→+∞

log Φ(y, t) = λ, t

μ − a.e. y ∈ Y.

In fact, there exist Y∗ ⊂ Y with μ(Y∗ ) = 1 and invariant, measurable line bundles {u± (y)}y∈Y∗ ⊂ R 2 \ {(0, 0)} such that Φ(y, t)u± (y) = u± (y · t), y ∈ Y∗ , t ∈ T, and log Φ(y, t)u± (y) = ±λ, t→∞ t lim

y ∈ Y∗ .

(9.2)

We say that the cocycle {Φ(y, t)}y∈Y,t∈T is elliptic if supt∈T Φ(y, t) < +∞ for all y ∈ Y ; hyperbolic if it admits an exponential dichotomy (or exponential splitting), parabolic if λ = 0 but the cocycle is not elliptic; and partially hyperbolic if λ > 0 but the cocycle is not hyperbolic. It is well known that if the cocycle is hyperbolic, then the line bundles {u± (y)} can be


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extended continuously to the entire space Y and the limits above exist everywhere on Y . In term of Sacker–Sell spectrum theory [53] for the almost periodic linear system (9.1), hyperbolicity corresponds to the case with two-points spectrum, partially hyperbolicity corresponds to the case with non-degenerate interval spectrum, and ellipticity and parabolicity correspond to cases with zero spectrum. Following works [7,29] for continuous projective bundle flow generated from the linear differential system (9.1) and work [4] for discrete projective bundle flow with one forcing frequency, we will give a complete classification of minimal sets of the projective bundle flow generated from a general almost periodic, sl(2, R)-valued cocycle in both continuous and discrete cases. Such a classification will be particularly useful in characterizing dynamical and topological complexities of a SNA in a such projective bundle flow (see recent work [35] and references therein). 9.1. A general classification of minimal sets Consider an almost periodic, sl(2, R)-valued cocycle {Φ(y, t)}y∈Y, t∈T and its generated linear skew-product flow πt : R 2 × Y → R 2 × Y : πt (v, y) = Φ(y, t)v, y · t . It is clear that the line bundle V (l, y) = (v, y): v is a vector in the line l through the origin is invariant to πt (or to the cocycle) in the sense that πt (V (l, y)) = V (φt (l, y)) for any line l through the origin and any y ∈ Y . Thus the cocycle generates a projective bundle flow (P 1 × Y, T). For simplicity, we parameterize P 1 by angle θ ∈ [0, 1] with 0, 1 being identified, i.e., we parameterize a line l through the origin with its angle Arg(l) = πθ . Then the projective bundle flow can be defined as Λt : P 1 × Y → P 1 × Y : Λt (θ, y) =

1 Arg Φ(y, t)v , y · t =: θ˜ (θ, y, t), y · t , π

θ ∈ R 1 (mod 1), y ∈ Y, t ∈ T,

where v is a vector in R 2 with angle = πθ and θ˜ (θ + 1, y, t) = θ˜ (θ, y, t) + 1. cos πθArg(v) Denote r(θ, y, t) = Φ(y, t) sin πθ . Lemma 9.1. Let (θ1 , y) = (θ2 , y) ∈ P 1 × Y . Then (θ1 , y), (θ2 , y) are proximal iff sup r(θ1 , y, t)r(θ2 , y, t) = +∞. t∈T

Proof. Without loss of generality, we let 0 < θ1 − θ2 < 1. By taking determinant on both hand sides of the identity

r(θ1 , y, t) cos π θ˜ (θ1 , y, t) r(θ1 , y, t) sin π θ˜ (θ1 , y, t)

r(θ2 , y, t) cos π θ˜ (θ2 , y, t) r(θ2 , y, t) sin π θ˜ (θ2 , y, t)

cos πθ1 = Φ(y, t) sin πθ1

cos πθ2 sin πθ2

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and using the fact that det Φ(y, t) ≡ 1, we have r(θ1 , y, t)r(θ2 , y, t) sin π θ˜ (θ1 , y, t) − θ˜ (θ2 , y, t) = sin π(θ1 − θ2 ) = 0,

y ∈ Y, t ∈ T.

(9.3)

Since (θ1 , y), (θ2 , y) are proximal iff either inft∈T (θ˜ (θ1 , y, t) − θ˜ (θ2 , y, t)) = 0 or supt∈T (θ˜ (θ1 , y, t) − θ˜ (θ2 , y, t)) = 1, the lemma immediately follows from (9.3). 2 Lemma 9.2. Consider the cocycle {Φ(y, t)}y∈Y,t∈T and its generated projective bundle flow (P 1 × Y, T). Then the followings are equivalent. (a) (b) (c) (d) (e)

The cocycle is elliptic; There exists y0 ∈ Y such that supt0 Φ(y0 , t) < +∞; There exist (θ1 , y0 ) = (θ2 , y0 ) ∈ P 1 × Y such that supt0 r(θi , y0 , t) < +∞, i = 1, 2; P 1 × Y is a distal extension of Y ; There exist (θi , y0 ) ∈ P 1 × Y , i = 1, 2, 3, which are pairwise distal.

Proof. It is clear that (a) ⇒ (b), (b) ⇒ (c), and (d) ⇒ (e). (b) ⇒ (a): Let K =: supt0 Φ(y0 , t) < +∞. Since det Φ(y, s) ≡ 1, K1 =: sups0 Φ −1 (y0 , s) < +∞. Using the cocycle property Φ(y0 · s, t) = Φ(y0 , t + s)Φ −1 (y0 , s),

s, t ∈ T,

we have that Φ(y0 · s, t) KK1 ,

s 0, s + t 0.

For any t ∈ T, y ∈ Y , we let {sn } be a positive sequence in T such that y0 · sn → y. It follows from the above inequality that Φ(y, t) KK1 , i.e., (a) holds. (c) ⇒ (b): We note that there is a constant c > 0 such that Φ(y0 , t) cos πθ1 cos πθ2 c r(θ1 , y0 , t) + r(θ2 , y0 , t) , sin πθ1 sin πθ2 from which (b) follows. (a) ⇒ (d): Let (θ1 , y), (θ2 , y) ∈ P 1 × Y be two distinct points. It follows from (a) that sup r(θ1 , y, t)r(θ2 , y, t) < +∞. t∈T

Hence by Lemma 9.1, (θ1 , y), (θ2 , y) are distal. (e) ⇒ (c): By Lemma 9.1, supt∈T {{r(θi , y0 , t)r(θj , y0 , t)} < +∞ for all 1 i < j 3. Consider the linear combination cos πθ1 cos πθ2 cos πθ3 = c1 Φ(y0 , t) + c2 Φ(y0 , t) , Φ(y0 , t) sin πθ1 sin πθ2 sin πθ3 where c1 , c2 are constants. Then


893

sup r 2 (θ1 , y, t) sup |c1 |r(θ1 , y, t)r(θ2 , y, t) + |c2 |r(θ1 , y, t)r(θ3 , y, t) t∈T

t∈T

|c1 | sup r(θ1 , y, t)r(θ2 , y, t) + |c2 | sup r(θ1 , y, t)r(θ3 , y, t) < ∞. t∈T

t∈T

It follows that {r(θ1 , y0 , t)} is bounded. Similarity, {r(θi , y0 , t)}, i = 2, 3 are bounded, i.e., (c) holds. 2 Proposition 9.1. If the cocycle {Φ(y, t)}y∈Y, t∈T is not elliptic, then its generated projective bundle flow has at most two minimal sets. Proof. Suppose that the projective bundle flow has three minimal sets Mi , i = 1, 2, 3. Let y0 ∈ Y and take (θi , y0 ) ∈ Mi , i = 1, 2, 3. Then {(θi , y0 ): i = 1, 2, 3} are pairwise distal. It follows from Lemma 9.2 that the cocycle is elliptic, a contradiction. 2 Lemma 9.3. Consider the cocycle {Φ(y, t)}y∈Y, t∈T and its generated projective bundle flow (P 1 × Y, T). Then the cocycle is hyperbolic iff supt∈T r(θ, y, t) = +∞ for all (θ, y) ∈ P 1 × Y . Proof. It is a special case of the main result in [52].

2

Theorem 9.1. Consider the cocycle {Φ(y, t)}y∈Y, t∈T and its generated projective bundle flow (P 1 × Y, T). Then the following holds: (1) If the cocycle is elliptic, then either P 1 × Y is minimal and distal or there is an integer N 1 such that P 1 × Y laminates into infinitely many minimal N –1 extensions of Y (hence they are almost periodic). (2) If the cocycle is hyperbolic, then (P 1 × Y, T) has precisely two minimal sets and each of them is a 1–1 extension of Y (hence they are almost periodic). Proof. (1) Since, by Lemma 9.2, P 1 × Y is a distal extension of Y , either (i) it is minimal and distal; or (ii) it laminates into infinitely many minimal sets [12]. In the case (ii), we have by Theorem 4 and distality that there is a positive integer N such that each minimal set is an N –1 extension of Y . (2) Let {u± (y)}y∈Y ⊂ R 2 be the continuous, invariant line bundles associated with hyperbolicity and let θ ± (y) = π1 Arg u± (y), y ∈ Y . Then M± =

± θ (y), y : y ∈ Y

are two minimal sets of (P 1 × Y, T) which are clearly 1–1 extensions of Y . By Proposition 9.1, the projective bundle flow (P 1 × Y, T) cannot have more than two minimal sets in this case. 2 We now exam minimal dynamics of the project bundle flow if the cocycle is either parabolic or partially hyperbolic. Lemma 9.4. The cocycle {Φ(y, t)}y∈Y, t∈T is either parabolic or partially hyperbolic iff there are (θ10 , y0 ), (θ20 , y0 ) ∈ P 1 × Y such that supt0 r(θ10 , y0 , t) = +∞ and supt∈T r(θ20 , y0 , t) < +∞.

894


Proof. The lemma is a direct consequence of Lemmas 9.2, 9.3.

2

We call an ordered pair {(θ+ , y), (θ− , y)} in P 1 × Y a Morse pair if lim sup t→∞

r(θ+ , y, t) = +∞. r(θ− , y, t)

This notion is a relaxed version of relative dichotomy or More decomposition in linear skewproduct flows. Let π : P 1 × Y → Y be the natural projection. Lemma 9.5. If the cocycle {Φ(y, t)}y∈Y, t∈T is not elliptic and some fiber of its generated projective bundle flow (P 1 × Y, T) over Y admits a distal pair, then each fiber of (P 1 × Y, T) over Y admits a Morse pair. Proof. Suppose for contradiction that there is a fiber π −1 (y0 ) which admits no Morse pair. Since some fiber of P 1 × Y over Y admits a distal pair, then all fibers of P 1 × Y over Y admit distal pairs. In particular, π −1 (y0 ) admits a distal pair, say {(θ01 , y0 ), (θ02 , y0 )}. It follows from Lemma 9.1 that sup r θ01 , y0 , t r θ02 , y0 , t < +∞.

(9.4)

t∈T

Since (θ01 , y0 ), (θ02 , y0 ) do not form a Morse pair in both orders, there are positive constants K1 , K2 such that K1 r θ01 , y0 , t r θ02 , y0 , t K2 r θ01 , y0 , t ,

t ∈ T.

It follows from (9.4) that both r(θ01 , y0 , t) and r(θ02 , y0 , t) are bounded. Hence by Lemma 9.2, the cocycle is elliptic, a contradiction. 2 Lemma 9.6. Let {(θ+0 , y 0 ), (θ−0 , y 0 )} be a Morse pair. Then the following holds: (1) For any θ = θ−0 , {(θ, y 0 ), (θ−0 , y 0 )} is a Morse pair. (2) If θ1 , θ2 = θ−0 , then (θ1 , y 0 ), (θ2 , y 0 ) are proximal. (3) Suppose supt∈T {r(θ−0 , y 0 , t)} < ∞. Then for any θ1 , θ2 = θ−0 , (θ1 , y 0 ), (θ−0 , y 0 ) are proximal iff (θ2 , y 0 ), (θ−0 , y 0 ) are proximal. In particular, for any θ = θ0− , (θ, y 0 ), (θ−0 , y 0 ) are proximal iff (θ+0 , y 0 ), (θ−0 , y 0 ) are proximal. Proof. (1) Let θ = θ−0 ∈ P 1 and consider the linear combination cos πθ+0 cos πθ−0 cos πθ 0 + c , Φ(y 0 , t) = c+ Φ(y 0 , t) Φ(y , t) − sin πθ sin πθ+0 sin πθ−0 where c+ , c− are constants. Since θ = θ−0 and c+ = 0, we have by (9.5) that

(9.5)


lim sup t→∞

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r(θ+0 , y 0 , t) |c | | = +∞, lim sup − |c + − r(θ−0 , y 0 , t) r(θ−0 , y 0 , t) t→∞ r(θ, y 0 , t)

i.e., {(θ, y 0 ), (θ−0 , y 0 )} is a Morse pair. (2) Let θ1 , θ2 = θ−0 and consider the linear combination cos πθ−0 cos πθ2 cos πθ1 = c1 Φ(y 0 , t) + c0 Φ(y 0 , t) , Φ(y 0 , t) sin πθ2 sin πθ1 sin πθ−0

(9.6)

where c1 , c0 are constants. Since θ2 = θ−0 and c1 = 0, we have by (9.3) that r(θ1 , y 0 , t)r(θ−0 , y 0 , t) admits a positive lower bound, say c(θ1 ). Then by (9.6), 0 0 r(θ1 , y 0 , t) 0 0 r θ1 , y , t r θ2 , y , t r θ1 , y , t r θ− , y , t max 0, |c1 | 0 − |c0 | r(θ− , y 0 , t) r(θ1 , y 0 , t) − |c0 | . c(θ1 ) max 0, |c1 | 0 r(θ− , y 0 , t)

0

It follows that supt∈T {r(θ1 , y 0 , t)r(θ2 , y 0 , t)} = +∞ since {(θ1 , y 0 ), (θ−0 , y 0 )} is a Morse pair. Hence by Lemma 9.1, (θ1 , y 0 ), (θ2 , y 0 ) are proximal. (3) By symmetry, it is sufficient to show that if (θ1 , y 0 ), (θ−0 , y 0 ) are proximal, then so are (θ2 , y 0 ), (θ−0 , y 0 ). Assume that (θ1 , y 0 ), (θ−0 , y 0 ) are proximal, i.e., supt∈T {r(θ1 , y 0 , t)r(θ−0 , y 0 , t)} = +∞. Then by (9.6), 2 r θ2 , y 0 , t r θ−0 , y 0 , t |c1 |r θ1 , y 0 , t r θ−0 , y 0 , t − |c0 |r θ−0 , y 0 , t . It follows that supt∈T {r(θ1 , y 0 , t)r(θ−0 , y 0 , t)} = +∞ since supt∈T {r(θ1 , y 0 , t)r(θ−0 , y 0 , t)} = +∞ and supt∈T {r(θ−0 , y 0 , t)} < ∞. By Lemma 9.1, (θ2 , y 0 ), (θ−0 , y 0 ) are proximal. 2 Theorem 9.2. Let the cocycle {Φ(y, t)}y∈Y, t∈T be either parabolic or partially hyperbolic. Then the following holds for its generated projective bundle flow (P 1 × Y, T): (1) (P 1 × Y, T) admits at most two minimal sets. (2) If (P 1 × Y, T) admits two minimal sets, then each minimal set is an almost 1–1 extension of Y (hence they are almost automorphic). (3) If (P 1 × Y, T) admits only one minimal set M, then precisely one of the following holds: (i) M is almost automorphic and is either an almost 1–1 or almost 2–1 extension of Y ; (ii) M is an everywhere non-locally connected Cantorian and is residually Li–Yorke chaotic; (iii) M is the entire space P 1 × Y and is residually Li–Yorke chaotic. Moreover, in cases (ii) and (iii), M is a proximal extension of Y which is not almost 1–1 (hence M is not almost automorphic). Proof. (1) is clear by Proposition 9.1.

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(2) It follows from Theorem 4 that each minimal set is an almost N –1 extension of Y for some positive integer N . We note that an N –1 extension of Y must be a distal extension. But by Lemma 9.2 there are no three points on a same fiber which can be pair-wise distal. Hence N = 1. (3) By Lemma 9.4, we let (θ+ (y0 ), y0 ), (θ− (y0 ), y0 ) ∈ P 1 × Y be such that supt0 r(θ+ (y0 ), y0 , t) = +∞ and supt∈T r(θ− (y0 ), y0 , t) < +∞. It is clear that {(θ+ (y0 ), y0 ), (θ− (y0 ), y0 )} is a Morse pair. Case 1. (θ+ (y0 ), y0 ), (θ− (y0 ), y0 ) are proximal. In this case, since supt∈T {r(θ− (y0 ), y0 , t)} < ∞, we have by Lemma 9.6(3) that any two points on the fiber p −1 (y0 ) are proximal. This particularly implies that M is a proximal extension of Y . Hence if M is point-distal, then it must be an almost 1–1 extension of Y . Now suppose that M is not point-distal. Then by Theorem 2, it must be residually Li–Yorke chaotic, and by Theorem 6, the corresponding projective bundle flow admits no mean motion (because M is not almost automorphic). Moreover, since M is not an almost N –1 extension of Y for any positive integer N , it follows from Theorem 3 that M is either a Cantorian or the entire phase space, and, in the case that M is a Cantorian, we have by Theorem 7 that it is everywhere non-locally connected. Case 2. (θ+ (y0 ), y0 ), (θ− (y0 ), y0 ) are distal. We note that supt∈T {r(θ− (y0 ), y0 , t)} < ∞. It follows from Lemma 9.6 that (θ, y0 ), (θ− (y0 ), y0 ) are distal if θ = θ− (y0 ) and (θ1 , y0 ), (θ2 , y0 ) are proximal if θ1 , θ2 = θ− (y0 ). According to a general result due to Auslander [3], there exists (θ∗ , y0 ) ∈ M such that (θ− (y0 ), y0 ), (θ∗ , y∗ ) are proximal. Since by Lemma 9.6, (θ, y0 ), (θ− (y0 ), y0 ) are distal for any θ = θ− (y0 ), we have that θ∗ = θ− (y0 ), i.e., (θ− (y0 ), y0 ) ∈ M. Applying the result of Auslander to (θ+ (y0 ), y0 ), we also find a point (θ (y0 ), y0 ) ∈ M such that (θ (y0 ), y0 ), (θ+ (y0 ), y0 ) are proximal. Clearly, θ (y0 ) = θ− (y0 ) and (θ (y0 ), y0 ), (θ− (y0 ), y0 ) are distal. Hence π −1 (y0 ) ∩ M admits a distal pair. It follows that all fibers π −1 (y) ∩ M, y ∈ Y , admit distal pairs. It now follows from Lemma 9.5 that each fiber π −1 (y) admits a Morse pair {(θ+ (y), y), (θ− (y), y)}. Hence by Lemma 9.6(2), (θ1 , y), (θ2 , y) are proximal for any θ1 , θ2 = θ− (y). Since π −1 (y) ∩ M admits a distal pairs, (θ− (y), y) ∈ M and there exists (θ (y), y) ∈ M such that (θ (y), y), (θ−0 (y), y) are distal. Let δ = inft∈T |θ˜ (θ (y0 ), y0 , t) − θ˜ (θ− (y0 ), y0 , t)|. It is clear that δ > 0. Claim 1. For any (θ, y) ∈ M with θ = θ− (y), (θ, y), (θ− (y), y) are distal and |θ − θ− (y)| δ. Since (θ (y0 ), y0 ) ∈ M, there exists a sequence {tn } ⊂ T such that limn→∞ Λtn (θ (y0 ), y0 ) = (θ, y) and limn→∞ Λtn (θ− (y0 ), y0 ) = (θ∗ (y), y) for some (θ∗ (y), y) ∈ M. Clearly, (θ, y), (θ∗ (y), y) are distal and |θ − θ∗ (y)| δ. Since (θ1 , y), (θ2 , y) are proximal for any θ1 , θ2 = θ− (y), we have θ∗ (y) = θ− (y). This proves the claim. Claim 2. There is a residual set Y0 ⊂ Y such that |π −1 (y) ∩ M| = 2 for all y ∈ Y0 . We use the argument in the proof of Theorem 7.4 in [29]. By Claim 1, there exist 0 θ1 θ2 < 1 + θ1 such that (θ1 , y0 ), (θ2 , y0 ) ∈ M and ([θ1 , θ2 ] × {y0 }) ∩ M = π −1 (y0 ) ∩ M \ {(θ− (y0 ), y0 )}. Let Y0 be the set of all continuity points of the upper semicontinuous map y → π −1 (y) ∩ M. Then Y0 is a residual subset of Y . For any y ∈ Y0 , since (θ1 , y0 ), (θ2 , y0 ) are proximal, there exists a sequence {tn } ⊂ T such that limn→∞ Λtn (θ1 , y0 ) = limn→∞ Λtn (θ2 , y0 ) = (θ (y), y) and limn→∞ Λtn (θ− (y0 ), y0 ) = (θ ∗ (y), y) for some (θ ∗ (y), y) ∈ M. Since (θ1 , y0 ), (θ− (y0 ), y0 ) are distal, so are (θ (y), y), (θ ∗ (y), y). Hence θ ∗ (y) = θ− (y). It follows from Lemma 5.3(2) that either limn→∞ Λtn ([θ1 , θ2 ] × {y0 }) = {(θ (y), y)} or limn→∞ Λtn ([θ2 , θ1 ] × {y0 }) = {(θ (y), y)} by taking subsequences if necessary.


897

Since θ− (y0 ) ∈ [θ2 , θ1 ] and limn→∞ Λtn (θ− (y0 ), y0 ) = (θ− (y), y) = (θ (y), y), we have limn→∞ Λtn ([θ1 , θ2 ] × {y0 }) = {(θ (y), y)}. Thus π −1 (y) ∩ M = lim π −1 (y0 · tn ) ∩ M ⊆ lim Λtn [θ1 , θ2 ] × {y0 } ∪ Λtn θ− (y0 ), y0 n→∞

= (θ (y), y), (θ− (y), y) .

n→∞

It follows that |π −1 (y) ∩ M| = 2 for all y ∈ Y0 . Now, we have by Claim 2 that M is an almost 2–1 extension of Y . To show that M is almost automorphic, we note that it easily follows from Claim 1 and Lemma 9.6(2) that the proximal relation P (M) is closed. Hence M/P (M) is a compact Hausdorff space and there is a natural flow (M/P (M), T) induced from the flow (P 1 × Y, T). By Lemma 9.1, M/P (M) is a 2–1 extension of Y , hence it is almost periodic minimal. Let p : M → M/P (M) be the natural projection. Then it follows from Claim 2 that p : (M, T) → (M/P (M), T) is an almost 1–1 extension. Hence M is almost automorphic. 2 In the partially hyperbolic case, more precise information can be obtained as follows. Theorem 9.3. Let the cocycle {Φ(y, t)}y∈Y, t∈T be partially hyperbolic. Then its generated projective bundle flow (P 1 × Y, T) admits a unique minimal set M. Moreover, the following holds: (a) M is characterized precisely by one of the case (i)–(iii) in Theorem 9.2 but it is nether almost periodic nor an almost 2–1 extension of Y ; (b) M is non-uniquely ergodic and admits precisely two ergodic sheets {(u± (y), y)}y∈Y ∗ as in (9.2); (c) There is a residual set Y0 ⊂ Y such that for each (θ, y) ∈ M ∩ p −1 (y), r(θ, y, t) oscillates between 0 and +∞ as t → ±∞. Proof. The fact that M cannot be an almost 2–1 extension of Y was proved in [29] for the linear system (9.1). (b) and (c) were given in Theorem 4.10 of [35] also for the linear system (9.1). The proof for the general situation follows from similar arguments. Since M is not uniquely ergodic, it cannot be almost periodic. 2 Examples of continuous, almost periodic, sl(2, R)-valued cocycles whose projective bundle flows have the property (i) stated in Theorem 9.2 are well-known (see [7,29]). Also there are many continuous, almost periodic, sl(2, R)-valued cocycles whose projective bundle flows have the property (iii) stated in Theorem 9.2 (see [31,42]). An interesting question is whether case (ii) in Theorem 9.2 can really occur in a projective bundle flow. The following result shows that the answer to this question is negative when the forcing space in a projective bundle flow is locally connected. Theorem 9.4. Let Y be locally connected and the cocycle {Φ(y, t)}y∈Y,t∈T be either parabolic or partially hyperbolic. Then its generated projective bundle flow (P 1 × Y, T) admits no Cantorian minimal set.

898


Proof. If (P 1 × Y, T) admits mean motion, then we have by Theorem 6 that all its minimal sets are almost automorphic hence they are not residually Li–Yorke chaotic. This show that case (ii) in Theorem 9.2 does not occur. In particular, (P 1 × Y, T) admits no Cantorian minimal set. If (P 1 × Y, T) admits no mean motion, then by Theorem 7(2) (Theorem 8.6) it is topologically transitive. It then follows from almost exact proof of Proposition 4.6 in [4] that if the entire phase space P 1 × Y is not minimal, then a minimal set M of (P 1 × Y, T) is either an almost 1–1 or an almost 2–1 extension of Y . In particular, M is not a Cantorian. 2 Remark. (1) Using the same argument as the above, one sees that if a projective bundle flow admits mean motion, then case (ii) in Theorem 9.2 cannot occur regardless whether the base Y is locally connected or not (this has already been shown in [7] for the continuous case). It then remains an open question whether case (ii) in Theorem 9.2 can occur in a projective bundle flow without mean motion when the base is not locally connected. We think that the answer to this question should be affirmative. (2) Suppose that the projective bundle flow of a partially hyperbolic, quasi-periodically forced cocycle {Φ(y, t)}y∈Y, t∈T admits a globally attracting SNA, say A. Then A cannot be the entire phase space, and by Theorem 9.3, A is made up by a unique minimal set M along with its “homoclinic orbits” (in the sense of proximality). Now, by Theorems 9.2, 9.4, M is non-almostperiodic, almost automorphic extension of Y . If we further assume that the rotation number of the projective bundle flow is rationally independent of the forcing frequencies, then by Theorem 8, M is everywhere non-locally connected. All these simply suggests an important role played by almost automorphic dynamics to such a SNA: topologically the minimal set in the SNA is everywhere non-local connected and an almost 1-cover of the forcing space, and dynamically the minimal set in the SNA is almost automorphic. 9.2. Cases with mean motion properties With the classification given in the above, it is important to know when or how often almost automorphic dynamics can occur in the projective bundle flow of an almost periodic, sl(2, R)valued cocycle in the non-parabolic case. Some affirmative answers to this problem was given in [35] with respect to extreme points of spectral gaps of the following almost periodic Schrödinger and Schrödinger-like operators: d2 Lq = − 2 + q(y · t) : L2 (R) → L2 (R); dt d − Q(y · t) : L2 R, R 2 → L2 R, R 2 ; LQ = J dt Lv = −A + v(y · n) : L2 (Z) → L2 (Z), where (Y, T) is almost periodic minimal for T = R or Z, q, v are continuous functions on Y , Q is a 2 × 2 matrix-valued continuous function on Y , J is the standard 2 × 2 symplectic matrix, and A is the operator defined by Az(n) = z(n + 1) + z(n − 1). Of course, when automorphic dynamics exist in the projective bundle flow of an almost periodic, sl(2, R)-valued cocycle in the non-hyperbolic case, it is also interesting to know whether the corresponding projective bundle flow admits mean motion.


899

Consider the spectral problem Lq x(t) = λx(t),

(9.7)

LQ X(t) = λX(t),

(9.8)

Lv z(n) = λz(n).

(9.9)

Each linear equation (9.7)–(9.9) generates an almost periodic, sl(2, T)-valued cocycle for T = R or Z in the natural way, which gives rise to a projective bundle flow Πλ = (P 1 × Y, T), where P 1 is parametrized by φ = − π2 Arg xx , φ = π2 Arg X, φ = − π2 Arg z(n+1) for (9.7)–(9.9) rez(n) spectively. For each L = Lq , LQ , Lv , according to the Gap labeling theorem [34], the rotation number ρ(λ) of Πλ is monotonically increasing and increases precisely on the spectrum ΣL of L which is contained in a half line [λ∗ , +∞). For each λ in the resolvent of L, it is well known that the corresponding cocycle is hyperbolic, hence Πλ admits exactly two minimal sets which are all 1–1 extensions of Y (hence they are almost periodic). Proposition 9.2. Consider L = Lq , LQ , Lv and let λ0 be a finite extreme point of a spectral gap (i.e., a maximal open interval in the resolvent of L). Then Πλ0 admits mean motion. Consequently, each minimal set of Πλ0 is almost automorphic and in fact an almost 1–1 extension of Y. Proof. We only give the proof for the case of Lq . The other two cases can be treated similarly. Observe from (9.7) that φ = − π2 Arg xx satisfies the equation φ =

λ − q(y · t) − 1 λ − q(y · t) + 1 + cos πφ. π π

(9.10)

We denote φ˜ λ (φ, y, t) as the solution of (9.10) corresponding to λ, y and with initial value φ. Let λ0 be the left end point of a spectral gap I in the resolvant of Lq . Then the rotation number of (9.10) is a constant over I¯, which we denote by ρ. By elementary theory of ordinary differential equations, we see from (9.10) that ∂ φ˜ λ (φ, y, t) 0 ∂λ

(9.11)

for all λ, φ ∈ R 1 , y ∈ Y , and t 0. For any (φ0 , y0 ) ∈ R 1 × Y and a given λ∗ ∈ I , we denote φ0 (t) = φ˜ λ0 (φ0 , y0 , t), φ∗ (t) = φ˜ λ∗ (φ0 , y0 , t). Using (9.11) and the comparison principle of scalar ordinary differential equations, it is easy to see that φ0 (t) φ∗ (t) for all t 0. It follows that φ0 (t) − φ0 − ρt φ∗ (t) − φ0 − ρt for all t 0. Since Πλ∗ admits almost periodic motion, it admits mean motion. Hence sup φ0 (t) − φ0 − ρt sup φ∗ (t) − φ0 − ρt < ∞. t0

t0

900


Since (φ0 , y0 ) is arbitrary, we have by Theorem 8.2(d) that Πλ0 admits mean motion. It follows from Theorem 6 that each minimal set of Πλ0 is almost automorphic. Since almost periodic minimal sets of Πλ∗ are all 1–1 extensions of Y , we have by Theorem 6(2) that ρ is contained in the frequency module of the forcing. Applying Theorem 6(2) again, we conclude that any minimal set of Πλ0 cannot be an almost 2–1 extension of Y . The case when λ0 is the right end point (including λ∗ ) of a spectral gap in the resolvant of Lq is similar. 2 Dynamics of Πλ when λ entering the spectrum through λ0 are expected to be more complicated due to the possible loss of mean motion property. This can be viewed as another intermittency phenomenon characterized by almost automorphic intermediate dynamics. Acknowledgments We would like to thank Professor Russell A. Johnson for valuable comments and suggestions, and also for bringing the work [4] to our attention. We are also grateful to an anonymous referee for corrections and helpful suggestions which lead to an improvement of the paper. References [1] E. Akin, E. Glasner, W. Huang, S. Shao, X. Ye, Sufficient conditions under which a transitive system is chaotic, preprint, 2008. [2] V.I. Arnold, Small divisors, I. On mappings of a circle onto itself, Izv. Akad. Nauk SSSR Ser. Math. 25 (1961) 21–86. [3] J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Stud., vol. 153, North-Holland, Amsterdam, 1988. [4] F. Béguin, S. Croviser, T.H. Jäger, F. Le Roux, Denjoy construction for fibered homeomorphism of the torus, Trans. Amer. Math. Soc., in press. [5] A. Berger, S. Siegmund, Y. Yi, On almost automorphic dynamics in symbolic lattices, Ergodic Theory Dynam. Systems 24 (3) (2004) 677–696. [6] K. Bjerklov, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems 25 (4) (2005) 1015–1045. [7] K. Bjerklov, R.A. Johnson, Minimal subsets of projective flows, Discrete Contin. Dyn. Sys. Ser. B 9 (3–4) (2008) 495–516. [8] F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li–Yorke pairs, J. Reine Angew. Math. 547 (2002) 51–68. [9] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971) 401– 414. [10] M. Denker, C. Grillenberger, C. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math., vol. 527, Springer-Verlag, New York, 1976. [11] R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958) 401–405. [12] R. Ellis, Lectures on Topological Dynamics, W.A. Benjamin, Inc., New York, 1969. [13] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Cambridge Stud. Adv. Math., vol. 88, Cambridge University Press, Cambridge, 2007. [14] U. Feudel, S. Kuznetsov, A. Pikovsky, Strange Nonchaotic Attactors, World Scientific, 2006. [15] H. Furstenberg, Strictly ergodicity and transformations of the torus, Amer. J. Math. 83 (1961) 573–601. [16] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981. [17] H. Furstenberg, B. Weiss, On almost 1–1 extensions, Israel J. Math. 65 (1989) 311–322. [18] E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 2003. [19] E. Glanser, Topological weak mixing and quasi-Bohr systems, Israel J. Math. 148 (2005) 277–304. [20] E. Glasner, J.P. Thouvenot, B. Weiss, Entropy theory without a past, Ergodic Theory Dynam. Systems 20 (2000) 1355–1370. [21] P. Glendinning, T.H. Jäger, G. Keller, How chaotic are strange non-chaotic attractors? Nonlinearity 19 (2006) 2005– 2022.


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On a parabolic logarithmic Sobolev inequality H. Ibrahim a,b,c , R. Monneau a,∗ a Université Paris-Est, CERMICS, Ecole des Ponts, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne,

77455 Marne-la-Vallée Cedex 2, France b LaMA-Liban, Lebanese University, P.O. Box 826 Tripoli, Lebanon c CEREMADE, Université Paris-Dauphine, Place De Lattre de Tassigny, 75775 Paris Cedex 16, France

Received 19 December 2008; accepted 12 January 2009 Available online 29 January 2009 Communicated by H. Brezis

Abstract In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191–200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono–Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems. © 2009 Elsevier Inc. All rights reserved. Keywords: Logarithmic Sobolev inequalities; Parabolic BMO spaces; Anisotropic Lizorkin–Triebel spaces; Harmonic analysis

1. Introduction and main results In [12], Kozono and Taniuchi showed an L∞ estimate of a given function by means of its BMO norm (space of functions of bounded mean oscillation) and the logarithm of its norm in some Sobolev space. In fact, they proved that for f ∈ Wps (Rn ), 1 < p < ∞, the following estimate * Corresponding author.

E-mail addresses: [email protected] (H. Ibrahim), [email protected] (R. Monneau). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.008

904

H. Ibrahim, R. Monneau / Journal of Functional Analysis 257 (2009) 903–930

holds (with log+ x = max(log x, 0)): f L∞ (Rn ) C 1 + f BMO(Rn ) 1 + log+ f Wps (Rn ) ,

sp > n,

(1.1)

for some constant C = C(n, p, s) > 0. The main advantage of the above estimate is that it was successfully applied (see [12, Theorem 2]) to extend the blow-up criterion of solutions to the Euler equations which was originally given by Beale, Kato and Majda in [1]. Inequality (1.1), as well as some variants of it, are shown (see [11,12,14]) using harmonic analysis on isotropic functional spaces of the Lizorkin–Triebel and Besov type. However, as is well known, it is important, say for parabolic partial differential equations to consider spaces that are anisotropic. Motivated by the study of the long-time existence of a certain class of singular parabolic coupled systems (see [8,9]), we show in this paper an analogue of the Kozono–Taniuchi inequality (1.1) but of the parabolic (anisotropic) type. Due to the parabolic anisotropy, we consider functional spaces on Rn+1 = Rn × R with the generic variable z = (x, t), where each coordinate xi , i = 1, . . . , n is given the weight 1, while the time coordinate t is given the weight 2. We now state the main results of this paper. The first result concerns a Kozono–Taniuchi parabolic type inequality on the entire space Rn+1 . Introducing parabolic bounded mean oscillation BMOp spaces, and parabolic Sobolev spaces W22m,m (for the definition of these spaces, see Definitions 2.1 and 2.2), we present our first theorem. Theorem 1.1 (Parabolic logarithmic Sobolev inequality on Rn+1 ). Let u ∈ W22m,m (Rn+1 ), m > n+2 4 . Then there exists a constant C = C(m, n) > 0 such that: uL∞ (Rn+1 ) C 1 + uBMOp (Rn+1 ) 1 + log+ uW 2m,m (Rn+1 ) .

(1.2)

2

The proof of Theorem 1.1 will be given in Section 2, and is based on an approach developed by Ogawa [14]. Let us mention that our proof in this paper is self-contained. The second result of this paper concerns a Kozono–Taniuchi parabolic type inequality on the bounded domain ΩT = (0, 1)n × (0, T ) ⊂ Rn+1 ,

T > 0.

More precisely, our next theorem reads: Theorem 1.2 (Parabolic logarithmic Sobolev inequality on a bounded domain). Let u ∈ W22m,m (ΩT ) with m > n+2 4 . Then there exists a constant C = C(m, n, T ) > 0 such that: uL∞ (ΩT ) C 1 + uBMOp (ΩT ) 1 + log+ uW 2m,m (Ω ) , 2

T

(1.3)

where · BMOp (ΩT ) = · BMOp (ΩT ) + · L1 (ΩT ) . The proof of Theorem 1.2 will be given in Section 3. 1.1. Brief review of the literature The brief review presented here only concerns logarithmic Sobolev inequalities of the elliptic type. Up to our knowledge, logarithmic Sobolev inequalities of the parabolic type have not been treated elsewhere in the literature.


905

The original type of the logarithmic Sobolev inequalities was found in Brezis and Gallouet [3] and Brezis and Wainger [4] where the authors investigated the relation between L∞ , Wrk and Wps and proved that there holds the embedding: r−1 f L∞ (Rn ) C 1 + log r 1 + f Wps (Rn ) ,

(1.4)

sp > n,

provided f Wrk (Rn ) 1 for kr = n. The estimate (1.4) was applied to prove global existence of solutions to the nonlinear Schrödinger equation (see [3,7]). Similar embedding for f ∈ (Wps (Rn ))n with div f = 0 was investigated by Beale, Kato and Majda in [1]. The authors showed that: ∇f L∞ C 1 + rot f L∞ 1 + log+ f W s+1 + rot f L2 , p

sp > n,

(1.5)

where they made use of this estimate in order to give a blow-up criterion of solutions to the Euler equations (see [1]). In [12], Kozono and Taniuchi showed their inequality (1.1) in order to extend the blow-up criterion of solutions to the Euler equations given in [1] (see [12, Theorem 2]). A generalized version of (1.1) in Besov spaces was given in Kozono et al. [11]. Finally, a sharp version of the logarithmic Sobolev inequality of the Beale–Kato–Majda and the Kozono– Taniuchi type in the Lizorkin–Triebel spaces was established by Ogawa in [14]. 1.2. Organization of the paper This paper is organized as follows. In Section 2, we recall basic tools used in our analysis, and give the proof of Theorem 1.1. In Section 3, we present the proof of Theorem 1.2, and as an application, we explain how to use the parabolic Kozono–Taniuchi inequality in order to prove the long-time existence of certain parabolic equations. 2. A parabolic Kozono–Taniuchi inequality on R n+1 This section is devoted to the proof of Theorem 1.1. 2.1. Preliminaries and basic tools 2.1.1. Parabolic BMOp and Sobolev spaces We start by recalling some definitions and introducing some notations. A generic point in Rn+1 will be denoted by z = (x, t) ∈ Rn × R, x = (x1 , . . . , xn ). Let S(Rn+1 ) be the usual Schwartz space, and S (Rn+1 ) the corresponding dual space. Let u ∈ S (Rn+1 ). For ξ = (ξ1 , . . . , ξn ) ∈ Rn and τ ∈ R we denote by F u(ξ, τ ) ≡ u(ξ, ˆ τ ), and F −1 u(ξ, τ ) ≡ u(ξ, ˇ τ) r the Fourier, and the inverse Fourier transform of u respectively. We also denote Dtr = ∂t∂ r , r ∈ N, and Dxs , s ∈ N, any derivative with respect to x of order s. The parabolic distance from z = (x, t) to the origin is defined by: [z] = max |x1 |, . . . , |xn |, |t|1/2 .

(2.1)

Let O ⊆ Rn+1 be an open set. The parabolic bounded mean oscillation space BMOp and the parabolic Sobolev space W22m,m are now recalled.

906


Definition 2.1 (Parabolic bounded mean oscillation spaces). A function u ∈ L1loc (O) is said to be of parabolic bounded mean oscillation, u ∈ BMOp (O) if we have: 1 uBMOp (O) = sup |u − uQ | < +∞. (2.2) Q⊂O |Q| Q

Here Q denotes an arbitrary parabolic cube Q = Qr = Qr (z0 ) = z ∈ Rn+1 ; [z − z0 ] < r ,

(2.3)

and uQ =

1 |Q|

(2.4)

u. Q

The functions in BMOp are defined up to an additive constant. We also define the space BMOp as: BMOp (O) = BMOp (O) ∩ L1 (O)

with · BMOp = · BMOp + · L1 .

Definition 2.2 (Parabolic Sobolev spaces). Let m be a non-negative integer. We define the parabolic Sobolev space W22m,m (O) as follows: W22m,m (O) = u ∈ L2 (O); Dtr Dxs u ∈ L2 (O), ∀r, s ∈ N such that 2r + s 2m . The norm of u ∈ W22m,m (O) is defined by: uW 2m,m (O) = 2

2m j =0

2r+s=j

Dtr Dxs uL2 (O) .

The next lemma concerns a Sobolev embedding of W22m,m . Lemma 2.3 (Sobolev embedding). (See [13, Lemma 3.3].) Let m be a non-negative integer satisfying m > n+2 4 . Then there exists a positive constant C depending on m and n such that for any 2m,m u ∈ W2 (O), the function u is continuous and bounded on O, and satisfies uL∞ (O) CuW 2m,m (O) .

(2.5)

2

2.1.2. Parabolic Lizorkin–Triebel and Besov spaces Here we give the definition of Lizorkin–Triebel spaces. These spaces are constructed out of the parabolic Littlewood–Paley decomposition that we recall here. Let ψ0 (z) ∈ C0∞ (Rn+1 ) be a function such that ψ0 (z) = 1 if [z] 1 and ψ0 (z) = 0 if [z] 2.

(2.6)

For such a function ψ0 , we may define a smooth, anisotropic dyadic partition of unity (ψj )j ∈N by letting ψj (z) = ψ0 2−j a z − ψ0 2−(j −1)a z if j 1.


907

Here a = (1, . . . , 1, 2) ∈ Rn+1 , and for η ∈ R, b = (b1 , . . . , bn , bn+1 ) ∈ Rn+1 , the dilation ηb z is defined by ηb z = (ηb1 z1 , . . . , ηbn zn , ηbn+1 zn+1 ). It is clear that ∞

ψj (z) = 1

for z ∈ Rn+1 ,

j =0

and supp ψj ⊂ z; 2j −1 [z] 2j +1 ,

j 1.

Define φj , j 0 as the inverse Fourier transform of ψj , i.e. φˆ j = ψj . It is worth noticing that φj (z) = 2(n+2)(j −1) φ1 2(j −1)a z

for j 1,

(2.7)

and that for any u ∈ S (Rn+1 ), u = (2π)−

(n+1) 2

∞

with convergence in S Rn+1 .

φj ∗ u

j =0

We now give the definition of the anisotropic Besov and Lizorkin–Triebel spaces. s (Rn+1 ) = B s , Definition 2.4 (Anisotropic Besov spaces). The anisotropic Besov space Bp,q p,q s ∈ R, 1 p ∞ and 1 q ∞ is the space of functions u ∈ S (Rn+1 ) with finite quasi-norms

s = uBp,q

∞

1/q 2

sqj

j =0

q φj ∗ uLp (Rn+1 )

(2.8)

and the natural modification for q = ∞, i.e. s uBp,∞ = sup 2sj φj ∗ uLp (Rn+1 ) .

(2.9)

j 0

Definition 2.5 (Anisotropic Lizorkin–Triebel spaces). The anisotropic Lizorkin–Triebel space s (Rn+1 ) = F s , s ∈ R, 1 p < ∞ and 1 q ∞ (or 1 q < ∞ and p = ∞) is the space Fp,q p,q of functions u ∈ S (Rn+1 ) with finite quasi-norms s uFp,q

∞

1/q sqj q = 2 |φj ∗ u| j =0

(2.10)

Lp (Rn+1 )

and the natural modification for q = ∞, i.e. s = sup 2sj |φj ∗ u| uFp,∞ j 0

Lp (Rn+1 )

.

(2.11)

908


A very useful space throughout our analysis will be the truncated anisotropic (parabolic) s that we define here. p,q Lizorkin–Triebel space F Definition 2.6 (Truncated anisotropic Lizorkin–Triebel space). The truncated anisotropic s (Rn+1 ) = F s , s ∈ R, 1 p < ∞ and 1 q ∞ (1 q < ∞ p,q p,q Lizorkin–Triebel space F if p = ∞) is the space of functions u ∈ S (Rn+1 ) with finite quasi-norms uF p,q s

∞

1/q = 2sqj |φj ∗ u|q j =1

(2.12)

Lp (Rn+1 )

and the natural modification for q = ∞, i.e. uF p,∞ = sup 2sj |φj ∗ u| s j 1

Lp (Rn+1 )

.

(2.13)

s and F s is that in F s we omit the term φ ∗ u and only take p,q p,q The basic difference between Fp,q 0 in consideration the terms φj ∗ u, j 1. Sobolev embeddings of parabolic Lizorkin–Triebel and Besov spaces are shown by the next two lemmas.

Lemma 2.7 (Embeddings of Besov spaces). (See [10, Theorem 7].) Let s, t ∈ R, s > t, and n+2 1 p, r ∞ satisfy: s − n+2 p = t − r . Then for any 1 q ∞ we have the following continuous embedding n+1 n+1 s t R

→ Br,q R . Bp,q

(2.14)

Lemma 2.8 (Sobolev embeddings). (See [15, Proposition 2].) Take an integer m 1. Then we have 2m 2m B2,1

→ W22m,m → B2,∞ .

(2.15)

2.2. Basic logarithmic Sobolev inequality In this subsection we show a basic logarithmic Sobolev inequality. In particular, we show the following lemma. Lemma 2.9 (Basic logarithmic Sobolev inequality). Let u ∈ W22m,m (Rn+1 ) for some m ∈ N, m > n+2 4 . Then there exists some constant C = C(m, n) > 0 such that uF 0

∞,1

1/2 . C 1 + uF 0 1 + log+ uW 2m,m ∞,2

(2.16)

2

Proof. First, let us mention that the ideas of the proof of this lemma are inspired from the proof of Ogawa [14, Corollary 2.4]. The proof is divided into three steps, and the constants in the proof may vary from line to line. Step 1 (Estimate of uF 0 ). Let γ > 0, and N ∈ N be two arbitrary variables. We compute: ∞,1


uF 0

∞,1

|φj ∗ u| 1j 2γ uF 0 , we choose 1 β < 2γ such that ∞,2

u γ F∞,2 ∈ N. N = log+ 2γ β u 0 F ∞,2

We then compute: N 1/2 uF 0 + 2−γ N uF γ

∞,2

∞,2

uF 0

∞,2

uF 0

∞,2

u γ 1/2 F∞,2 1 + log+ γ β 2 β uF 0 ∞,2

uF γ 1/2

1 2 ∞,2 + log+ β log 2γ uF 0

Cγ uF 0

∞,2

∞,2

1 + log+

uF γ 1/2 ∞,2

uF 0

,

∞,2

hence we also have (2.17) with a different constant Cγ . Step 3 (Estimate of uF γ

∞,2

and conclusion). Noting the inequality

1/2

y C(1 + x(log(e + y))1/2 ) x log e + x Cx(log(e + y))1/2

for 0 < x 1, for x > 1,

we deduce from (2.17) that: uF 0

∞,1

1/2 , C 1 + uF 0 1 + log+ uF γ ∞,2

∞,2

(2.18)

910


where the constant C depends also on γ . We now estimate the term uF γ . Choose γ such that ∞,2

0 < γ < 2m − Call α = 2m −

n+2 2 ,

n+2 . 2

we compute: uF γ

∞,2

1/2 2j γ 2 = 2 |φ ∗ u| j

L∞

j 0

2

2j (γ −α)

1/2 sup 2αj |φj ∗ u| j 0

j 0 α CuB∞,∞ .

L∞

(2.19)

It is easy to check (see (2.14), Lemma 2.7, and (2.15), Lemma 2.8) that we have the continuous embeddings 2m α W22m,m → B2,∞

→ B∞,∞ .

Therefore (from inequality (2.19)) we get: uF γ

∞,2

CuW 2m,m ,

hence the result directly follows from (2.18).

2

2

2.3. Proof of Theorem 1.1 In this subsection we present the proof of several lemmas leading to the proof of Theorem 1.1. We start with the following lemma concerning mean estimates of functions on parabolic cubes. Call Q2j ⊂ Rn+1 , j 0, any arbitrary parabolic cube of radius 2j (see (2.3) for the definition of parabolic cubes). For the sake of simplicity, we denote Qj = Q2j

for all j ∈ Z.

(2.20)

Our next lemma reads: Lemma 2.10 (Mean estimates on parabolic cubes). Let u ∈ BMOp (Rn+1 ). Take Qj ⊂ Qj +1 , j 0 (Qj and Qj +1 do not necessarily have the same center). Then we have (with the notation (2.4)): |uQj +1 − uQj | 1 + 2n+2 uBMOp .

(2.21)

More generally, we have for any Qj ⊆ Qk , j, k ∈ Z: |uQk − uQj | (k − j ) 1 + 2n+2 uBMOp .

(2.22)


911

Proof. We easily remark that: j +1 Q = 2n+2 Qj . We compute: 1 |uQj +1 − uQj | = j |Q |

1 |Qj |

|uQj +1 − uQj | Qj

|u − uQj | + Qj

uBMOp +

2n+2 |Qj +1 |

1 |Qj |

|u − uQj +1 | Qj

|u − uQj +1 | Qj +1

uBMOp + 2n+2 uBMOp 1 + 2n+2 uBMOp , 2

which immediately gives (2.21), and consequently (2.22).

The following two lemmas are of notable importance for the proof of the logarithmic Sobolev inequality (1.2). In the first lemma we bound the terms φj ∗ u for j 1, while, in the second lemma, we give a bound on φ0 ∗ u. Lemma 2.11 (Estimate of φj ∗ uL∞ (Rn+1 ) for j 1). Let u ∈ BMOp (Rn+1 ). Then there exists a constant C = C(n) > 0 such that: u ∗ φj L∞ (Rn+1 ) CuBMOp (Rn+1 )

for any j 1,

(2.23)

where (φj )j 1 is the sequence of functions given in (2.7). Proof. We will show that (φj ∗ u)(z) CuBMO p

for z = 0.

(2.24)

The general case with z ∈ Rn+1 could be deduced from (2.24) by translation. Throughout the proof, we will sometimes omit (when there is no confusion) the dependence of the norm on the space Rn+1 . The proof is divided into three steps. Step 1 (Decomposition of (φj ∗ u)(0) on parabolic cubes). Since φˆ j is supported in {z ∈ Rn+1 ; 2j −1 [z] 2j +1 } then φˆ j (0) = 0 = Rn+1 φj . Using this equality, we can write: (φj ∗ u)(0) =

φj (−z) u(z) − uQ1−j dz,

Rn+1

where Q1−j is the parabolic cube defined by (2.20) and centered at 0. This implies that

912


(φj ∗ u)(0)

A1

φj (−z)u(z) − u

Q

1−j dz

Q1−j A2

φj (−z)u(z) − u

+

Q

1−j dz .

(2.25)

Rn+1 \Q1−j

Step 1.1 (Estimate of A1 ). From (2.7), the term A1 can be estimated as follows:

A1 2(n+2)(j −1) φ1 L∞

u(z) − u

Q1−j

dz

Q1−j

2(n+2)(j −1) Q1−j φ1 L∞ uBMO

p

|Q1 |φ1 L∞ uBMOp , hence A1 C0 uBMOp

with C0 = |Q1 |φ1 L∞ (Rn+1 ) .

(2.26)

Step 2 (Estimate of A2 ). We rewrite A2 as the following series:

A2 = 2(n+2)(j −1)

(j −1)a φ1 −2 z u(z) − u

Q1−j

−∞ 0,

φ1 (z) C1 [z]m

for all [z] 1.

(2.28)

The asymptotic behavior of φ1 shown by (2.28) leads to the following decomposition of the term A2 : A3

A2 C1 2

(n+2)(j −1)

−∞ 0 such that we have: φ0 ∗ uL∞ C 1 + uBMOp 1 + log+ uW 2m,m .

n+2 4 .

Then there

(2.31)

2

Proof. The constants that will appear may differ from line to line, but only depend on n and m. The proof of this lemma combines somehow the proof of Lemmas 2.9 and 2.11. We write down uQ1 as a finite sum of a telescopic sequence for N 1: uQ1 = (uQ1 − uQ2 ) + · · · + (uQN−1 − uQN ) + uQN . From Lemma 2.10, we deduce that: |uQ1 | C(N − 1)uBMOp + |uQN |.

914


Remark that applying Cauchy–Schwarz inequality, we get |uQN |

1 |QN |

1/2

|u|

1/2

u2

QN

12

QN

,

QN

then we obtain n+2 . with γ = |uQ1 | C NuBMOp + 2−γ N uW 2m,m 2 2

(2.32)

Following similar arguments as in the proof of Lemma 2.9, we may optimize (2.32) in N , we finally get: |uQ1 | C 1 + uBMOp 1 + log+ uW 2m,m . (2.33) 2

We now estimate |(φ0 ∗ u)(z)| for z = 0. Again, the same estimate could be obtained for any z ∈ Rn+1 by translation. We write φ0 (−z)u(z) (φ0 ∗ u)(0) = Rn+1

φ0 (−z) u(z) − uQ1 +

= Rn+1

φ0 (−z)uQ1

Rn+1 B1

φ0 (−z) u(z) − uQ1 +

=

B3

B2

φ0 (−z) u(z) − uQ1 + φ0 (−z)uQ1 ,

Rn+1 \Q1

Q1

Rn+1

where |B1 | CuBMOp ,

(2.34)

and, from (2.33), |B3 | C 1 + uBMOp 1 + log+ uW 2m,m .

(2.35)

2

In order to estimate B2 , we argue as in Step 2 of Lemma 2.11. In fact we have: φ0 (−z)|u k+1 − u 1 | φ0 (−z) u(z) − uQk+1 + |B2 | Q Q k1 k+1 k Q \Q

2

k1 k+1 k Q \Q

k+1 (1 + k) uBMOp sup φ0 (−z) Q

k+1 k k1 Q \Q

n+2

k1

2

−(m−(n+2))

(1 + k) |Q1 |uBMOp ,

(2.36)


915

where for the last line we have used the fact that |φ0 (z)| [z]Cm for [z] 1. Of course the above series converges if we choose m > n + 2. From (2.34), (2.35) and (2.36), the result follows. 2 0 (Rn+1 ), then u ∈ Corollary 2.13 (A control of uF 0 ). Let u ∈ BMOp (Rn+1 ) ∩ F ∞,1 ∞,2 0 n+1 F (R ) and we have: ∞,2

uF 0

∞,2

1/2

1/2 F∞,1

CuBMOp u 0 ,

(2.37)

where C = C(n) > 0 is a positive constant. Proof. Using (2.23), we compute: uF 0

∞,2

1/2 2 = |φj ∗ u|

L∞

j 1

1/2

1/2 ∞ sup φj ∗ uL |φj ∗ u| j 1

L∞

j 1

1/2 F∞,1

CuBMOp u 0 , which terminates the proof.

2

Remark 2.14. From [2], it seems that BMOp spaces can be characterized in terms of parabolic Lizorkin–Triebel spaces. In the case of elliptic spaces, it is a well-known result (see [6,16]) which allows to simplify the proof of the Kozono–Taniuchi inequality. We can now give the proof of our first main result (Theorem 1.1). Proof of Theorem 1.1. Using (2.16) and (2.37), we obtain: uF 0

∞,1

1/2 1/2 1/2 . C 1 + uBMOp u 0 1 + log+ uW 2m,m F∞,1

(2.38)

2

Notice that the constant C can always be chosen such that C 1. If uF 0 1, we evidently ∞,1 have: uF 0

∞,1

If uF 0

∞,1

C C 1 + uBMOp 1 + log+ uW 2m,m .

(2.39)

2

1/2 F∞,1

> 1, then, dividing (2.38) by u 0 , we can easily deduce inequality (2.39). Using

the fact that uL∞ C

φj ∗ uL∞ C φ0 ∗ uL∞ + uF 0 ,

j 0

and using inequalities (2.31) and (2.39), we directly get into the result.

∞,1

2

916


3. A parabolic Kozono–Taniuchi inequality on a bounded domain The goal of this section is to present, on the one hand, the proof of Theorem 1.2. On the other hand, at the end of this section, we give an application where we show how to use inequality (1.3) in order to maintain the long-time existence of solutions to some parabolic equations. Let us indicate that throughout this section, the positive constant C = C(T ) > 0 may vary from line to line. 3.1. Proof of Theorem 1.2 In order to simplify the arguments of the proof, we first show Theorem 1.2 in the special case when n = m = 1. Then we give the principal ideas how to prove the result in the general case. Call I = (0, 1)

and ΩT = I × (0, T ),

we first show the following proposition: Proposition 3.1 (Theorem 1.2, case: n = m = 1). Let u ∈ W22,1 (ΩT ). Then there exists a constant C = C(T ) > 0 such that: uL∞ (ΩT ) C 1 + uBMOp (ΩT ) 1 + log+ uW 2,1 (Ω ) . 2

T

(3.1)

As a similar inequality of (3.1) is already shown on R2 (see inequality (1.2)), the idea of the proof of (3.1) lies in using (1.2) for a special extension of the function u ∈ W22,1 (ΩT ) to the entire space R2 . For this reason, we demand that the extended function stays in W22,1 (R2 ) which is done via the following arguments. Remark first that the function u can be extended by continuity to the boundary ∂ΩT of ΩT . Take u˜ as the function defined over T = (−1, 2) × (−T , 2T ) Ω as follows: u(x, ˜ t) =

−3u(−x, t) + 4u(− x2 , t) for − 1 < x < 0, 0 t T , 3−x −3u(2 − x, t) + 4u( 2 , t) for 1 < x < 2, 0 t T ,

(3.2)

and u(x, ˜ t) =

u(x, −t) for − T < t 0, u(x, 2T − t) for T t < 2T .

(3.3)

A direct consequence of this extension is the following lemma. Lemma 3.2 (L1 estimate of u). ˜ Let u˜ be the function defined by (3.2) and (3.3). Then there exists a constant C = C(T ) > 0 such that: u ˜ L1 (Ω T ) CuL1 (ΩT ) .

(3.4)


Proof. The proof of this lemma is direct by the extension.

917

2

T ), Another important consequence of the extension (3.2) and (3.3) is the fact that u˜ ∈ W22,1 (Ω and that we have (see for instance [5]) u ˜ W 2,1 (Ω ) CuW 2,1 (Ω ) , 2

T

2

T

C = C(T ) > 0.

(3.5)

T defined by: Let Z1 ⊂ Z2 be the two subsets of Ω Z1 = (x, t); −1/4 < x < 5/4 and − T /4 < t < 5T /4 , and Z2 = (x, t); −3/4 < x < 7/4 and − 3T /4 < t < 7T /4 . Taking the cut-off function Ψ ∈ C0∞ (R2 ), 0 Ψ 1 satisfying: Ψ (x, t) =

1 for (x, t) ∈ Z1 , 0 for (x, t) ∈ R2 \ Z2 ,

(3.6)

we can easily deduce from (3.5) that Ψ u˜ ∈ W22,1 (R2 ), and Ψ u ˜ W 2,1 (R2 ) CuW 2,1 (Ω ) . 2

2

(3.7)

T

Since Ψ u˜ ∈ W22,1 (R2 ), we can apply inequality (1.2) to the function Ψ u, ˜ and, having (3.7) in hands, the proof of Proposition 3.1 directly follows if we can show that Ψ u ˜ BMOp (R2 ) CuBMOp (ΩT ) ,

(3.8)

and this will be done in the forthcoming arguments. 3.1.1. Proof of Proposition 3.1 In all what follows, it will be useful to deal with an equivalent norm of the BMOp space. This norm is given by the following lemma. Lemma 3.3 (Equivalent BMOp norms). Let u ∈ BMOp (O), O ⊆ Rn+1 is an open set. The parabolic BMOp norm of u given by (2.2) is equivalent to the following norm for which we keep the same notation:

1 |u − c| , Q given by (2.3). (3.9) uBMOp (O) = sup inf Q⊂O c∈R |Q| Q

Proof. The proof of this lemma is direct. It suffices to see that for any c ∈ R, we have: 1 |u − c|, |u − uQ | |u − c| + |c − uQ | |u − c| + |Q| Q

918


which immediately gives:

|u − uQ | 2 Q

|u − c|, Q

hence 1 2|Q|

1 c∈R |Q|

|u − uQ | inf Q

and the equivalence of the two norms follows.

1 |Q|

|u − c| Q

|u − uQ |,

(3.10)

Q

2

From now on, and for the sake of simplicity, we will denote: 1 −u= u. |Q| Q

Q

The following lemma gives an estimate of infc∈R − Q |u − c| on small parabolic cubes. Lemma 3.4. Let f ∈ L1loc (R2 ). Take Qr ⊆ Q2r two parabolic cubes of R2 . We do not require that the cubes have the same center. Then we have: inf − |f − c| 8 inf − |f − c|.

c∈R

c∈R

Qr

(3.11)

Q2r

Proof. For c ∈ R, we compute: |Q2r | − |f − c| − |f − c| 8 − |f − c|. |Qr | Qr

Q2r

Taking the infimum of both sides we arrive to the result.

Q2r

2

The next lemma gives an estimate of infc∈R − Qr |u˜ − c| on small parabolic cubes in T = (−1, 2) × (0, T ). Ω Define the term r0 > 0 as the greatest positive real number such that there exists Qr0 ⊆ ΩT , i.e., r0 = sup r > 0; r 1/2 and r 2 T /2 . We show the following:

(3.12)


919

Fig. 1. Analysis on cubes intersecting {x = 0}.

T ). Let u˜ be the function defined by (3.2) Lemma 3.5 (Estimates on small parabolic cubes in Ω and (3.3). Take any parabolic cube Qr satisfying: T Qr ⊆ Ω

with r r1 and 2r1 = r0 ,

(3.13)

where r0 is given by (3.12). Then there exists a universal constant C > 0 such that: inf − |u˜ − c| CuBMOp (ΩT ) .

c∈R

(3.14)

Qr g

Proof. Call ΩTd and ΩT the right and the left neighbor sets of ΩT defined respectively by: ΩTd = (−1, 0) × (0, T )

g

and ΩT = (1, 2) × (0, T ).

First let us mention that if the cube Qr lies in ΩT then inequality (3.14) is evident (see the equivalent definition (3.9) of the parabolic BMOp norm). Two remaining cases are to be considered: g either Qr intersects the set {x = 0} ∪ {x = 1}, or Qr lies in ΩTd ∪ ΩT . Our assumption (3.13) on g the radius of the parabolic cube makes it impossible that the cube Qr meets ΩTd and ΩT at the same time. Therefore, and in order to make the proof simpler, we only consider the following g cases: either Qr intersects the set {x = 0}, or Qr lies in ΩT . The proof is then divided into three main steps: Step 1 (Qr intersects the line {x = 0}). Step 1.1 (First estimate). Again the assumption (3.13) imposed on the radius r makes it possible T of radius 2r, which is symmetric with respect to embed Qr in a larger parabolic cube Q2r ⊆ Ω to the line {x = 0} (see Fig. 1). Then the center of the cube Q2r should be also on the same line,

920


but we do not require that the two cubes Qr and Q2r have centers with the same ordinate t. Now, using Lemma 3.4, we deduce that: inf − |u˜ − c| 8 inf − |u˜ − c|,

c∈R

(3.15)

c∈R

Qr

Q2r

and hence in order to conclude, we need to estimate the right-hand side of the above inequalg ity with respect to uBMOp (ΩT ) . Call Qd2r and Q2r the right and the left sides of Q2r defined respectively by: Qd2r = Q2r ∩ ΩT

g

g

and Q2r = Q2r ∩ ΩT .

Also call Qtrans 2r ⊆ ΩT , the translation of the cube Q2r by the vector (2r, 0), i.e. Qtrans 2r = (2r, 0) + Q2r . For c ∈ R, we compute:

|u˜ − c| = g Q2r

Q2r

|u˜ − c| +

g Q2r

|u − c|,

Qd2r

|u˜ − c| +

|u − c|,

(3.16)

Qtrans 2r

where we have used the fact that u˜ = u on ΩT , and that Qd2r ⊆ Qtrans 2r . Step 1.2 (Estimate of on

g ΩT ):

g

Q2r

g

Q2r

|u˜ − c|). We compute (using the definition (3.2) of the function u˜

u(x, ˜ t) − c dx dt =

−3u(−x, t) + 4u(−x/2, t) − c dx dt

g

Q2r

3

u(−x, t) − c dx dt + 4

g

3 Qd2r

u(−x/2, t) − c dx dt

g

Q2r

Q2r

u(x, t) − c dx dt + 8

u(x, t) − c dx dt,

¯ Qd2r

where ¯ Qd2r = (x/2, t); (x, t) ∈ Qd2r ⊆ Qd2r ⊆ Qtrans 2r .

(3.17)


921

From (3.17) we easily deduce that:

|u˜ − c| 11 g Q2r

|u − c|,

Qtrans 2r

and hence (using (3.16)), we finally get:

|u˜ − c| 12

|u − c|.

(3.18)

Qtrans 2r

Q2r

Since |Q2r | = |Qtrans 2r |, inequality (3.18) gives − |u˜ − c| 12 − |u − c|. Qtrans 2r

Q2r

Since Qtrans is a parabolic cube in ΩT , taking the infimum over c ∈ R of the above inequality, 2r we obtain: inf − |u˜ − c| 12uBMOp (ΩT ) . (3.19) c∈R

Q2r

From (3.15) and (3.19), we deduce (3.14). g

Step 2 (Qr ⊆ ΩT ). Let 0 < a0 < b0 < 1 and 0 < a1 < b1 < T be such that Qr = (−b0 , −a0 ) × (a1 , b1 ). For any c ∈ R, we compute:

u(x, ˜ t) − c dx dt =

Qr

−3u(−x, t) + 4u(−x/2, t) − c dx dt

Qr

3

u(x, t) − c dx dt + 8

u(x, t) − c dx dt

(3.20)

Qsr¯

Qsr

with (see Fig. 2), Qsr

= (a0 , b0 ) × (a1 , b1 ) ⊆ ΩT

and

Qsr¯

=

a0 b0 , 2 2

× (a1 , b1 ) ⊆ ΩT .

We remark that Qsr is a parabolic cube in ΩT , while Qsr¯ is not (its aspect ratio is different). In ¯ ¯ fact Qsr¯ could be embedded in a parabolic cube Qsr¯ ⊆ Qsr¯ ⊆ ΩT , where Qsr¯ is simply a space

922


g

Fig. 2. Analysis on cubes Qr ⊆ ΩT .

translation of Qsr . In particular we have: ¯ |Qr | = Qsr = Qsr¯ .

(3.21)

The above arguments, together with (3.20) give:

|u˜ − c| 3

|u − c| + 8

Qsr

Qr

|u − c|.

(3.22)

Qsr¯¯

Taking the infimum in c ∈ R for both sides of inequality (3.22), leads to inf − |u˜ − c| 11uBMOp (ΩT ) ,

c∈R

(3.23)

Qr

which implies (3.14). Step 3 (Conclusion). As it was already mentioned at the beginning of the proof, the case where the parabolic cube Qr meets the line {x = 1} or lies completely in ΩTd , could be treated using identical arguments. Therefore, for all small parabolic cubes Qr satisfying (3.13), inequality (3.14) is always valid, and this terminates the proof of Lemma 3.5. 2 A generalization of Lemma 3.5 is now given. T ). Let u˜ be the function defined by (3.2) Lemma 3.6 (Estimates on small parabolic cubes in Ω T satisfying: and (3.3). Take any parabolic cube Qr ⊆ Ω r r2

√ with r2 2 = r1 ,

(3.24)


923

Fig. 3. Qr ∩ {t = T } = ∅.

where r1 is given by (3.13). Then there exists a universal constant C > 0 such that: inf − |u˜ − c| CuBMOp (ΩT ) .

c∈R

(3.25)

Qr

Sketch of the proof. The arguments leading to the proof of this lemma are already contained in √ T , we enter directly (since r r 2 r1 ) to the proof of Lemma 3.5. First notice that if Qr ⊆ Ω the framework of Lemma 3.5, and hence (3.25) is direct. Because r r1 , remark that there exists T . Therefore it is impossible that a cube Qr obtained by a time translation of Qr such that Qr ⊆ Ω Qr meets at the same time (−1, 2) × (T , 2T ) and (−1, 2) × (−T , 0). For this reason, we either consider parabolic cubes intersecting {t = T } (see Fig. 3), or parabolic cubes in (−1, 2)×(T , 2T ) (see Fig. 4). Case Qr ∩ {t = T } = ∅. In this case, we first embed Qr in a larger parabolic cube Qr √2 which is symmetric with respect to the line {t = T }, so the center of this cube lies in {t = T }. We now repeat the same arguments as in Step 1 of Lemma 3.5, using in particular the symmetry (3.3) of the function u˜ with respect to {t = T }, and the fact that we can consider the cube √ = 0, −2r 2 + Q √ Qtrans r 2 r 2 such that

T , √ ⊆ Qr ⊆ Ω Qtrans 1 r 2

√ are already controlled by (3.14). for some cube Qr1 . Indeed, estimates on all such cubes Qtrans r 2

924


Fig. 4. Qr ∩ {t = T } = ∅.

Case Qr ∩ {t = T } = ∅. In this case we repeat the same arguments as in Step 2 of Lemma 3.5. Indeed, in the present case, it is even simpler since the function u˜ is symmetric with respect to {t = T }. 2 We now show how to prove estimate (3.8). Proof of estimate (3.8). The parabolic BMOp norm (3.9) of Ψ u˜ could be estimated taking the supremum of − Qr |Ψ u˜ − (Ψ u) ˜ Qr |, Qr ⊆ R2 , over small parabolic cubes (Qr with r r2 /2), and big parabolic cubes (Qr with r > r2 /2). The proof is then divided into two steps. Step 1 (Analysis on big parabolic cubes Qr , r > r2 /2). We compute, using the fact that Ψ = 0 on R2 \ Z2 , and Ψ 1 on R2 (see (3.6)): 2 ˜ Qr 2 − |Ψ u| − Ψ u˜ − (Ψ u) ˜ |Qr | Qr

Qr

22 r23

Q r ∩Z 2

|u| ˜

Q r ∩Z 2

|u| ˜

22 r23

|u| ˜ CuL1 (ΩT ) .

(3.26)

T Ω

Step 2 (Analysis on small parabolic cubes Qr , r r2 /2). From the definition (3.24) of r2 , and the T . construction (3.6) of the function Ψ , we deduce that if Qr intersects Z2 then forcedly Qr ⊆ Ω If not, i.e. Qr ∩ Z2 = ∅ then Ψ = 0 on Qr , and therefore: ˜ Qr = 0. (3.27) − Ψ u˜ − (Ψ u) Qr


925

T . Then we have only to consider Qr ⊆ Ω Step 2.1 (First estimate). Using (3.10), we get ˜ Qr 2 inf − |Ψ u˜ − c| 2 − |Ψ u˜ − c0 ΨQr |, − Ψ u˜ − (Ψ u) c∈R

Qr

Qr

(3.28)

Qr

for any fixed constant c0 ∈ R. Remark that we can write: Ψ u˜ − c0 ΨQr = (Ψ − ΨQr )u˜ + (u˜ − c0 )ΨQr .

(3.29)

Hence, we deduce that ˜ Qr Cr − |u| ˜ + 2 inf − |u˜ − c0 | − Ψ u˜ − (Ψ u) c0 ∈R

Qr

Qr

Qr

˜ + 2CuBMOp (ΩT ) , Cr − |u|

(3.30)

Qr

where for the first line we have used that fact that Ψ 1 and that Ψ is Lipschitz, and for the second line we have used (3.25). ˜ We have Step 2.2 (Estimate of − Qr |u|). − |u| ˜ |u˜ Qr | + − |u˜ − u˜ Qr | Qr

Qr

|u˜ Qr | + 2 inf − |u˜ − c| c∈R

Qr

|u˜ Qr | + 2CuBMOp (ΩT ) ,

(3.31)

where for the second line, we have used (3.10), while for the third line, we have used (3.25). T : Remark that from the proof of Lemma 2.10 with n = 1, we have for Q2j r ⊆ Q2j +1 r ⊆ Ω |u˜ Q2j r

3 − u˜ Q2j +1 r | − |u˜ − u˜ Q2j r | + 2 − |u˜ − u˜ Q2j +1 r | Q2j r

2 1 + 23

Q2j +1 r

sup

T ,ρ2j +1 r Qρ ⊆Ω

2C 1 + 23 uBMOp (ΩT ) ,

inf − |u˜ − c|

c∈R

Qρ

926


where we have used (3.10) for the second line, and, for the third line, we have used (3.25) assuming 2j +1 r r2 . Defining j0 = min j ∈ N; r2 /2 2j r < r2 , and using a telescopic sequence, we can deduce that |u˜ Qr − u˜ Q

2j0 r

| j0 2C 1 + 23 uBMOp (ΩT ) C 1 + | log r| uBMOp (ΩT ) .

(3.32)

Moreover, we have |u˜ Q

2j0 r

|

1 |Qr2 /2 |

|u| ˜ CuL1 (ΩT ) ,

(3.33)

T Ω

where we have used (3.4) for the second inequality. From (3.30), (3.32) and (3.33), we get: − |u| ˜ C uL1 (ΩT ) + 1 + | log r| uBMOp (ΩT )

(3.34)

Qr

for some constant C > 0. Step 2.3 (Conclusion for r r2 /2). Finally, putting together (3.30) and (3.34), we deduce that ˜ Qr C r| log r| + 1 uBMOp (ΩT ) + uL1 (ΩT ) − Ψ u˜ − (Ψ u) Qr

C uBMOp (ΩT ) + uL1 (ΩT ) ,

(3.35)

where in the second line, we have used that r ∈ (0, 1), and that r| log r| is bounded. Step 3 (General conclusion). Putting together (3.26), (3.27) and (3.35), we get (3.8).

2

We are now ready to show the proof of Proposition 3.1. Proof of Proposition 3.1. Applying estimate (2.37), with m = n = 1, to the function Ψ u˜ ∈ W22,1 (R2 ) ⊆ L∞ (R2 ), we get: ˜ L∞ (ΩT ) Ψ u ˜ L∞ (R2 ) C 1 + Ψ u ˜ BMOp (R2 ) 1 + log+ Ψ u ˜ W 2,1 (R2 ) . uL∞ (ΩT ) = Ψ u 2

Here, we have also used the fact that Ψ = 1 over ΩT (see (3.6)). Using (3.7), (3.8) and the above inequality, we directly get (3.1). 2


927

3.1.2. Ideas of the proof of Theorem 1.2 One of the main motivations for starting with the detailed proof of Proposition 3.1 (a simplified version of Theorem 1.2) is that it was used to show [8, Theorem 1.1]. The other motivation is that the arguments of the proof of Theorem 1.2 are all contained in the proof of Proposition 3.1. It suffices to make the following generalizations that we list below. T ) Extension of u. ˜ In order to extend the function u ∈ W22m,m (ΩT ) to the function u˜ ∈ W22m,m (Ω T = (−1, 2)n × (−T , 2T ), we first make the extension separately and successively with with Ω respect to the spatial variables xi , with i = 1, . . . , n. Then we make the extension with respect to the time variable that is treated somehow differently. Fix (x2 , . . . , xn , t) ∈ (0, 1)n−1 × (0, T ), the spatial extension of u in x1 is as follows: 2m−1 u(x ˜ 1 , . . .) = with λj =

1 2j

for − 1 < x1 < 0,

j =0

cj u(−λj x1 , . . .)

j =0

cj u(1 + λj (1 − x1 ), . . .) for 1 < x1 < 2,

2m−1

(3.36)

, and where we require that: 2m−1

cj (−λj )k = 1 for k = 0, . . . , 2m − 1.

j =0

The above inequalities can be regarded as a linear system whose associated matrix is of the Vandermonde type and hence invertible. This ensures the existence of the constants cj , j = 0, . . . , 2m − 1, and therefore the above extension (3.36) gives sense. After doing the extension with respect to x1 , the extension with respect to x2 is done in the same way by varying the x2 and fixing all other variables. This is repeated successively until the xn variable. For the time variable, we also use the same extension (3.36). Indeed, in this case, we may only sum up to m − 1 in (3.36). The cut-off function Ψ . For the definition of the cut-off function Ψ , we first define the two sets: Z1 = (x1 , . . . , xn , t); ∀i = 1, . . . , n, −1/4 < xi < 5/4 and − T /4 < t < 5T /4 and Z2 = (x1 , . . . , xn , t); ∀i = 1, . . . , n, −3/4 < xi < 7/4 and − 3T /4 < t < 7T /4 . The function Ψ is then defined as Ψ ∈ C0∞ (Rn+1 ) with 0 Ψ 1 and Ψ (x, t) =

1 for (x, t) ∈ Z1 , 0 for (x, t) ∈ R2 \ Z2 .

(3.37)

Generalization of Lemma 3.6. An analogue estimate of (3.25) could be obtained for (n + 1)-di T = (−1, 2)n × (−T , 2T ). It suffices to replace r2 satisfying mensional parabolic cubes Qr ⊆ Ω (3.24), by the radius

928


rn rn+1 = √ , 2 where rn is defined recursively as follows: rj +1 = rj /2 for 0 j n − 1. Using the above generalizations, the proof of Theorem 1.2 follows, line by line, the proof of Proposition 3.1. 3.2. Application of the parabolic Kozono–Taniuchi inequality In this subsection, we show how to apply the parabolic Kozono–Taniuchi inequality in order to give some a priori estimates for the solution of certain parabolic equations. These a priori estimates provide a good control on the solution in order to avoid singularities at a finite time, and hence serve for the long-time existence. The application that will be given here deals with a model that can be considered as a toy model. Indeed, this is a simplification of the one treated in [8], where a rigorous proof of the long-time existence of solutions of a singular parabolic coupled system was presented (see [8, Theorem 1.1]). Consider, for 0 < a < 1, the following parabolic equation: ⎧ ⎨ ut (x, t) − uxx (x, t) = sin ux (x, t)ux (x + a, t) + sin log ux (x, t) u(x + 1, t) = u(x, t) + 1 on R × (0, ∞), ⎩ ux (x, 0) δ0 > 0 on R,

on R × (0, ∞), (3.38)

the following proposition can be established: Proposition 3.7 (Gradient estimate). Let v = ux and m(t) = minx∈R v(x, t). If u ∈ C ∞ (R × [0, ∞)) is a smooth solution of (3.38), then, for some constant C = C(t) > 0 we have: mt −Cm |log m| + 1 ,

∀t 0.

(3.39)

Remark 3.8. Inequality (3.39) directly implies that for every t 0 we have m(t) > 0. This is important to avoid the logarithmic singularity in (3.38) when v = ux = 0. Remark 3.9. The proof of the above proposition goes along the same lines as the proof of [8, Theorem 1.1]. For this reason we only present a heuristic proof explaining only the basic ideas. The interested reader could see the full details in [8]. Ideas of the proof of Proposition 3.7. Heuristically, the proof is divided into the following four steps. In what follows all the constants can depend on the time t, but are bounded for any finite t. Step 1 (First estimate from below on the gradient). Writing down the equation satisfied by v: ⎧ vt (x, t) − vxx (x, t) = cos v(x, t)v(x + a, t) vx (x, t)v(x + a, t) + v(x, t)vx (x + a, t) ⎪ ⎪ ⎪ ⎪ vx (x, t) ⎨ on R × (0, ∞), + cos log v(x, t) (3.40) v(x, t) ⎪ ⎪ ⎪ v(x + 1, t) = v(x, t) on R × (0, ∞), ⎪ ⎩ v(x, 0) δ0 > 0 on R,


929

we can show that for every t 0: mt −mG

with G(t) = maxvx (x, t). x∈R

(3.41)

Step 2 (Estimate of vx BMOp ). Using the fact that u(x + 1, t) = u(x, t) + 1, and that the righthand term of the first equation of (3.38) is bounded, we apply the BMO theory for parabolic equation to (3.38) and hence we obtain, for some positive constant c1 > 0: vx BMOp ((0,1)×(0,t)) c1

for any t > 0.

However, the Lp theory for parabolic equation applied to (3.38) gives, for some positive constant c2 > 0: vx L1 ((0,1)×(0,t)) c2

for any t > 0.

Finally, the above two inequalities give: vx BMOp ((0,1)×(0,t)) c1 + c2

for any t > 0.

(3.42)

Step 3 (Estimate of vx W 2,1 ). Let w = vx , we write down the equation satisfied by w: 2

⎧ wt (x, t) − wxx (x, t) ⎪ ⎪ 2 ⎪ ⎪ ⎪ = − sin v(x, t)v(x + a, t) v(x + a, t)vx (x, t) + v(x, t)vx (x + a, t) ⎪ ⎪ ⎪ ⎪ + cos v(x, t)v(x + a, t) v(x + a, t)vxx (x, t) + 2vx (x, t)vx (x + a, t) ⎪ ⎪ ⎪ ⎪ ⎪ v 2 (x, t) ⎨ + v(x, t)vxx (x + a, t) − sin log v(x, t) x2 v (x, t) ⎪ " ⎪ 2 ⎪ ⎪ vxx (x, t) vx (x, t) ⎪ ⎪ − 2 on R × (0, ∞), + cos log v(x, t) ⎪ ⎪ v(x, t) v (x, t) ⎪ ⎪ ⎪ ⎪ w(x + 1, t) = w(x, t) on R × (0, ∞), ⎪ ⎪ ⎩ w(x, 0) = vx (x, 0) on R.

(3.43)

Using the Lp theory for parabolic equations (with various values of p) to (3.38), (3.40) and (3.43), we deduce, for some other positive constant c > 0, that: vx W 2,1 ((0,1)×(0,t)) 2

c

for any t > 0.

m2 (t)

(3.44)

Step 4 (Conclusion). Applying the parabolic Kozono–Taniuchi inequality (3.1) to the function vx , using in particular (3.42) and (3.44), we deduce that: G C 1 + | log m| , which, together with (3.43), directly gives the result.

2

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Acknowledgments This work was supported by the contract ANR MICA (2006–2009). The authors would like to thank M. Jazar for his encouragement during the preparation of this work. The first author would like to thank B. Kojok for some discussions. References [1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61–66. [2] M. Bownik, Anisotropic Triebel–Lizorkin spaces with doubling measures, J. Geom. Anal. 17 (2007) 387–424. [3] H. Brézis, T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980) 677–681. [4] H. Brézis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980) 773–789. [5] L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 1998. [6] M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990) 34–170. [7] N. Hayashi, W. von Wahl, On the global strong solutions of coupled Klein–Gordon–Schrödinger equations, J. Math. Soc. Japan 39 (1987) 489–497. [8] H. Ibrahim, M. Jazar, R. Monneau, Dynamics of dislocation densities in a bounded channel. Part I: Smooth solutions to a singular coupled parabolic system, preprint hal-00281487. [9] H. Ibrahim, M. Jazar, R. Monneau, Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions, C. R. Math. Acad. Sci. Paris, Ser. I 346 (2008) 945–950. [10] J. Johnsen, W. Sickel, A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms, J. Funct. Spaces Appl. 5 (2007) 183–198. [11] H. Kozono, T. Ogawa, Y. Taniuchi, Navier–Stokes equations in the Besov space near L∞ and BMO, Kyushu J. Math. 57 (2003) 303–324. [12] H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191–200. [13] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ura´lceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith, Transl. Math. Monogr., vol. 23, American Mathematical Society, Providence, RI, 1967. [14] T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal. 34 (2003) 1318–1330 (electronic). [15] B. Stöckert, Remarks on the interpolation of anisotropic spaces of Besov–Hardy–Sobolev type, Czechoslovak Math. J. 32 (107) (1982) 233–244. [16] H. Triebel, Theory of Function Spaces. II, Monogr. Math., vol. 84, Birkhäuser-Verlag, Basel, 1992.


Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices ✩ Jaegil Kim a,∗ , Han Ju Lee b,∗ a Department of Mathematics, Kent State University, Kent, OH 44240, USA b Dongguk University, Department of Mathematics Education, 26, Pil-dong 3-ga, Chung-gu, Seoul, 100-715,

Republic of Korea Received 11 July 2008; accepted 20 November 2008 Available online 17 December 2008 Communicated by K. Ball

Abstract Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb (K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of P(n X), then the set of all strongly norm attaining elements in P(n X) is dense. In particular, the set of all points at which the norm of P(n X) is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner’s graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to n∞ . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm. © 2008 Elsevier Inc. All rights reserved. Keywords: Peak points; Peak functions; Polynomial numerical index

✩

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2007412-J02301). * Corresponding authors. E-mail addresses: [email protected] (J. Kim), [email protected] (H.J. Lee). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.024

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J. Kim, H.J. Lee / Journal of Functional Analysis 257 (2009) 931–947

1. Introduction and preliminaries Let K be a complete metric space and X a (real or complex) Banach space. We denote by Cb (K : X) the Banach space of all bounded continuous functions from K to X with the supremum norm. A nonzero function f ∈ Cb (K : X) is said to be a strong peak function at t ∈ K if every sequence {tn } in K with limn f (tn ) = f converges to t. Given a subspace A of Cb (K : X), a point t ∈ K called a strong peak point for A if there is a strong peak function f in A with f = f (t). We denote by ρA the set of all strong peak points for A. Let BX (resp. SX ) be the unit ball (resp. sphere) of a Banach space X. A nonzero function f ∈ Cb (BX : Y ) is said to strongly attain its norm at x if for every sequence {xn } in BX with limn f (xn ) = f , there exist a scalar λ with |λ| = 1 and a subsequence of {xn } which converges to λx. Given a subspace A of Cb (BX : Y ), x ∈ BX is called a strongly norm-attaining point of A if there exists a nonzero function f in A which strongly attains its norm at x. Denote by ρA ˜ the set of all strongly norm-attaining points of A. For complex Banach spaces X and Y , we may use the following two subspaces of Cb (BX : Y ): Ab (BX : Y ) = f ∈ Cb (BX : Y ): f is holomorphic on the interior of BX , Au (BX : Y ) = f ∈ Ab (BX : Y ): f is uniformly continuous on BX . We shall denote by A(BX : Y ) either Ab (BX : Y ) or Au (BX : Y ). In case that Y is the complex filed C, we write A(BX ), Au (BX ) and Ab (BX ) instead of A(BX : C), Au (BX : C) and Ab (BX : C) respectively. If X and Y are Banach spaces, an k-homogeneous polynomial P from X to Y is a mapping such that there is a k-linear continuous mapping L from X × · · · × X to Y such that P (x) = L(x, . . . , x) for every x ∈ X. P(k X : Y ) denote the Banach space of all k-homogeneous polynomials from X to Y , endowed with the polynomial norm P = supx∈BX P (x). We also say that P : X → Y is a polynomial, and write P ∈ P(X : Y ) if P is a finite sum of homogeneous polynomials from X into Y . In particular, replace P(k X : Y ) by P(k X) and P(X : Y ) by P(X) when Y is a scalar field. We refer to [8] for background on polynomials. The (polynomial) numerical index of a Banach space is a constant relating to the concepts of the numerical radius of functions on X. Actually, for each f ∈ Cb (BX : X), the numerical radius v(f ) is defined by v(f ) = sup x ∗ f (x): x ∗ (x) = 1, x ∈ SX , x ∗ ∈ SX∗ , where X ∗ is the dual space of X. For every integer k 1, the k-polynomial numerical index of a Banach space X is the constant defined by n(k) (X) = inf v(P ): P = 1, P ∈ P k X : X . If k = 1, in particular, then it is called the numerical index of X and we write n(X). For more recent results about numerical index, see a survey paper [15] and references therein. Let us briefly see the contents of the paper. In Section 2, using the variational method in [7], we show that the set ρA is a norming subset of A if and only if the set of all strong peak functions


933

in A is a dense Gδ subset of A, when A is a closed subspace Cb (K : X) which contains all elements of the form t → (x ∗ f (t))m x for all x ∈ X, x ∗ ∈ X ∗ , f ∈ A and integers m 1. Using this theorem and the variational methods, we investigate the denseness of the set of strong peak holomorphic functions and the denseness of the set of numerical strong peak functions on certain Banach spaces. In Section 3, we also apply the variation method to investigate the denseness of the set of strongly norm attaining polynomials when the set of all uniformly strongly exposed points of a Banach space X is a norming subset of P(n X). As a direct corollary, the set of all points at which the norm of P(n X) is Fréchet differentiable is a dense Gδ subset if the set of all uniformly strongly exposed points of a Banach space X is a norming subset of P(n X). In the last part, we will use the graph theory to get some strongly norm-attaining points or complex extreme points. Reisner gave a one-to-one correspondence between n-dimensional some Banach spaces and certain graphs with n vertices. In detail he give a characterization of all finite dimensional real CL-spaces with an absolute norm using the graph-theoretic terminology. It gives a geometric picture of extreme points of the unit ball of CL-spaces and plays an important role to find the strongly norm-attaining points of P(k X). Moreover we can find all complex extreme points on a complex CL-space with an absolute norm. These strongly normattaining points or complex extreme points help answering a problem about the numerical index of a Banach space. We give a partial answer to Problem 43 in [15]: Characterize the complex Banach spaces X satisfying n(k) (X) = 1 for all k 2. We show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to n∞ . For later use, recall the definitions of real and complex extreme points of a unit ball. Let X be a real or complex Banach space. Recall that x ∈ BX is said to be an extreme point of BX if whenever y + z = 2x for some y, z ∈ BX , we have x = y = z. Denote by ext(BX ) the set of all extreme points of BX . When X is a complex Banach space, an element x ∈ BX is said to be a complex extreme point of BX if sup0θ2π x + eiθ y 1 for some y ∈ X implies y = 0. The set of all complex extreme points of BX is denoted by extC (BX ). 2. Denseness of the set of strong peak functions Let X be a Banach space and A a closed subspace Cb (K : X). A subset F of K is said to be a norming subset for A if for each f ∈ A, we have f = sup f (t): t ∈ F . Following the definition of Globevnik in [11], the smallest closed norming subset of A is called the Shilov boundary for A and it is shown in [6] that if the set of strong peak functions is dense in A then the Shilov boundary of A exists and it is the closure of ρA. The variation method used in [7] gives the partial converse of the above mentioned result. Theorem 2.1. Let A be a closed subspace Cb (K : X) which contains all elements of the form t → (x ∗ f (t))m x for all x ∈ X, x ∗ ∈ X ∗ , f ∈ A and integers m 1. Then the set ρA is a norming subset of A if and only if the set of all strong peak functions in A is a dense Gδ subset of A.

934


Proof. Let d be the complete metric on K. Fix f ∈ A and > 0. For each n 1, set Un = g ∈ A: ∃z ∈ ρA with (f − g)(z) > sup (f − g)(x): d(x, z) > 1/n . Then Un is open and dense in A. Indeed, fix h ∈ A. Since ρA is a norming subset of A, there is a point w ∈ ρA such that (f − h)(w) > f − h − /2.

(2.1)

Choose a peak function q ∈ A at w with q(w) = 1 and w ∗ ∈ SX∗ such that w ∗ q(w) = 1. Then it is easy to see that w ∗ ◦ q is also a strong peak function in Cb (K). So there is an integer m 1 such that |w ∗ q(x)|m < 1/3 for all x ∈ K with d(x, w) > 1/n. Now define the function by m (f − h)(w) p(t) = − w ∗ q(t) · , (f − h)(w)

∀t ∈ K.

Set g(x) = h(x) + · p(x). Then f (w) − h(w) − p(w) = f (w) − h(w) + and g − h . Eq. (2.1) shows that (f − g)(w) = f (w) − h(w) − p(w) = f (w) − h(w) + > f − h + /2 sup (f − h)(x) − p(x): d(x, w) > 1/n = sup (f − g)(x): d(x, w) > 1/n . Therefore g ∈ Un . By the Baire category theorem there is a g ∈ Un with g < , and we shall show that f − g is a strong peak function. Indeed, g ∈ Un implies that there is zn ∈ X such that (f − g)(zn ) > sup (f − g)(x): d(x, zn ) > 1/n . Thus d(zp , zn ) 1/n for every p > n, and hence {zn } converges to a point z, say. Suppose that there is another sequence {xk } in BX such that {(f − g)(xk )}k converges to f − g. Then for each n 1, there is Mn 1 such that for every m Mn , (f − g)(xm ) > sup (f − g)(x): d(x, zn ) > 1/n . Then d(xm , zn ) 1/n for every m Mn . Hence {xm }m converges to z. This shows that f − g is a strong peak function at z. By Proposition 2.19 in [14], the set of all strong peak functions in A is a Gδ subset of A. This proves the necessity. Concerning the converse, it is shown in [6] that if the set of all strong peak functions is dense in A, then the set of all strong peak points is a norming subset of A. This completes the proof. 2 Let BX be the unit ball of the Banach space X. Recall that the point x ∈ BX is said to be a smooth point if there is a unique x ∗ ∈ BX∗ such that Re x ∗ (x) = 1. We denote by sm(BX ) the set


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of all smooth points of BX . We say that a Banach space is smooth if sm(BX ) is the unit sphere SX of X. The following corollary shows that if ρA is a norming subset, then the set of smooth points of BA is dense in SA . Corollary 2.2. Suppose that K is a complete metric space and A is a closed subalgebra of Cb (K). If ρA is a norming subset of A, then the set of all smooth points of BA contains a dense Gδ subset of SA . Proof. It is shown in [6] that if f ∈ A is a strong peak function and f = 1, then f is a smooth point of BA . Then Theorem 2.1 completes the proof. 2 Recall that a Banach space X is said to be a locally uniformly convex if whenever there is a sequence {xn } ∈ BX with limn xn + x = 2 for some x ∈ SX , we have limn xn − x = 0. It is shown in [6] that if X is locally uniformly convex, then the set of norm attaining elements is dense in A(BX ). That is, the set consisting of f ∈ A(BX ) with f = |f (x)| for some x ∈ BX is dense in A(BX ). The following corollary gives a stronger result. Notice that if X is locally uniformly convex, then every point of SX is the strong peak point for A(BX ) [11]. Theorem 2.1 and Corollary 2.2 imply the following. Corollary 2.3. Suppose that X is a locally uniformly convex, complex Banach space. Then the set of all strong peak functions in A(BX ) is a dense Gδ subset of A(BX ). In particular, the set of all smooth points of BA(BX ) contains a dense Gδ subset of SA(BX ) . It is shown [5] that the set of all strong peak points for A(BX ) is dense in SX if X is an order continuous locally uniformly c-convex sequence space. (For the definition see [5].) Then by Theorem 2.1, we get the denseness of the set of all strong peak functions. Corollary 2.4. Let X be an order continuous locally uniformly c-convex Banach sequence space. Then the set of all strong peak functions in Au (BX : X) is a dense Gδ subset of Au (BX : X). Let Π(X) = {(x, x ∗ ) ∈ BX × BX∗ : x ∗ (x) = 1} be the topological subspace of BX × BX∗ , where BX (resp. BX∗ ) is equipped with norm (resp. weak-∗ compact) topology. The numerical radius of holomorphic functions was deeply studied in [12] and the denseness of numerical radius holomorphic functions is studied on the classical Banach spaces [2]. Recently the numerical strong peak function is introduced in [14] and the denseness of holomorphic numerical strong peak functions in A(BX : X) is studied in various Banach spaces. The function f ∈ A(BX : X) is said to be a numerical strong peak function if there is (x, x ∗ ) such that limn |xn∗ f (xn )| = v(f ) for some {(xn , xn∗ )}n in Π(X) implies that (xn , xn∗ ) converges to (x, x ∗ ) in Π(X). The function f ∈ A(BX : X) is said to be numerical radius attaining if there is (x, x ∗ ) in Π(X) such that v(f ) = |x ∗ f (x)|. An element in the intersection of the set of all strong peak functions and the set of all numerical strong peak functions is called a norm and numerical strong peak function of A(BX : X). Using the variational method again we obtain the following. Proposition 2.5. Suppose that the set Π(X) is complete metrizable and the set Γ = {(x, x ∗ ) ∈ Π(X): x ∈ ρA(BX ) ∩ sm(BX )} is a numerical boundary. That is, v(f ) = sup(x,x ∗ )∈Γ |x ∗ f (x)|

936


for each f ∈ A(BX : X). Then the set of all numerical strong peak functions in A(BX : X) is a dense Gδ subset of A(BX : X). Proof. Let A = A(BX : X) and let d be a complete metric on Π(X). In [14], it is shown that if Π(X) is complete metrizable, then the set of all numerical peak functions in A is a Gδ subset of A. We need prove the denseness. Let f ∈ A and > 0. Fix n 1 and set Un = g ∈ A: ∃(z, z∗ ) ∈ Γ with z∗ (f − g)(z) > sup x ∗ (f − g)(x): d (x, x ∗ ), (z, z∗ ) > 1/n . Then Un is open and dense in A. Indeed, fix h ∈ A. Since Γ is a numerical boundary for A, there is a point (w, w ∗ ) ∈ Γ such that ∗ w (f − h)(w) > v(f − h) − /2. Notice that d((x, x ∗ ), (w, w ∗ )) > 1/n implies that there is δn > 0 such that x − w > δn . Choose a peak function p ∈ A(BX ) such that p = 1 = |p(w)| and |p(x)| < 1/3 for x − w > δn and |w ∗ (f − h)(w) − p(w)| = |w ∗ f (w) − w ∗ h(w)| + |p(w)| = |w ∗ f (w) − w ∗ h(w)| + . Put g(x) = h(x) + · p(x)w. Then ∗ w (f − g)(w) = w ∗ f (w) − w ∗ h(w) − p(w) = w ∗ f (w) − w ∗ h(w) + > v(f − h) + /2 sup x ∗ (f − h)(x) − p(x)x ∗ (w): d (x, x ∗ ), (w, w ∗ ) > 1/n = sup x ∗ (f − g)(x): d (x, x ∗ ), (w, w ∗ ) > 1/n . That is, g ∈ Un . By the Baire category theorem there is a g ∈ Un with g < , and we shall show that f − g is a strong peak function. Indeed, g ∈ Un implies that there is (zn , zn∗ ) ∈ Γ such that ∗ z (f − g)(zn ) > sup x ∗ (f − g)(x): d (x, x ∗ ), (zn , z∗ ) > 1/n . n n Thus d((zp , zp∗ ), (zn , zn∗ )) 1/n for every p > n, and hence {(zn , zn∗ )} converges to a point (z, z∗ ), say. Suppose that there is another sequence {(xk , xk∗ )} in Π(X) such that {|xk∗ (f − g)(xk )|} converges to v(f − g). Then for each n 1, there is Mn 1 such that for every m Mn , ∗ x (f − g)(xm ) > sup x ∗ (f − g)(x): d (x, x ∗ ), (zn , z∗ ) > 1/n . m n ∗ ), (z , z∗ )) 1/n for every m M . Hence {(x , x ∗ )} converges to (z, z∗ ). This Then d((xm , xm n n n m m shows that f − g is a numerical strong peak function at (z, z∗ ). 2

Corollary 2.6. Suppose that X is separable, smooth and locally uniformly convex. Then the set of norm and numerical strong peak functions is a dense Gδ subset of Au (BX : X).


937

Proof. It is shown [14] that if X is separable, Π(X) is complete metrizable. In view of [22, Theorem 2.5], Γ is a numerical boundary for Au (BX : X). Hence Proposition 2.5 shows that the set of all numerical strong peak functions is dense in Au (BX : X). Finally, Corollary 2.3 implies that the set of all norm and numerical peak functions is a dense Gδ subset of Au (BX : X). 2 Corollary 2.7. Let X be an order continuous locally uniformly c-convex, smooth Banach sequence space. Then the set of all norm and numerical strong peak functions in Au (BX : X) is a dense Gδ subset of Au (BX : X). Proof. Notice that X is separable since X is order continuous. Hence the set of all smooth points of BX is dense in SX by the Mazur theorem and Π(X) is complete metrizable [14]. In view of [22, Theorem 2.5], Γ is a numerical boundary for Au (BX : X). Hence Proposition 2.5 shows that the set of all numerical strong peak functions is a dense Gδ subset of Au (BX : X). Theorem 2.1 also shows that the set of all strong peak functions is a dense Gδ subset of Au (BX : X). This completes the proof. 2 3. Denseness of strongly norm attaining polynomials Recall that the norm · of a Banach space is said to be Fréchet differentiable at x ∈ X if x + δy + x − δy − 2x = 0. δ y=1

lim sup

δ→0

It is well known that the set of Fréchet differentiable points in a Banach space is a Gδ subset [3, Proposition 4.16]. Ferrera [9] shows that in a real Banach space X, the norm of P(n X) is Fréchet differentiable at Q if and only if Q strongly attains its norm. A set {xα } of points on the unit sphere SX of X is called uniformly strongly exposed (u.s.e.), if there are a function δ() with δ() > 0 for every > 0, and a set {fα } of elements of norm 1 in X ∗ such that for every α, fα (xα ) = 1, and for any x, x 1 and Re fα (x) 1 − δ() imply x − xα . Lindenstrauss [21, Proposition 1] showed that if BX is the closed convex hull of a set of u.s.e. points, then X has property A, that is, for every Banach space Y , the set of norm-attaining elements is dense in L(X, Y ), the Banach space of all bounded operators from X into Y . The following theorem gives stronger result. Theorem 3.1. Let F be the real or complex scalar field and X, Y Banach spaces over F. For k 1, suppose that the u.s.e. points {xα } in SX is a norming subset of P(k X). Then the set of strongly norm attaining elements in P(k X : Y ) is dense. In particular, the set of all points at which the norm of P(n X) is Fréchet differentiable is a dense Gδ subset. Proof. Let {xα } be a u.s.e. points and {xα∗ } be the corresponding functional which uniformly strongly exposes {xα }. Let A = P(k X : Y ), f ∈ A and > 0. Fix n 1 and set

Un = g ∈ A: ∃z ∈ ρA with (f − g)(z) > sup (f − g)(x): inf x − λz > 1/n . |λ|=1

938


Then Un is open and dense in A. Indeed, fix h ∈ A. Since {xα } is a norming subset of A, there is a point w ∈ {xα } such that (f − h)(w) > f − h − δ(1/n). Set g(x) = h(x) − · p(x)k

f (w) − h(w) , f (w) − h(w)

where p is a strongly exposed functional at w such that |p(x)| < 1 − δ(1/n) for inf|λ|=1 x − λw > 1/n, p(w) = 1. Then g − h and (f − g)(w) = f (w) − h(w) + p(w)k f (w) − h(w) = f (w) − h(w) + f (w) − h(w) > f − h + 1 − δ(1/n) k f (w) − h(w) (f − h)(x) − p(x) sup : inf x − λw > 1/n f (w) − h(w) |λ|=1

= sup (f − g)(x): inf x − λw > 1/n . |λ|=1

That is, g ∈ Un . By the Baire category theorem there is a g ∈ Un with g < , and we shall show that f − g is a strong peak function. Indeed, g ∈ Un implies that there is zn ∈ X such that

(f − g)(zn ) > sup (f − g)(x): inf x − λzn > 1/n . |λ|=1

Thus inf|λ|=1 zp − λzn 1/n for every p > n, and inf|λ|=1 |zp∗ (zn ) − λ| = 1 − |zn∗ (zp )| 1/n for every p > n. So limn infp>n |zn∗ (zp )| = 1. Hence there is a subsequence of {zn } which converges to z, say by [1, Lemma 6]. Suppose that there is another sequence {xk } in BX such that {(f − g)(xk )} converges to f − g. Then for each n 1, there is Mn such that Mn n and for every m Mn ,

(f − g)(xm ) > sup (f − g)(x): inf x − λzn > 1/n . |λ|=1

Then inf|λ|=1 xm − λzn 1/n for every m Mn . So inf|λ|=1 xm − λz inf|λ|=1 xm − λzn + z − zn 2/n for every m Mn . Hence we get a convergent subsequence of xn of which limit is λz for some λ ∈ SC . This shows that f − g strongly norm attains at z. It is shown in [6] that the norm is Fréchet differentiable at P if and only if whenever there are sequences {tn }, {sn } in BX and scalars α, β in SF such that limn αP (tn ) = limn βP (sn ) = P , we get lim sup αQ(tn ) − βQ(sn ) = 0. n

Q=1

(3.1)


939

We have only to show that every nonzero element P in A which strongly attains its norm satisfies the condition (3.1). Suppose that P strongly attains its norm at z and P = 0. For each Q ∈ A, there is a k-linear form L such that Q(x) = L(x, . . . , x) for each x ∈ X. The polarization identity [8] shows that Q L (k k /k!)Q. Then for each x, y ∈ BX , Q(x) − Q(y) nLx − y and k+1 Q(x) − Q(y) k Qx − y. k!

Suppose that there are sequences {tn }, {sn } in BX and scalars α, β in SF such that limn αP (tn ) = limn βP (sn ) = P , then for any subsequences {sn } of {sn } and {tn } of {tn }, there are convergent further subsequences {tn

} of {tn } and {sn

} of {sn } and scalars α

and β

in SF such that limn tn

= α

z and limn sn

= β

z. Then αP (α

z) = βP (β

z) = 1. So α(α

)k = β(β

)k . Then we get

lim sup αQ tn

− βQ sn

lim sup αQ tn

− αQ(α

z) + βQ(β

z) − βQ sn

n

Q=1

n

Q=1

lim n

k k+1 t

− α

z + β

z − s

= 0. n n k!

This implies that limn supQ=1 |αQ(tn ) − βQ(sn )| = 0. Therefore the norm is Fréchet differentiable at P . This completes the proof. 2 Remark 3.2. Suppose that the BX is the closed convex hull of a set of u.s.e. points, then the set of u.s.e. points is a norming subset of X ∗ = P(1 X). Hence the elements in X ∗ at which the norm of X ∗ is Fréchet differentiable is a dense Gδ subset. 4. Polynomial numerical index and graph theory A norm · on Rn or Cn is said to be an absolute norm if (a1 , . . . , an ) = (|a1 |, . . . , |an |) for every scalar a1 , . . . , an , and (1, 0, . . . , 0) = · · · = (0, . . . , 0, 1) = 1. We may use the fact that the absolute norm is less than or equal to the 1 -norm and it is nondecreasing in each variable. A real or complex Banach space X is said to be a CL-space if its unit ball is the absolutely convex hull of every maximal convex subset of the unit sphere. In particular, if X is finite dimensional, then it is equivalent to the condition that |x ∗ (x)| = 1 for every x ∗ ∈ ext BX∗ and every x ∈ ext BX [20]. Let X be an n-dimensional Banach space with an absolute norm · . In this section, X as a vector space is considered Rn or Cn and we denote by {ej }nj=1 and {ej∗ }nj=1 the canonical basis and the coordinate functionals, respectively. We also denote by ext BX the set of all extreme points of BX . Now define the following mapping between n-dimensional Banach spaces with an absolute norm and certain graphs with n vertices: X → G = G(X), where G is a graph with the vertex set V = {1, 2, . . . , n} and the edge set E = {(i, j ): / BX }. For example, if X = n1 , then G(X) is a complete graph of order n, that is, a e i + ej ∈

940


graph in which every pair of n vertices is connected by an edge. Conversely, G(X) is a graph in which no pairs of n vertices are connected by an edge if X = n∞ . Using these graphs and their theory, Reisner gave the exact characterization of all finite dimensional CL-spaces with an absolute norm. Prior to Reisner’s theorem, we give some basic definitions in the graph theory. Given a graph G = (V , E), we say that σ ⊂ V is a clique of G if the edge set E of G contains all pairs consisting of any two vertices in σ . Conversely, τ ⊂ V is called a stable set of G if E contains no pairs consisting of two vertices in τ . A graph G is said to be perfect if ω(H ) = χ(H )

for every induced subgraph H of G,

where ω(G) denotes the clique number of G (the largest cardinality of a clique of G) and χ(G) is the chromatic number of G (the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color). Theorem 4.1. (See [23, Reisner].) Let X be a finite dimensional Banach space with an absolute norm. Then X is a CL-space if and only if G = G(X) is a perfect graph and there exists a unique common element between every maximal clique and each maximal stable set of G. In particular, for every n-dimensional CL-space X, the following characterizations of the set of extreme points of BX and BX∗ hold respectively: (1) (x1 , x2 , . . . , xn ) ∈ ext BX if and only if |xj | ∈ {0, 1} for all j = 1, 2, . . . , n and the support {j ∈ V : xj = 0} is a maximal stable set of G. (2) (x1 , x2 , . . . , xn ) ∈ ext BX∗ if and only if |xj | ∈ {0, 1} for all j = 1, 2, . . . , n and the support {j ∈ V : xj = 0} is a maximal clique of G. In this theorem, the maximality of cliques and stable sets comes from the partial order of inclusion. Let X be a finite dimensional CL-space. If τ is a maximal clique of G = G(X), then a sub|τ | to 1 . Indeed, since the absolute norm space span{ej : j ∈ τ } of X is isometrically isomorphic is less than or equal to the 1 -norm, we have j ∈τ aj ej j ∈τ |aj | for every scalar aj . For the inverse inequality, let xτ∗ = j ∈τ sign(aj ) · ej∗ . Then xτ∗ is in ext BX∗ by Theorem 4.1 and hence xτ∗ ( j ∈τ aj ej ) = j ∈τ |aj |, which completes the proof. Remark 4.2. Originally, Reisner just proved the above theorem for the real case. However, it can be extended to the general case (real or complex). For this, we need the following comments and proposition. There is a natural one-to-one correspondence between the absolute norm of Rn and the one of Cn . Specifically, given real Banach space X = (Rn , · ) with an absolute norm, we can find the complexification X˜ = (Cn , · C ) of X defined by (z1 , . . . , zn )C := (|z1 |, . . . , |zn |) for ˜ Then X˜ is clearly the complex Banach space with the absolute norm. each (z1 , . . . , zn ) ∈ X. Moreover we get the following basic proposition. Proposition 4.3. Let X be a real Banach space with an absolute norm and X˜ the complexification of X. Then, for an element x of X, x ∈ ext BX if and only if x ∈ ext BX˜ . In particular, X is a CLspace if and only if X˜ is a CL-space.


941

Proof. The sufficiency is clear. Suppose that x is an extreme point of BX and 2x = y + z for some y, z in BX˜ . We claim that y and z are in X. For the contrary, suppose that some j th-coordinate of y is not a real number. That is, ej∗ (y) = a + bi, a, b ∈ R and b = 0. Since 2x = y + z = Re(y) + Re(z) + i(Im(y) + Im(z)), we have Re(y) = Re(z) = x, where Re(x) = nk=1 Re(ek∗ (x))ek and Im(x) = nk=1 Im(ek∗ (x))ek . Take a positive real number δ less √ than a 2 + b2 − |a|. Note that x ± δej y 1 by the basic property of an absolute norm. So 2x = (x + δej ) + (x − δej ) contradicts to the fact that x is an extreme point of BX . 2 Using the graph-theoretic technique on CL-spaces, we are about to find the strongly norm attaining points of ρP( ˜ m X). For this, let us consider the following lemma. Lemma 4.4. Let Y = N 1 and let m be a positive integer. For any j1 , j2 , . . . , jm ∈ {1, 2, . . . , N }, define an m-homogeneous polynomial Qj1 ,j2 ,...,jm of Y by Qj1 ,j2 ,...,jm (x) =

m

ej∗k (x) +

k=1

Then Qj1 ,j2 ,...,jm attains its norm only at its norm at m1 m k=1 ejk .

c m

m

ej∗ (x)

.

j ∈{j1 ,...,jm }

m

k=1 ejk , |c| = 1. Hence Qj1 ,j2 ,...,jm

strongly attains

Proof. For positive integers m1 , . . . , mn , consider the product (x1 , . . . , xn ) →

n

xkmk

k=1

on the compact subset Rn+ ∩ Sn1 . Then it is easy to see by induction that the product has the m1 mn ∗ , . . . , m1 +···+m ). Hence the norm of the polynomial m unique maximum at ( m1 +···+m k=1 ejk (x) n n ∗ is attained only at x = (x1 , . . . , xN ), where |xj | = m1 m k=1 ej (ejk ) for each 1 j N . Notice N ∗ also that the norm of the polynomial ( j =1 en (x))m is attained only at x = (x1 , . . . , xN ), where sign(x1 ) = · · · = sign(xN ) and x ∈ SN . Hence Qj1 ,j2 ,...,jm attains its norm only at mc m k=1 ejk 1 for some c ∈ SC . This completes the proof. 2 Theorem 4.5. Let X be a finite dimensional CL-space with an absolute norm (real or complex) ˜ m X) whenever y1 , y2 , . . . , ym are extreme and let m be a positive integer. Then m1 m j =1 yj ∈ ρP( points of BX whose coordinates are nonnegative real numbers. Proof. Denote by M(G) the family of all maximal cliques of G = G(X) and let y1 , y2 , . . . , ym be extreme points of BX whose coordinates are nonnegative real numbers. For each J ∈ M(G), define the m-homogeneous polynomial QJ and linear functional LJ as the following QJ = Qj1 ,j2 ,...,jm ,

LJ =

ej∗ ,

j ∈{j1 ,...,jm }

where each jk (k = 1, 2, . . . , m) is a unique common element between a maximal clique J and the support of an extreme point yk . Now define an m-homogeneous polynomial Q of X by

942


Q=

QJ +

m

J ∈M(G)

LJ

.

J ∈M(G)

For every maximal clique J of G, denote by PJ the projection of X onto span{ej : j ∈ J }. Then, it is clear that PJ

1 yj m m

=

j =1

1 1 PJ (yj ) = e jk . m m m

m

j =1

k=1

Notice that QJ ◦ PJ = QJ and LJ ◦ PJ = LJ for each J ∈ M(G). It follows by Theorem 5.1 that

1 Q yj m m

=

j =1

QJ

J ∈M(G)

=

QJ

+

j =1

QJ PJ

J ∈M(G)

=

m

J ∈M(G)

=

1 yj m

m

j =1

m

+

QJ +

1 yj m

m

j =1

LJ PJ

LJ

1 yj m m

m

j =1

J ∈M(G)

J ∈M(G)

m

+

k=1

J ∈M(G)

LJ

J ∈M(G)

1 yj m

1 e jk m

1 e jk m m

m

k=1

m

LJ

J ∈M(G)

= Q, and that Q attains its norm at m1 m j =1 yj . Then we claim that the above polynomial Q strongly attains its norm at m1 m j =1 yj . Now, suppose that |Q(y)| = Q and we need to show that y = mc m y for some c ∈ SC . Then we j j =1 have the following inequalities: m Q(y) = QJ (PJ y) + LJ (PJ y) J ∈M(G)

J ∈M(G)

m QJ (PJ y) + LJ (PJ y) J ∈M(G)

J ∈M(G)

J ∈M(G)

QJ +

m

LJ

J ∈M(G)

= Q. Hence |QJ (PJ y)| = QJ , LJ (PJ y) = LJ for each J∈ M(G) and sign LJ (PJ y) are all the same for all J ∈ M(G). By Theorem 5.1, PJ y = cmJ m k=1 ejk for each J ∈ M(G). Since sign(LJ (PJ y)) = cJ and they are all the same for all J ∈ M(G), take cJ = c for all J ∈ M(G).


943

Since each maximal clique J induces every extreme point xJ∗ of BX∗ , the above equation implies that xJ∗ (y) = xJ∗ (PJ y) = xJ∗ PJ

c yj m m

= xJ∗

j =1

Consequently y =

c m

m

j =1 yj .

This completes the proof.

m c yj . m j =1

2

It was shown in [19] that a necessary and sufficient condition for a Banach space with the Radon–Nikodym Property to have the polynomial numerical index one can be stated as follows: n(k) (X) = 1 if and only if ∗ x (x) = 1 for all x ∈ ρP ˜ k X and x ∗ ∈ ext BX∗ .

(4.1)

The following theorem is a partial answer to Problem 43 in [15]: Characterize the complex Banach spaces X satisfying n(k) (X) = 1 for all k 2. Theorem 4.6. Let X be an n-dimensional complex Banach space with an absolute norm and let k be an integer greater than or equal to 2. Then n(k) (X) = 1 if and only if X is isometric to n∞ . Proof. Suppose that n(k) (X) = 1. Since n(k) (X) = 1 implies n(X) = 1 (i.e. X is a CL-space), we can apply Theorem 4.1. To show that X is isometric to n∞ , it suffices to prove that BX has only one extreme point whose coordinates are nonnegative real number. Suppose that there exist distinct two extreme points x, y of BX whose coordinates are all nonnegative. Because the supports of x and y are maximal stable sets, we can choose i, j ∈ {1, . . . , n} such that (i, j ) is in the edge set E = E(G) and ei∗ (x) = 1 = ej∗ (y), ej∗ (x) = 0 = ei∗ (y). Take a maximal clique τ of G = G(X) containing (i, j ). Then, by Theorem 4.1, i is a unique common element between the maximal clique τ and the support of x. Similarly, j is a unique common element between τ and the support of y. Now consider an xτ∗ ∈ ext BX∗ defined by xτ∗ (ek ) =

−1, if k = i, 1, if k ∈ τ \{i}, 0, if k ∈ / τ.

Then, using Theorem 4.1, we get xτ∗

x+y 2

=

xτ∗ (x) + xτ∗ (y) −1 + 1 = = 0. 2 2

However, since x+y ˜ 2 X) by Theorem 4.5, this contradicts to (4.1). 2 ∈ ρP( For the converse, note that ρP( ˜ k X) ⊂ extC BX by [19, Proposition 2.1]. It is also easy to see that extC BX = ext BX when X = n∞ . After all, it follows from (4.1) that n(k) (n∞ ) = 1. 2

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5. Characterization of complex extreme points Theorem 5.1. Let X = (Cn , · ) be an n-dimensional complex CL-space with an absolute norm. Then an element (a1 , a2 , . . . , an ) in X is a complex extreme point of BX if and only if (|a1 |, |a2 |, . . . , |an |) is a convex combination of extreme points of BX whose coordinates all are positive real numbers. In short, Rn+ ∩ extC BX = co(Rn+ ∩ ext BX ). Proof. Let (a1 , a2 , . . . , an ) be a complex extreme point of BX . If X˜ = (Rn , · ), then all of BX by Proposition extreme points of BX˜ are extreme points m 4.3. So an element x := λ y where (|a1 |, |a2 |, . . . , |an |) has an expression x = m j j j =1 j =1 λj = 1, λj > 0 and each yj is an extreme point of BX whose coordinates are real numbers. Now we claim that all coordinates of each yj are positive. For the contradiction, suppose that the first coordinate of some yj is negative. More specifically, assume that the first coordinate of yj is −1 if 1 j r and 1 otherwise. Then |a1 | < 1. Note that for yj = yj + 2e1 , j = 1, 2, . . . , r, r m 1, |a2 |, . . . , |an | = λj yj + λj yj j =1

j =r+1

∈ co(ext BX ) = BX . It follows that for all ε ∈ C with |ε| < 1 − |a1 |, x + εe1 = |a1 | + ε, |a2 |, . . . , |an | 1, |a2 |, . . . , |an | 1. So (|a1 |, |a2 |, . . . , |an |) is not a complex extreme point, m which is a contradiction. λ y where For the converse, let x = m j =1 j j j =1 λj = 1 and yj ∈ ext BX for all j = 1, 2, . . . , m. Suppose that x is not a complex extreme point of BX . Then there exists a nonzero y in X such that x + εy ∈ BX whenever |ε| 1. Take a maximal clique τ in V containing some vertices which are nonzero coordinates of y. Consider the projection Pτ of X onto the linear span Y of {ej : j ∈ τ }. Then it follows from assumption that Pτ x is also not a complex extreme point of BY . Moreover, we can check that Pτ x is on the unit sphere of X. Indeed, if an extreme point / τ , then we have xτ∗ (yj ) = 1 xτ∗ of BX∗ is defined by xτ∗ (ej ) = 1 for j ∈ τ and xτ∗ (ej ) = 0 for j ∈ for all j = 1, 2, . . . , m by Theorem 4.1. Consequently, xτ∗ (Pτ x) = xτ∗ (x) =

m j =1

λj xτ∗ (yj ) =

m

λj = 1.

j =1 |τ |

Now, from the fact that Y = span{ej : j ∈ τ } is isometrically isomorphic to 1 since τ is a clique, we get that every element of norm one in Y is a complex extreme point of BY [10]. This is a contradiction. 2 The above characterization of complex extreme points has some application in the theory about the numerical index of Banach spaces. Specifically we want to apply to the analytic numerical index of X defined by


945

na (X) = inf v(P ): P = 1, P ∈ P(X : X) . It is easily observed that 0 na (X) n(k) (X) n(X) 1 for every k 2. Note that a necessary and sufficient condition for a finite dimensional complex Banach space to have the analytic numerical index 1 can be stated using complex extreme points as follows (see [19]): na (X) = 1 if and only if ∗ x (x) = 1 for all x ∈ extC BX and x ∗ ∈ ext BX∗ .

(5.1)

After applying the characterization of complex extreme points to the above theorem, we have the following corollary by the same argument as the proof of Theorem 4.6. Corollary 5.2. Let X be an n-dimensional complex Banach space with an absolute norm. Then na (X) = 1 if and only if X is isometric to n∞ . As an immediate consequence of the above theorem, we get a corollary. For further details, we need some definitions about the Daugavet property. A function Φ ∈ ∞ (BX , X) is said to satisfy the (resp. alternative) Daugavet equation if the norm equality I d + Φ = 1 + Φ resp. sup I d + ωΦ = 1 + Φ ω∈SC

holds. If this happens for all weakly compact polynomials in P(X : X), we say that X has the (resp. alternative) p-Daugavet property. Similarly, X is said to have the k-order Daugavet property if the Daugavet equation is satisfied for all rank-one k-homogeneous polynomials in P(k X : X). Corollary 5.3. Let X be a finite dimensional complex Banach space with an absolute norm. Then the followings are equivalent: (a) (b) (b ) (c) (c ) (d) (e)

X has the alternative p-Daugavet property. X has the k-order Daugavet property for some k 2. X has the k-order Daugavet property for every k 2. n(k) (X) = 1 for some k 2. n(k) (X) = 1 for every k 2. na (X) = 1. X = n∞ .

Proof. Theorems 4.6 and 5.2 imply (c) ⇔ (e) and (d) ⇔ (e) respectively. (b) ⇒ (c) or (b ) ⇒ (c ) is induced by [4, Proposition 1.3]. It is also easy to check that (b) ⇒ (c) ⇔ (e) ⇑ ⇑ (a) ⇒ (b ) ⇒ (c ) ⇐ (d) Corollary 2.10 in [4] shows (e) ⇒ (a). The proof is done.

2

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Remark 5.4. Let X be a finite dimensional (real or complex) Banach space with an absolute norm. Then it is impossible that X has the p-Daugavet property. Indeed, if X is a real (resp. complex) Banach space, then n(k) (X) = 1 for all k 2 and X = R (resp. X = n∞ for some n) (see [19] and Corollary 4.6). Then it is easy to see that both R and complex n∞ do not have p-Daugavet property. It is worth to note that the complex extreme points play an important role in the study of norming subsets of A(BX ). In particular, ρA(BX ) = extC BX when X is a finite dimensional complex Banach space (see [5]). Using the result in [13], we obtain the following corollary. For the further references about complex convexity and monotonicity, see [16–18]. Corollary 5.5. Let X = (Rn , · ) be an n-dimensional real CL-space with an absolute norm. Then an element (|a1 |, |a2 |, . . . , |an |) in SX is a point of upper monotonicity of X if and only if (|a1 |, |a2 |, . . . , |an |) is a convex combination of extreme points of BX whose coordinates all are positive real numbers. Acknowledgment The latter part of this paper is based on the first named author’s master thesis which is supervised by Prof. Yun Sung Choi. Furthermore, the authors wish to thank him for his comments about this subject. References [1] María D. Acosta, Denseness of numerical radius attaining operators: Renorming and embedding results, Indiana Univ. Math. J. 40 (3) (1991) 903–914. [2] María D. Acosta, Sung Guen Kim, Denseness of holomorphic functions attaining their numerical radii, Israel J. Math. 161 (2007) 373–386. [3] Yoav Benyamini, Joram Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Amer. Math. Soc. Colloq. Publ., vol. 48, American Mathematical Society, Providence, RI, 2000. [4] Yun Sung Choi, Domingo García, Manuel Maestre, Miguel Martín, The Daugavet equation for polynomials, Studia Math. 178 (1) (2007) 63–82. [5] Yun Sung Choi, Kwang Hee Han, Han Ju Lee, Boundaries for algebras of holomorphic functions on Banach spaces, Illinois J. Math. 51 (3) (2007) 883–896. [6] Yun Sung Choi, Han Ju Lee, Hyun Gwi Song, Bishop’s theorem and differentiability of a subspace of Cb (K), preprint, http://arxiv.org/abs/0708.4069, 2007. [7] Robert Deville, Gilles Godefroy, Václav Zizler, A smooth variational principle with applications to Hamilton– Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1) (1993) 197–212. [8] Seán Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer Monogr. Math., Springer-Verlag London Ltd., London, 1999. [9] Juan Ferrera, Norm-attaining polynomials and differentiability, Studia Math. 151 (1) (2002) 1–21. [10] Josip Globevnik, On complex strict and uniform convexity, Proc. Amer. Math. Soc. 47 (1975) 175–178. [11] Josip Globevnik, Boundaries for polydisc algebras in infinite dimensions, Math. Proc. Cambridge Philos. Soc. 85 (2) (1979) 291–303. [12] Lawrence A. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971) 1005–1019. [13] Henryk Hudzik, Agata Narloch, Relationships between monotonicity and complex rotundity properties with some consequences, Math. Scand. 96 (2) (2005) 289–306. [14] Sung Guen Kim, Han Ju Lee, Norm and numerical peak holomorphic functions on Banach spaces, preprint, http:// arxiv.org/abs/0706.0574, 2007. [15] Vladimir Kadets, Miguel Martín, Rafael Payá, Recent progress and open questions on the numerical index of Banach spaces, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (1–2) (2006) 155–182.


[16] [17] [18] [19] [20]

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Han Ju Lee, Monotonicity and complex convexity in Banach lattices, J. Math. Anal. Appl. 307 (1) (2005) 86–101. Han Ju Lee, Complex convexity and monotonicity in quasi-Banach lattices, Israel J. Math. 159 (2007) 57–91. Han Ju Lee, Randomized series and geometry of Banach spaces, preprint, http://arxiv.org/abs/0706.3740, 2007. Han Ju Lee, Banach spaces with polynomial numerical index 1, Bull. Lond. Math. Soc. 40 (2008) 193–198. Åsvald Lima, Intersection properties of balls in spaces of compact operators, Ann. Inst. Fourier (Grenoble) 28 (1978) 35–65. [21] Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963) 139–148. [22] Ángel Rodríguez Palacios, Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space, J. Math. Anal. Appl. 297 (2) (2004) 472–476, special issue dedicated to John Horváth. [23] Shlomo Reisner, Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases, J. London Math. Soc. (2) 43 (1) (1991) 137–148.


Composition formulas in the Weyl calculus Toshiyuki Kobayashi a , Bent Ørsted b , Michael Pevzner c , André Unterberger c,∗ a Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan b Matematisk Institut, Byg. 430, Ny Munkegade, 8000 Aarhus C, Denmark c Laboratoire de Mathématiques, CNRS FRE 3111, Université de Reims, Moulin de la Housse, B.P. 1039,

F-51687 Reims Cedex 2, France Received 25 December 2008; accepted 30 December 2008 Available online 26 January 2009 Communicated by Paul Malliavin

Abstract In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in R n × R n ) as integral superpositions of homogeneous ones, of degrees lying on the complex line with real part −n. It extends the one known in the one-dimensional case in connection with automorphic pseudodifferential analysis. © 2009 Elsevier Inc. All rights reserved. Keywords: Weyl calculus; Composition formulas; Principal series representations; Automorphic symbols and triple products

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposing the action of the symplectic group on L2 (Rn × Rn ) ε ,ε ;ε The integral kernel Jν 1,ν 2;ν (Y, Z; X) . . . . . . . . . . . . . . . . . . . . 1 2 Hyperplane waves and rays . . . . . . . . . . . . . . . . . . . . . . . . . . . Some one-dimensional preparation . . . . . . . . . . . . . . . . . . . . . .

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949 952 960 968 974


E-mail addresses: [email protected] (T. Kobayashi), [email protected] (B. Ørsted), [email protected] (M. Pevzner), [email protected] (A. Unterberger). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.023

T. Kobayashi et al. / Journal of Functional Analysis 257 (2009) 948–991

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6. Another composition of Weyl symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 7. Irreducibility of the decomposition of L2 (R2n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991

1. Introduction No symbolic calculus of operators is more popular or better known than the Weyl calculus. It is the one that associates to a function S = S(x, ξ ) of n + n variables, lying in S(Rn × Rn ), the operator Op(S), called the operator with symbol S, defined by the equation Op(S)u (x) =

S

Rn ×Rn

x +y , η e2iπx−y,η u(y) dy dη: 2

(1.1)

such a linear operator extends as a continuous operator from S (Rn ) to S(Rn ) while, in the case when S ∈ S (Rn × Rn ), one can still define Op(S) as a linear operator from S(Rn ) to S (Rn ); also, Op sets up an isometry from L2 (Rn × Rn ) onto the space of Hilbert–Schmidt operators on L2 (Rn ). The sharp composition S1 # S2 of two symbols, say lying in S(Rn × Rn ), is that which makes the formula Op(S1 )Op(S2 ) = Op(S1 # S2 ),

(1.2)

in which the left-hand side denotes the usual composition of operators, valid. The image of the Heisenberg representation is the group of unitary transformations exp(2iπ(η, Q − y, P − t)) of L2 (Rn ), as made meaningful by Stone’s theorem, where the j th component of the vector Q = (Q1 , . . . , Qn ) is the multiplication by the j th coordinate xj , 1 ∂ n n n 2 P = (P1 , . . . , Pn ) with Pj = 2iπ ∂xj , and y, η ∈ R , t ∈ R. Introducing on (R × R ) the symplectic form [,] such that (x, ξ ), (y, η) = −x, η + y, ξ ,

(1.3)

let us use on Rn × Rn the symplectic Fourier transformation F defined by the equation (F S)(X) =

S(Y )e−2iπ[X,Y ] dY,

(1.4)

Rn ×Rn

which commutes with all symplectic linear transformations of the variable in Rn × Rn . Another, fully equivalent, way to define the Weyl calculus is by means of the equation Op(S) =

(F S)(y, η) exp 2iπ η, Q − y, P dy dη.

Rn ×Rn

The first covariance rule of the Weyl calculus is the observation that

(1.5)

950


exp 2iπ η, Q − y, P Op(S) exp −2iπ η, Q − y, P = Op (x, ξ ) → S(x − y, ξ − η) .

(1.6)

One way to emphasize this action on symbols of the group of translations of R2n is to decompose in a systematic way the space of symbols L2 (R2n ) with respect to this action. Now, the operators which commute with it are just the partial differential operators with constant coefficients: the generalized joint eigenfunctions of these are exactly the exponentials X = (x, ξ ) → e2iπ[A,X] with A ∈ R2n , and the sought-after decomposition of a symbol is provided by the symplectic Fourier transformation. On the other hand, if A = (y, η), the operator with symbol e2iπ[A,X] is none other than the operator exp(2iπ(η, Q − y, P )), so that Heisenberg’s commutation relation, expressed in Weyl’s exponential version, takes the form 1 ,X]

e2iπ[A

2 ,X]

# e2iπ[A

1 ,A2 ]

= eiπ[A

1 +A2 ,X]

e2iπ[A

.

(1.7)

Before coming to the point of the present work, let us briefly recall a few immediate consequences of this relation. First, one has (say, when S1 and S2 lie in S(R2n )), using (1.5), the integral composition formula S1 (Y )S2 (Z)e−4iπ[Y −X,Z−X] dY dZ (1.8) (S1 # S2 )(X) = 22n R2n ×R2n

or (a fully equivalent one) (S1 # S2 )(X) = exp(iπL) S1 (Y )S2 (Z)

(Y = Z = X)

(1.9)

with (setting Y = (y, η), Z = (z, ζ )) n ∂2 1 ∂2 − . iπL = + 4iπ ∂yj ∂ζj ∂zj ∂ηj

(1.10)

j =1

Expanding the exponential into a series, one obtains the so-called Moyal formula (S1 # S2 )(x, ξ )

β α (−1)|α| 1 |α|+|β| ∂ α ∂ β ∂ ∂ = S1 (x, ξ ) S2 (x, ξ ). α!β! 4iπ ∂x ∂ξ ∂x ∂ξ

(1.11)

This formula is an exact one in the case when the two operators under consideration are differential operators, which means exactly that their symbols (of course, not in S(R2n )) are polynomial with respect to the variables ξ , with coefficients depending on x in a smooth, but otherwise fairly arbitrary way; it is also exact when one of the two symbols is a polynomial in (x, ξ ). As it turns out, this version of the composition formula is the only universally known one. Indeed, it has considerable importance in applications of pseudodifferential analysis to partial differential equations: classes of symbols for which the above formula, without being an exact one, still has some asymptotic value, provide a good proportion of the auxiliary operators needed for the solution of PDE problems. In a conclusion, however, we shall illustrate on one example


951

while this may sometimes fail and call instead for the composition formula which is the object of the present paper. Our derivation of (1.8) was obtained as the result of pairing the concept of sharp composition of symbols with the decomposition of symbols according to the action by translations of the group R2n : the success of this point of view was essentially dependent on the fact that this action is an ingredient of the covariance formula (1.6). This takes us to the aim of the present paper: to take advantage of the other covariance property of the Weyl calculus—to be recalled now—and follow the same policy. Recall that the metaplectic representation Met in L2 (Rn ) is a certain unitary representation [15] of the twofold cover of the symplectic group Sp(n, R), which consists of all linear transformations g of Rn × Rn such that [gX, gY ] = [X, Y ] for every pair (X, Y ) of points of Rn × Rn : it acts irreducibly on each of the two subspaces of L2 (Rn ) consisting of functions with a given parity. Unitary transformations in the image of the metaplectic representation also act as automorphisms of the space S(Rn ) or of the space S (Rn ): moreover, if such a unitary transformation U lies above g ∈ Sp(n, R), and if S ∈ S (R2n ), one has the covariance formula U Op(S)U −1 = Op S ◦ g −1 .

(1.12)

In full analogy with the procedure adopted above in connection with the Heisenberg representation, we now start from a decomposition of the phase space representation (g, S) → S ◦ g −1 of Sp(n, R) in L2 (R2n ) into irreducibles: this is just the same as decomposing functions in L2 (R2n ) as integral superpositions of functions homogeneous of a given degree, and with a given parity. Our main result is the formula which takes the place of (1.7): it decomposes the sharp product of two symbols h1 and h2 , homogeneous of degrees −n − iλ1 and −n − iλ2 and with parities characterized by indices δ1 and δ2 , as an integral superposition of functions homogeneous of degrees −n − iλ, with the parity δ ≡ δ1 + δ2 . It involves the integral kernel

−n−iλ+iλ1 −iλ2

−n−iλ−iλ1 +iλ2

−n+iλ+iλ1 +iλ2

2 2 2

[Y, X]

[X, Z]

[Z, Y ]

, ε ε ε 2

1

(1.13)

a product of three signed powers, obtained from the decomposition into homogeneous components with respect to the three variables of the integral kernel which occurs in the composition formula (1.8). Some preparation is needed in order to give this kernel a genuine meaning as a distribution, not only as a partially defined function. The principle of the proof of the new composition formula is simple, and relies on the decomposition of symbols into hyperplane waves, and the dual notion of rays. Its main difficulty lies in the singular nature of such distributions, which are nevertheless the only ones, sufficiently general, for which explicit computations are possible. In the one-dimensional case, the integral kernel above reduces to a function −1−iλ+iλ1 −iλ2 2

J (x, y, z) = |x − y|ε2

−1−iλ−iλ1 +iλ2 2

|z − x|ε1

−1+iλ+iλ1 +iλ2 2

|y − z|ε

(1.14)

of three real variables, and the composition formula was treated along these lines in [12, Section 17]. It is true that the proof, in the higher-dimensional case, is actually, for the main part, a reduction to the one-dimensional case: but signed powers of linear forms with exponents lying on the line −n + iR, the consideration of which is necessary for spectral-theoretic reasons,

952


are more singular distributions when n 2, which has made some technical improvements necessary. It may be interesting to recall briefly what can be done in the one-dimensional case in relation to automorphic distribution theory. In the automorphic situation, the integral kernel (1.14) enables one to build new nonholomorphic modular forms from given pairs of such. In [11], one of this paper’s authors introduced the notion of automorphic distribution: this is a distribution in R2 invariant under linear changes of coordinates associated to elements of some arithmetic subgroup of SL(2, R), for instance SL(2, Z). This concept is equivalent—in a non-trivial way—to the Lax–Phillips notion of pairs of non-holomorphic modular forms, as introduced in their scattering theory [7] for the automorphic wave equation. Automorphic distributions can be taken as symbols in the Weyl calculus and, at the price of important difficulties, the one-dimensional case of the analysis of sharp-products in the present paper can be developed in the automorphic environment. Things are more interesting, in some sense, since besides a continuous part, in which Eisenstein distributions serve as generalized eigenfunctions, the automorphic Euler operator has a discrete spectrum, and the corresponding eigendistributions are cusp-distributions. Finding the appropriate composition formulas calls for the explicit computation of integrals of J (x, y, z) against three non-holomorphic modular forms, in the realization of these as distributions on the line invariant under representations taken from the principal series of the arithmetic subgroup of SL(2, R) under consideration: this has been completed up to some large extent, for the case of the full modular group, in [12] (cf. in particular Section 16), and it provides a pseudodifferentialtheoretic approach to such notions as L-functions, convolution L-functions, etc. As a preparation for automorphic pseudodifferential analysis, and in view of other applications as well, either to arithmetic or to quantization theory, a study of the integral kernel (1.14) had been made in [11]. It has also been considered recently in [8], in the automorphic case (for its own sake, not in connection with pseudodifferential analysis), and we take it from the references there that, outside the automorphic environment, it had already appeared in [9]: note that the objects called automorphic distributions in [8] are not the same as those in [11,12] (they are close to what was called modular distributions in [11]). Obviously, it would be of great interest to push the present composition formula for n-dimensional pseudodifferential analysis up to an automorphic environment, despite the great difficulties experienced with automorphic pseudodifferential analysis in the one-dimensional case. In any case, linking pseudodifferential analysis to harmonic analysis, then to modular form theory (also the subject of [13], though the connection between these domains is different there) is certain to bring rewards in the future. In a non-automorphic environment, the basic idea put forward in the present paper, namely that of building composition formulas from the pairing of covariance with the decomposition of representations into irreducibles, may also [12, Section 19] be of use whenever some symbolic calculus of operators is examined, thus finding its place within quantization theory in general. Rn × R n ) 2. Decomposing the action of the symplectic group on L2 (R Consider the linear space Rn × Rn with its canonical symplectic form (1.3) and measure dx dξ : we also set, when convenient, X = (x, ξ ). The symplectic group G = Sp(n, R) is the group of linear transformations g of Rn × Rn which preserve the symplectic form, i.e., satisfy the identity [gX, gY ] = [X, Y ] for any pair X, Y of points of R2n . The phase space representation of G in L2 (R2n ) is defined by the action (g, h) → g.h such that (g.h)(X) = h(g −1 X). It is unitary, and since all linear transformations on Rn × Rn preserve the parity of functions and commute


953

with the Euler operator 2iπE =

xj

∂ ∂ + ξj +n ∂xj ∂ξj

(2.1)

(the additional constant turns E into a formally self-adjoint operator on L2 (Rn × Rn )), the (extension of the) phase space representation under study preserves the linear space of functions on R2n \ {0} homogeneous of a given degree, and with a given parity. Given h ∈ L2 (R2n ), we first decompose it into its even and odd parts. Then, setting for every real number s = 0 and α ∈ C |s|α0 = |s|α ,

|s|α1 = sα = |s|α sign s,

(2.2)

we may write h=

∞

hiλ,δ dλ,

(2.3)

δ=0,1 −∞

provided we set 1 hiλ,δ (X) = 4π

∞ |t|δn−1+iλ h(tX) dt.

(2.4)

−∞

Then, hiλ,δ is homogeneous of degree −n − iλ and has the parity associated to δ: we shall refer to the pair (−n − iλ, δ) as the type of hiλ,δ . More generally, we may consider on R2n \ {0} functions of type (−n − ν, δ) for an arbitrary complex parameter ν. So as to cut down, as is needed, the dimension by 1, one may realize functions of a given type as sections of some appropriate line bundle over the projective space P2n−1 (R). We first need to introduce the so-called tautological bundle EC over P2n−1 (R), the fibre of which above a point p(θ ) (p being the canonical map: R2n \ {0} → P2n−1 (R)) is the complex line Cθ in C2n . Incidentally, note that the total space of the real line analogue ER of this bundle is just the blown 2n which is used consistently for desingularization purposes, as will be the case in next up space R section. A canonical set of charts of P2n−1 (R) is obtained in the following way: given a vector θ S ∈ R2n \ {0}, set ΩS = {θ ∈ R2n : [θ, S] = 0} and, in ωS = p(ΩS ), take the chart p(θ ) → [θ,S] , which identifies ωS with the affine hyperplane MS = {X ∈ R2n : [X, S] = 1}. Above MS , a section of EC can be identified with a complex-valued function fS , associating to such a function the section X → fS (X)X. Note that, if X ∈ MS satisfies [X, T ] = 0 for some new vector X T ∈ R2n \ {0}, the points X ∈ MS and [X,T ] ∈ MT are truly the images, under the charts associated with S and T , of the same point in P2n−1 (R). Identifying fS (X)X with fT (Y )Y , where we X have set Y = [X,T ] , leads to the compatibility condition fT

X [X, T ]

= [X, T ]fS (X),

which defines the transition functions of the line bundle EC .

(2.5)

954

T. Kobayashi et al. / Journal of Functional Analysis 257 (2009) 948–991 μ

More generally, given (μ, δ) with μ ∈ C and δ = 0 or 1, define the signed power |(EC )|δ of EC by taking the corresponding signed powers of the transition functions: then, a section of the μ line bundle |(EC )|δ is associated to a set (fS ) of functions, fS defined in MS , satisfying the requirement that fT

X [X, T ]

μ

= [X, T ] δ fS (X)

(2.6)

whenever X ∈ M0 and [X, T ] = 0. Then, a function h of type (−n − ν, δ) can be identified with the section of |(EC )|n+ν characterized by the fact that, for every S ∈ R2n \ {0}, fS is the δ restriction of h to MS . Conversely, any function f in MS uniquely lifts as a function f in the part of R2n \ {0} consisting of vectors θ such that [θ, S] = 0, to wit the one defined by the equation

−n−ν

f (θ ) = [θ, S] δ f

θ . [θ, S]

(2.7)

The representation πν,δ from the full, non-unitary principal series of Sp(n, R) is by definition the restriction of the phase space representation of Sp(n, R) (again, this is defined by the assignment (g, h) → h ◦ g −1 ) to the space of functions in R2n \ {0} of type (−n − ν, δ). It will be convenient—but there is a price to pay—not to have to change the hyperplane MS consistently, and we denote as M0 the one which should really be denoted as Me1 (where e1 is the first vector from the canonical basis of Rn × Rn ), i.e., the one consisting of vectors X = (x; ξ ) ∈ Rn × Rn such that ξ1 = 1. Starting from (2.7) and using the fact that f is of type (−n − ν, δ), together with the relation [g −1 X, e1 ] = [X, ge1 ], one obtains the relation

−n−ν f πν,δ (g)f (X) = [X, ge1 ] δ

g −1 X . [X, ge1 ]

(2.8)

dx−b As an example, when n = 1 and g = ac db , starting from X = 1x , so that g −1 X = −cx+a , one x obtains, after one has abbreviated f 1 as f (x), the relation πν,δ (g)f (x) = | − cx + a|δ−1−ν f

dx − b . −cx + a

(2.9)

Still specializing, for the time being, in the hyperplane M0 , we set X = (x; ξ ) = (x1 , x∗ ; ξ1 , ξ∗ ),

(2.10)

and denote as hiλ,δ the restriction of hiλ,δ to M0 (it is the same as the function which would have been denoted as (hiλ,δ )e1 in the less specialized setting above). One has the reciprocal equations

hiλ,δ (x; ξ∗ ) = hiλ,δ (x; 1, ξ∗ ), x ξ∗ −n−iλ . hiλ,δ ; hiλ,δ (x; ξ ) = |ξ1 |δ ξ1 ξ1

(2.11)


955

Remark 2.1. Under the preceding pair of equations, the functions hiλ,δ and hiλ,δ are virtually indistinguishable, once the type (−n − iλ, δ) has been fixed. Using the second notion will be useful in connection with all concepts using integrals, such as integral operators, norms, . . . . However, the first point of view is more intrinsic, and is especially useful (since some singularities could lie “at infinity” relative to the chosen hyperplane M0 ) when, as will be the case in Section 4, we need to extend the representation πν,δ or the intertwining operator to be introduced below to a distribution setting. Proposition 2.1. The space L2 (R2n ) can be decomposed as the Hilbert direct integral ⊕ 2n L R ∼ Hiλ,δ dλ, 2

(2.12)

δ=0,1

if one denotes as Hiλ,δ the inverse image under the map hiλ,δ → hiλ,δ of the space L2 (M0 ; dx dξ∗ ): the decomposition is provided by (2.3), and it commutes with the phase space representation of G in L2 (R2n ). Proof. What remains to be done is proving the equation h2L2 (R2n )

∞ 2 h 2 = 4π iλ,δ L (M ) dλ, 0

(2.13)

δ=0,1 −∞

using on M0 the measure dx dξ∗ . Indeed, with h(δ) = heven or hodd according to the parity of δ, set (2.14) φX (s) = e2πns h(δ) e2πs X , s ∈ R, X ∈ R2n \ {0}, so that φˆ X (λ) = hiλ,δ (X).

(2.15)

The one-dimensional Fourier inversion formula then yields (2.3) (of course, using the Mellin transform rather than coupling a Fourier transform with the change of variable t = e2πs would be more natural: the choice really depends on your familiarity with the inversion formula in both cases). Next, using (2.11) and the Plancherel formula for the Fourier transformation,

∞ h(δ) 2L2 (R2n )

= 4π

e2πs ds −∞

R2n−1

∞ = 4π

h(δ) x; e2πs , ξ∗ 2 dx dξ∗

ds −∞

φ(x;1,ξ ) (s) 2 dx dξ∗ ∗

R2n−1

= 4π

∞ dx dξ∗

R2n−1

−∞

φˆ (x,1,ξ ) (s) 2 ds ∗

956


= 4π

∞ dx dξ∗

(2.16)

−∞

R2n−1

which proves (2.13).

hiλ,δ (x; ξ∗ ) 2 ds,

2

The decomposition above gives right to the series (πiλ,δ )λ∈R, δ=0,1 of representations of G in L2 (M0 ), a special case of the representations πν,δ already considered; it suffices to set

πiλ,δ (g)hiλ,δ = fiλ,δ

(2.17)

if h ∈ L2 (R2n ), g ∈ G, f = h ◦ g −1 . Each representation πiλ,δ (g) is unitary as a consequence of Proposition 2.1: to show that πiλ,δ (g)hiλ,δ = hiλ,δ for every λ such that hiλ,δ ∈ L2 (M0 ),

not only almost every λ, it suffices to start from a dense space of functions h such that hiλ,δ depends in a continuous way on λ, which is ensured for instance when h lies in S(R2n ). Recall (cf. Remark 2.1) that we also set πiλ,δ (g)hiλ,δ = fiλ,δ . In Section 7, it will be proved that most representations πiλ,δ are irreducible. Remark 2.2. When integrating on MS , we shall have to worry a lot about singularities: but we shall never have to worry about the contribution to integrals of the part of this hyperplane away from some compact subset because, in reality, we shall be dealing with integrals on the compact space P2n−1 (R) and (say, with the help of partitions of unity), we could always, replacing the integral under consideration by a finite sum of integrals taken on distinct hyperplanes, replace for each term the integral by the integral taken on some compact subset of the corresponding hyperplane. The (symplectic) Fourier transform of a function homogeneous of degree −n−iλ with a given parity is homogeneous of degree −n + iλ, and has the same parity, so that, given h ∈ L2 (R2n ), one has F hiλ,δ = (F h)−iλ,δ :

(2.18)

consequently, the representations πiλ,δ and π−iλ,δ are unitarily equivalent. Definition 2.2. The (unitary) intertwining operator θiλ,δ is the one characterized by the validity of the equation θiλ,δ hiλ,δ = (F h)−iλ,δ

(2.19)

for every h ∈ L2 (R2n ). We also set (cf. Remark 2.1)

θiλ,δ hiλ,δ = (F h)−iλ,δ .

(2.20)

The proof that θiλ,δ preserves the L2 -norm for every λ, not only almost every λ, is the same as the one which, in connection with the definition of πiλ,δ , followed (2.17). It is easy to make the unitary intertwining operator θiλ,δ associated to (2.18) explicit in terms of the coordinates on M0 . Indeed, starting from (2.11), one can write


957

(F h)−iλ,δ (x; ξ∗ ) = (F hiλ,δ )(x; 1, ξ∗ ) y η∗ exp 2iπ x1 η1 + x∗ , η∗ − y1 − y∗ , ξ∗ dy dη1 dη∗ = |η1 |δ−n−iλ hiλ,δ ; η1 η1 = |η1 |δn−1−iλ hiλ,δ (y; η∗ ) exp 2iπη1 x1 + x∗ , η∗ − y1 − y∗ , ξ∗ dy dη1 dη∗ . (2.21) Making a one-dimensional Fourier transformation explicit, this gives another approach to the intertwining operator θiλ,δ from πiλ,δ to π−iλ,δ : the operator θiλ,δ is defined formally as the operator with integral kernel 1

kiλ,δ (x, ξ∗ ; y, η∗ ) = i δ π 2 −n+iλ

( n−iλ+δ )

2

x1 − y1 + x∗ , η∗ − y∗ , ξ∗ −n+iλ . δ 1−n+iλ+δ ( ) 2

(2.22)

Note that, while Definition 2.2 is a rigorous definition of the intertwining operator, (2.22) can only be used after some preparation, which will be done in Section 3. While X = (x; ξ ) (or Y = (y; η), . . .) will always denote a generic point in R2n , we shall draw attention to points (x; 1, ξ∗ ) = (x1 , x∗ ; 1, ξ∗ ) of M0 by denoting them as X∗ : similarly, Y∗ = (y; 1, η∗ ). Given X∗ ∈ M0 , we set X∗∗ = (x∗ ; ξ∗ ), so that one can also identify X∗ with (x1 , X∗∗ ). We abbreviate the measure dx dξ∗ on M0 as dm(X∗ ). On R2n−2 , one can also consider the symplectic form obtained from an appropriate restriction of the one available on R2n , i.e., set [X∗∗ , Y∗∗ ] = −x∗ , η∗ + y∗ , ξ∗ ,

(2.23)

while, on M0 , one must define [X∗ , Y∗ ] =

(x1 , x∗ ); (1, ξ∗ ) , (y1 , y∗ ); (1, η∗ )

= −x1 + y1 − x∗ , η∗ + y∗ , ξ∗ .

(2.24)

One may then rewrite (2.22) as 1

(θiλ,δ f )(X∗ ) = i δ π 2 −n+iλ

( n−iλ+δ ) 2 ( 1−n+iλ+δ ) 2

[Y∗ , X∗ ] −n+iλ f (Y∗ ) dm(Y∗ ). δ

(2.25)

M0

The intertwining operator may be better understood after some transformation. Denote as F1 the usual Fourier transformation as applied when emphasis is set on the first variable only of a function of several variables. Given a function f on M0 , write it as hiλ,δ , which, according to (2.11), is possible in a unique way for a given pair (iλ, δ), so that the left-hand side of (2.21) is just (θiλ,δ f )(x; ξ∗ ) according to (2.18). Starting from (2.21), one can then write, if n 2,

958


(F1 θiλ,δ f )(t, x∗ ; ξ∗ ) = (F1 θiλ,δ f )(t, X∗∗ ) = |t|δn−1−iλ f (y1 , Y∗∗ ) exp −2iπt y1 + [X∗∗ , Y∗∗ ] dy1 dY∗∗ M0

(F1 f )(t, Y∗∗ ) exp −2iπt[X∗∗ , Y∗∗ ] dY∗∗ .

= |t|δn−1−iλ

(2.26)

R2n−2

In this definition of the intertwining operator, θiλ,δ appears as the “product” of a one-dimensional intertwining operator with respect to the first variable and of a Fourier transformation in R2n−2 : only, some rescaling, by the variable dual to the first one, is performed with respect to the last 2n − 2 variables. As a straightforward application of this equation, note the formula, in which δ2 := δ1 + δ, −i(λ1 +λ)

(F1 θiλ1 ,δ1 θiλ,δ f )(t, X∗∗ ) = |t|δ2

(F1 f )(t, X∗∗ ):

(2.27)

hence, the composition of the two intertwining operators under consideration reduces to an intertwining operator with respect to the first variable, with integral kernel (x1 , X∗∗ ), (y1 , X∗∗ ) 1

→ i δ2 π − 2 +i(λ1 +λ)

( 1−i(λ12+λ)+δ2 ) 2 ( i(λ1 +λ)+δ ) 2

1 +λ) δ(X∗∗ − Y∗∗ ). |x1 − y1 |δ−1+i(λ 2

(2.28)

At this point, it may be useful to clarify the respective roles of the coordinates ξ1 and x1 , as they occur in what precedes. Isolating the coordinate ξ1 is tantamount to singling out the affine hyperplane M0 , the equation of which is [X, e1 ] = 1, while [X, e1 ] = ξ1 generally. The expres∂f sion ∂x , for f ∈ C ∞ (M0 ), is then the image of f under a canonical operator on M0 , since 1 it may be thought of as the Poisson bracket of the function X → ξ1 with an arbitrary smooth extension of f to the whole of R2n . One may interpret the convolution operator the integral kernel of which is given in (2.28) as a function (a signed power, of course), in the sense of func1 ∂ tional calculus, of the operator 2iπ ∂x1 . On the other hand, the coordinate x1 is not intrinsically attached to M0 : with the help of a well-chosen symplectic transformation preserving the coordinate ξ1 , it can be transformed to the sum of x1 and of an arbitrary linear combination of x 2 , . . . , x n , ξ 1 , . . . , ξn . Note if f ∈ L2 (M0 ) the relation πiλ,δ (g)f = π−iλ,δ (g)f¯

(2.29)

from which, polarizing the identity which expresses that πiλ,δ is unitary, we obtain the identity

f2 (X)f1 (X∗ ) dm(X∗ ) = M0

M0

π−iλ,δ (g)f2 (X∗ ) πiλ,δ (g)f1 (X∗ ) dm(X∗ )

(2.30)


959

involving a pair (f1 , f2 ) of functions in L2 (M0 ): this can also be regarded as a particular case of (2.27), to the effect that the inverse of the isometry θiλ,δ is θ−iλ,δ . Assuming convergence, one can extend (2.30) as

π−ν,δ (g)f2 (X∗ ) πν,δ (g)f1 (X∗ ) dm(X∗ ).

f2 (X∗ )f1 (X∗ ) dm(X∗ ) = M0

(2.31)

M0

We now introduce the integral kernel obtained from the decomposition into homogeneous components of the integral kernel e4iπ[Y,X] e4iπ[X,Z] e4iπ[Z,Y ] which occurs in the composition formula (1.8). Consider on R2n × R2n × R2n the (almost everywhere defined only) function

α

α

α

(Y, Z; X) → [Y, X] ε 1 [X, Z] ε 2 [Z, Y ] ε 3 , 2

1

(2.32)

where the exponents and indices of parity are given. It is of type (α1 + α3 , ε + ε2 mod 2), resp. (α2 + α3 , ε + ε1 mod 2), resp. (α1 + α2 , ε1 + ε2 mod 2) with respect to Y , resp. Z, resp. X. Given a triple (ν1 , ν2 , ν) of complex numbers, and a triple (δ1 , δ2 , δ) of numbers equal to 0 or 1, satisfying the relation δ ≡ δ1 + δ2 mod 2, the system of equations ε2 + ε ≡ δ1 ,

ε1 + ε ≡ δ2 ,

ε1 + ε2 ≡ δ

(2.33)

for ε, ε1 , ε2 mod 2 has two solutions, obtained as ε ≡ j + δ,

ε1 ≡ j + δ1 ,

ε2 ≡ j + δ2

(2.34)

with j = 0 or 1. Then, the types of the function above with respect to Y, Z, X will be (−n + ν1 , δ1 ), (−n + ν2 , δ2 ) and (−n − ν, δ) if and only if α1 =

−n − ν + ν1 − ν2 , 2

−n − ν − ν1 + ν2 , 2

α2 =

α3 =

−n + ν + ν1 + ν2 . 2

(2.35)

Hence, provided that (2.33) is satisfied, the integral kernel

−n−ν+ν1 −ν2

−n−ν−ν1 +ν2

−n+ν+ν1 +ν2

;ε 2 2 2

[X, Z]

[Z, Y ]

Jνε11,ν,ε22;ν (Y, Z; X) = [Y, X] ε ε ε 2

1

(2.36)

in (R2n \ {0}) × (R2n \ {0}) × (R2n \ {0}) satisfies the covariance relation ;ε (Y, Z; X) πν,δ (g) X → Jνε11,ν,ε22;ν ;ε (Y, Z; X) . = π−ν1 ,δ1 g −1 ⊗ π−ν2 ,δ2 g −1 (Y, Z) → Jνε11,ν,ε22;ν

(2.37)

We may also restrict this integral kernel to M0 × M0 × M0 : the relation of covariance is preserved, though with a slightly different understanding (cf. (2.17)). In next section, we shall see, after we have given the integral kernel so obtained a meaning in an appropriate distribution sense, 2 ;ε not only as a partially defined function, that if one denotes as Jεν11 ,ε ,ν2 ;ν the associated operator, thought of as being defined by the equation

960


ε1 ,ε2 ;ε Jν1 ,ν2 ;ν (f1 , f2 ) (X∗ ) ;ε = Jνε11,ν,ε22;ν (Y∗ , Z∗ ; X∗ )f1 (Y∗ )f2 (Z∗ ) dm(Y∗ ) dm(Z∗ ),

(2.38)

M 0 ×M 0

one has the covariance identity ε1 ,ε2 ;ε 2 ;ε πν,δ (g) Jεν11 ,ε ,ν2 ;ν (f1 , f2 ) = Jν1 ,ν2 ;ν πν1 ,δ1 (g)f1 , πν2 ,δ2 (g)f2 ,

(2.39)

formally immediate from (2.37) and (2.31). In the case when f1 = (h1 )ν1 ,δ1 and f2 = (h2 )ν2 ,δ2 ,

ε1 ,ε2 ;ε 2 ;ε we can, and shall sometimes, write Jεν11 ,ε ,ν2 ;ν ((h1 )ν1 ,δ1 , (h2 )ν2 ,δ2 ) for Jν1 ,ν2 ;ν (f1 , f2 ). Also, as explained in Remark 2.1, the result can be regarded as a function in R2n \ {0} of type (−n − ν, δ) rather than, again, as being defined only on M0 . ε ,ε ;ε

3. The integral kernel Jν11,ν22;ν (Y, Z; X) In all this section, we deal with functions of a given type in their realizations as functions 2 ;ε on M0 . Rather than trying to define Jεν11 ,ε ,ν2 ;ν (f1 , f2 ), as in (2.38), as a function of X∗ , we lower our requirements, only trying to define the expression

2 ;ε Jεν11 ,ε ,ν2 ;ν (f1 , f2 ), f ;ε = Jνε11,ν,ε22;ν (Y∗ , Z∗ ; X∗ )f1 (Y∗ )f2 (Z∗ )f (X∗ ) dm(Y∗ ) dm(Z∗ ) dm(Z∗ )

(3.1)

M 0 ×M 0 ×M 0 2 ;ε for appropriate triples (f1 , f2 , f ). This is of course tantamount to a reinterpretation of Jεν11 ,ε ,ν2 ;ν ∞ as a distribution of some kind, a notion dependent on that of C -vectors of the representations πν1 ,δ1 , πν2 ,δ2 , π−ν,δ involved (the sign change in the last subscript is an effect of duality: cf. (2.30)). First, we observe that, though the representation πν,δ is not unitary unless ν is pure imaginary, it is still useful to regard it as a representation in some Hilbert space, to wit the one defined by the equation

f 2ν =

f (X∗ ) 2 |X∗ |2 Re ν dm(X∗ ):

(3.2)

M0

here, |X∗ |2 = |x|2 + 1 + |ξ∗ |2 when X∗ = (x; 1, ξ∗ ). We now show that, for any given g ∈ Sp(n, R), the transformation πν,δ (g) is a bounded endomorphism of the Hilbert space Hν thus defined. First, Y :=

g −1 X lies in M0 [X, ge1 ]

if X ∈ R2n and [X, ge1 ] = 0:

(3.3)


961

indeed, recall that ξ1 = [X, e1 ] if X = (x; ξ ) and that [X, ge1 ] = [g −1 X, e1 ]. Recalling the recipe, just before (2.30), which served as a definition of πν,δ (g), we first extend f , initially defined on M0 , as a function f in R2n \ {0}, setting x ξ∗ −n−ν , (3.4) f ; 1, f (x; ξ1 , ξ∗ ) = |ξ1 |δ ξ1 ξ1 so that

−n−ν

f g −1 .(x; ξ1 , ξ∗ ) = [X, ge1 ] δ f

g −1 X , [X, ge1 ]

(3.5)

and

−n−ν πν,δ (g)f (X∗ ) = [X∗ , ge1 ] δ f (Y∗ )

(3.6)

−1

g X∗ dm(Y∗ ) with Y∗ = [X . The next thing to do is to compute the Jacobian dm(X when X∗ lies in M0 : ∗ ,ge1 ] ∗) to this effect, the simplest way is to use the unitarity of π0,δ , to wit the relation

[X∗ , ge1 ] −2n f (Y∗ ) 2 dm(X∗ ) =

f (X∗ ) 2 dm(X∗ ), (3.7)

M0

M0

finding

−2n dm(Y∗ ) = [X∗ , ge1 ]

dm(X∗ ).

(3.8)

Then, with the help of the same change of variables, one has more generally πν,δ (g)f 2 = ν

[X∗ , ge1 ] −2n−2 Re ν f (Y∗ ) 2 |X∗ |2 Re ν dm(X∗ )

M0

=

[X∗ , ge1 ] −2 Re ν f (Y∗ ) 2 |X∗ |2 Re ν dm(Y∗ )

M0

= M0

|X∗ | |g −1 X∗ |

2 Re ν

f (Y∗ ) 2 |Y∗ |2 Re ν dm(Y∗ ),

(3.9)

an expression which we want to bound in terms of f 2ν . It suffices to observe that the ratio 2 Re ν is bounded for X ∈ M , the bound depending of course on g. Hence, π ∗| ( |g|X ∗ 0 ν,δ is a −1 X | ) ∗ representation by means of bounded operators in Hν . This makes it possible, in the usual way, to define the space of C ∞ vectors of the given representation. that the Lie algebra of the symplectic group consists of block A B Recalling with B and C symmetric, one sees that the space of infinitesimal operators matrices C −A of the phase space representation of Sp(n, R) in L2 (R2n ) is generated by the vector fields ξj ∂x∂ k + ξk ∂x∂ j , xj ∂x∂ k − ξk ∂ξ∂ j , xj ∂ξ∂ k + xk ∂ξ∂ j , the values of which at each point (x; ξ ) with ξ1 = 1

962


generate the linear subspace of R2n tangent to M0 . It follows that the space of C ∞ -vectors of the representation πν,δ consists of C ∞ functions in the usual sense. This condition is of course not sufficient: there are conditions “at infinity” best rephrased by simply changing the hyperplane M0 to an appropriate finite collection of hyperplanes MS , as will be seen for instance in the proof of Lemma 4.1. ;ε Proposition 3.1. When Re ν1 = Re ν2 = n and Re ν = −n, the function Jνε11,ν,ε22;ν (Y∗ , Z∗ ; X∗ ) as defined in (2.36) is a bounded function. One can extend its meaning as a distribution in M0 × M0 × M0 , holomorphic with respect to ν1 , ν2 , ν in the open subset of C3 defined, recalling (2.33) and (2.34), by the conditions

n + ν − ν1 + ν2 n + ν + ν1 − ν2 = ε2 + 1, ε2 + 3, . . . ; = ε1 + 1, ε1 + 3, . . . ; 2 2 n − ν − ν1 − ν2 = ε + 1, ε + 3, . . . , 2

(3.10)

together with the fact that at least one of three following conditions should hold: 3n + ν − ν1 − ν2 =

1, 3, . . . , 2j + 2, 2j + 6, . . .

and n + ν = δ + 1, δ + 3, . . .

(3.11)

or any of the conditions obtained from (3.11) by changing (ν, ν1 , ν2 ; δ, δ1 , δ2 ) to (−ν1 , −ν, ν2 ; δ1 , δ, δ2 ) or to (−ν2 , ν1 , −ν; δ2 , δ1 , δ). When n = 1, one can delete the condition 3 + ν − ν1 − ν2 = 1, 3, . . . from (3.11). Something entirely similar holds after one has replaced M0 by MS for an arbitrary S ∈ R2n \ {0}. In view of the inclusion C ∞ (πν,δ ) ⊂ C ∞ (M0 ) and of Remark 2.2, this will automatically make it a continuous trilinear form on the space of (f1 , f2 , f ) ∈ C ∞ (πν1 ,δ1 ) × C ∞ (πν2 ,δ2 ) × C ∞ (π−ν,δ ). Setting, when ν1 , ν2 , ν satisfy (3.10) and (3.11), and f1 , f2 , f are C ∞ functions with compact support in M0 , 2 ;ε Jεν11 ,ε ,ν2 ;ν (f1 , f2 ; f ) ;ε Jνε11,ν,ε22;ν (Y∗ , Z∗ ; X∗ )f1 (Y∗ )f2 (Z∗ )f (X∗ ) dm(Y∗ ) dm(Z∗ ) dm(X∗ ), =

M 0 ×M 0 ×M 0

(3.12) one has the covariance relation ε1 ,ε2 ;ε 2 ;ε Jεν11 ,ε ,ν2 ;ν πν1 ,δ1 (g)f1 , πν2 ,δ2 (g)f2 ; π−ν,δ (g)f = Jν1 ,ν2 ;ν (f1 , f2 ; f )

(3.13)

for every symplectic transformation g such that the transformed versions of f1 , f2 , f also have compact support in M0 . Proof. The “integral” on the right-hand side of (3.12) is of course a usual notation for what is in effect the result of testing a certain distribution on the function f1 ⊗ f2 ⊗ f . Before coming to the proof, let us indicate that one should not worry about the condition of compact support: in the


963

way explained in Remark 2.2, one can dispense with it, only replacing the domain of integration M0 × M0 × M0 by a finite collection of domains MS × MS × MS . ;ε When Re ν1 = Re ν2 = n and Re ν = −n, all exponents in definition (2.36) of Jνε11,ν,ε22;ν (Y∗ , Z∗ ; X∗ ) have real part zero, so that the first point is obvious. To define when possible, in the distribution sense, complex powers of possibly vanishing functions can often be done by using Hironaka’s desingularization theorem [4], in particular, when necessary (this will be the case here because we wish to find the poles as they appear in conditions (3.10) and (3.11)) explicit blow-up transformations: the idea was used in general, and applied toward a shorter proof of a classical theorem in partial differential equations, in [1,3]. We shall use it here, following its use in the one-dimensional case in [8]. Recall that one can define the direct image of a distribution under any C ∞ proper map. Our point is to give products of signed powers of the three functions 1 := [Y∗ , X∗ ] = x1 − y1 + x∗ , η∗ − y∗ , ξ∗ , 2 := [X∗ , Z∗ ] = z1 − x1 + z∗ , ξ∗ − x∗ , ζ∗ , 3 := [Z∗ , Y∗ ] = y1 − z1 + y∗ , ζ∗ − z∗ , η∗

(3.14)

a meaning for generic values of the parameters. Note that it is not necessary to desingularize fully the variety of zeros of the product 1 2 3 , only to reach a situation in which we are dealing locally with products of signed powers of functions with linearly independent differentials at common zeros. Considering only the partial derivatives with respect to x1 , y1 , z1 , one observes that a linear relation between the differentials of these three functions cannot hold unless it consists in the fact that the sum of the three differentials is zero: computing then the partial derivatives with respect to ξ∗ , η∗ , ζ∗ , finally with respect to x∗ , y∗ , z∗ , one sees that the three differentials are linearly dependent if and only if X∗∗ = Y∗∗ = Z∗∗ with the notation of Section 2. In the open set where this condition is not satisfied, one can complete the set of three functions under consideration into a local coordinate system in R2n , and the proposition follows in this case from the following well-known fact from the theory of distributions in one variable [10]: the function ν → |x|δ−1−ν , a locally summable function if Re ν < 0, extends as a distribution2 ;ε valued holomorphic function of ν for ν = δ, δ + 2, . . . . This gives the distribution Jεν11 ,ε ,ν2 ;ν a 1 −ν2 1 +ν2 (local) meaning provided that n+ν+ν = ε1 + 1, ε1 + 3, . . . , n+ν−ν = ε2 + 1, ε2 + 3, . . . 2 2 n−ν−ν1 −ν2 and = ε + 1, ε + 3, . . . . 2 When the condition X∗∗ = Y∗∗ = Z∗∗ is satisfied, saying that [Z∗ , Y∗ ] is zero is the same as saying that y1 = z1 , and there are two analogous statements related to the last two equations. At points where none of the three functions under consideration vanishes, there is of course no problem. Near points where only, say, the first function [Z∗ , Y∗ ] vanishes, it can be taken as one of a set of local coordinates, and the distribution under examination makes sense whenever n−ν−ν1 −ν2 = ε + 1, ε + 3, . . . . The only problem remains near points at which X∗∗ = Y∗∗ = Z∗∗ 2 and x1 = y1 = z1 i.e., X∗ = Y∗ = Z∗ . We thus need to tame the three functions under consideration near a point such as (X∗0 , X∗0 , X∗0 ), and there is no loss of generality in assuming that X∗0 = en+1 , the (n + 1)th vector from the canonical basis of Rn × Rn , since a symplectic transformation preserving the linear form X → ξ1 can take us to this case.

964


We first replace the triple (Y∗ , Z∗ , X∗ ) ∈ M0 × M0 × M0 by the set of points (T1 , T2 ; x1 ; 3 Y∗∗ , Z∗∗ , X∗∗ ) in R2 × R × (R2n−2 ) , with T1 = 1 (Y∗ , Z∗ , X∗ ),

T2 = 2 (Y∗ , Z∗ , X∗ ).

(3.15)

That these equations define, near (X∗0 , X∗0 , X∗0 ), an admissible new set of coordinates, follows the fact that 1 and 2 have linearly independent partial differentials with respect to the pair (y1 , z1 ). 2 of P1 (R) × R2 Next, we blow up the (T1 , T2 )-plane around 0, replacing it by the subspace R consisting of pairs (τ, T ) such that, in the case when T = 0, τ is the image of T under the canonical projection map p : R2 \ {0} → P1 (R). Generally setting τ = p(θ ), the domain ωj of 2 consisting of P1 (R) characterized by the condition θj = 0 gives rise to the domain Ωj of R pairs (τ, T ) such that either Tj = 0 and p(T ) = τ or T = 0 and τ ∈ ωj . The domains Ω1 and Ω2 2 and taking in Ω1 the set of coordinates cover R (τ2 , T1 ) =

θ2 , T1 , θ1

(3.16)

θ1 , T2 , θ2

(3.17)

and in Ω2 the set of coordinates (τ1 , T2 ) =

2 into a smooth manifold. The projection map φ : (τ, T ) → T is proper since the one turns R inverse image of a point T = 0 reduces to the point (p(T ), T ), while that of 0 is Σ = P1 (R)×{0}. 2 × R × (R2n−2 )3 of the three In Ω1 , one has 1 = T1 , 2 = τ2 T1 , so that the pullbacks in R functions under consideration express themselves as

1 = T1 ,

2 = τ2 T1 ,

3 = −(1 + τ2 )T1 + [X∗∗ , Y∗∗ − Z∗∗ ] − [Y∗∗ , Z∗∗ ].

(3.18)

The differentials of 1 and 2 are not linearly independent when T1 = 0, but the differentials of T1 and τ2 are, which is sufficient as a start. We must now insert a lemma, in order to take care of the extra terms in 3 . Lemma 3.2. Consider on R2n × R2n × R2n the function F (Y, Z, X) = [X, Y − Z] − [Y, Z],

(3.19)

6n which is critical exactly at points (−X 0 , −X 0 , X 0 ), where it vanishes. Consider the blow-up R 6n

6n of R at such a point, and the pullback F in R of the function F . Locally around any point lying in the inverse image of (−X 0 , −X 0 , X 0 ), one can find two smooth real-valued functions R expresses itself as RS 2 . and S such that F


965

Proof. First, observe the identity F −X 0 + Y, −X 0 + Z, X 0 + X = F (Y, Z, X),

(3.20)

6n obtained as the result so that there is no loss of generality in assuming that X 0 = 0. The space R 6n of blowing up R around 0 is covered by a family (Ωj )1j 6n of open sets with the following properties: for each j , there is a function Sj taken from the set of canonical coordinates of one of the three vectors Y, Z, X such that, within Ωj , the equation Sj = 0 defines the inverse image ˙ X, ˙ each P6n−1 (R) × {0} of 0 ∈ R6n ; next, there is a set of smooth vector-valued functions Y˙ , Z, ˙ ˙ ˙ of which has 2n components, such that the identities Y = Sj Y , Z = Sj Z, X = Sj X hold, and ˙ X˙ the coordinate which, of such that, deleting from the set of components of the vectors Y˙ , Z, necessity, is the constant 1, one obtains a family of functions which, when completed by the function Sj , constitutes an admissible set of coordinates in Ωj . Then, one may write ˙ Y˙ − Z] ˙ − [Y˙ , Z] ˙ , ˙ X) ˙ = Sj2 [X, (Sj , Y˙ , Z, F

(3.21)

and it suffices to observe that the second factor is a function without critical point. Indeed, assuming for instance that the coordinate Sj has been taken from the components of Y (it would be fully similar if it had been taken from any of the other two remaining vectors), the equation (Y˙ )j = 1 shows that the partial derivatives of φ˜ with respect to the coordinates in X˙ or Z˙ “conjugate with respect to the symplectic form” to (Y˙ )j are not zero. 2 End of proof of Proposition 3.1. Applying Lemma 3.2 with n − 1 substituted for n, we may rewrite (3.18), more precisely the pullbacks of the three functions there to a new blown-up space, as

1 = T1 ,

2 = τ2 T1 ,

3 = −(1 + τ2 )T1 + RS 2 ,

(3.22)

where the four functions T1 , τ2 , R, S have linearly independent differentials. The differential d3 is a linear combination of d1 and d3 exactly at points where S = 0, but let us not forget the origin (3.16) of the coordinate T1 , which implies that there is no loss of generality in assuming that we are near a point where T1 = 0 as well. In the open set where 1 + τ2 does not vanish, we may take 3 to the form −T1 + RS 2 , and we blow up the plane of the variables T1 , S around 0: this amounts, with new variables, to setting in appropriate domains either S = T1 S or T1 = ST1 , finding either −T1 + RS 2 = T1 (−1 + RT1 S 2 ) or −T1 + RS 2 = S(−T1 + RS). In the first case we are dealing with a pair of functions, the first of which is T1 and the second is the product of T1 by a function which, at points where it vanishes, has a differential linearly independent from dT1 . In the second case, we still have to desingularize the pair of functions (ST1 , S(−T1 + RS) or, setting aside the factors S in the product of signed powers to be analyzed, the triple of functions (S, T1 , −T1 + RS). Again, we blow up the (T1 , S)-space, which amounts to setting either S = T1 S , in which case the triple becomes (T1 S , T1 , T1 (−1 + RS ), or T1 = ST1 , in which case the triple becomes (S, ST1 , S(−T1 + R)), a satisfactory situation.

966


Finally, we must place ourselves near a point where T1 and 1 + τ2 vanish. We may then forget about 2 entirely, and we blow up the variables T1 , 1 + τ2 , S near 0. In local charts, this makes up one of the three following possibilities: 1 + τ 2 = T1 σ 2 , T1 = (1 + τ2 )T1 ,

S = T1 S ,

S = (1 + τ2 )S ,

T1 = ST1 ,

1 + τ2 = Sσ2 ,

3 = T12 −σ2 + RS 2 , 3 = (1 + τ2 )2 −T1 + RS 2 , 3 = S 2 −σ2 T1 + R .

(3.23)

In the first (resp. third) case, a product of signed powers of T1 and 3 becomes a product of signed powers of T1 and −σ2 + RS 2 (resp. a product of signed powers of S, of T1 and −σ2 T1 + R), a satisfactory situation since we are dealing in each case with two functions with linearly independent differentials. This is not the case on the second line, in which, after leaving the factors 1 + τ2 aside, we have to consider the pair of functions T1 and −T1 + RS 2 : these do not have linearly independent differentials; however, this pair can be desingularized since we are back to the situation examined above, relative to the pair (T1 , −T1 + RS 2 ). 2 ;ε We are now in a position to define locally the distribution Jεν11 ,ε ,ν2 ;ν as the direct image, under a proper map, of a distribution of the kind

−n−ν+ν1 −ν2 2

|1 |ε2

−n−ν−ν1 +ν2 2

|2 |ε1

−n+ν+ν1 +ν2 2

|3 |ε

,

(3.24)

where the factors 1 , 2 , 3 really denote the initial functions 1 , 2 , 3 after they have been pulled back in one of the appropriate ways just described: only, we here dispense with the collection of superscripts which has been used before in order to keep track of the number of blow-ups needed. In case the reader should worry about it, the fact that the subscript ε2 should be associated to 1 , not 2 , is not a blunder: the index δ1 is actually that which must be associated to 1 , and we recall (2.33). The important fact is that, in local charts, the functions 1 , 2 , 3 are all built as powers of the same set of functions with linearly independent differentials. Recall from (2.35) that α1 =

−n − ν + ν1 − ν2 , 2

α2 =

−n − ν − ν1 + ν2 , 2

α3 =

−n + ν + ν1 + ν2 . 2

(3.25)

To find the poles, as a distribution-valued function of ν1 , ν2 , ν, of the distribution (3.24), we must go back to the desingularizing operations and keep track of the signed powers involved in −1−μ each case, starting from the fact that |f |δ makes sense as a distribution, assuming that f has no critical zero, when μ = δ, δ + 2, . . . . As already said, when none of the three functions 1 , 2 , 3 vanishes, there is of course no condition on the exponents involved, and when just one of them vanishes (the case discussed between (3.14) and (3.15)), we must assume −α1 = ε2 + 1, ε2 + 3, . . . ;

−α2 = ε1 + 1, ε1 + 3, . . . ;

−α3 = ε + 1, ε + 3, . . . .

(3.26)

Next, we go to our discussion following (3.22). Forgetting the factors without zeros, the product of signed powers we are led to is of one of the following species, in which we introduce the new letter V , S , T1 , . . . for each of the functions, with differentials independent from the other ones at points where they vanish, such as −1 + RT1 S 2 , which have appeared in the discussion:


|T1 |αε21 |τ2 T1 |αε12 |T1 V |αε 3 or

α1

T S T τ2 T S T α2 T S T V α3 1

1 ε2

1

1 ε1

1

1

2 α1 2 α2 2 α3

S T τ2 S T S V

or

ε

or ε

α |T1 |αε21 |τ2 T1 |αε12 T12 V ε 3 or

α1 α2 2 α3

ST ST S V

or 1 ε2 1 ε1 ε

α

α

α α |1 + τ2 |αε21 |1 + τ2 |αε12 (1 + τ2 )2 ε 3 T1 S T1 ε 1 T1 S T1 ε 2 T1 S T1 V ε 3 2 1

α

α

α

α

|1 + τ2 |αε21 |1 + τ2 |αε12 (1 + τ2 )2 ε 3 S 2 T1 ε 1 S 2 T1 ε 2 S 2 V ε 3 . 1 ε2

967

1 ε1

2

1

or (3.27)

Besides, we must not forget that all these local forms are only available in some domains above parts of Ω1 , not Ω2 (cf. (3.16)), so we must complete the preceding list with the one obtained from it by exchanging the two pairs (ε2 , ν1 ) and (ε1 , ν2 ). All lines are treated in the same way: let us consider the last one, which happens to make all possible demands on the exponents, and let us rewrite it as

α +α +α2 |1 + τ2 |αε11+ε |1 + τ2 |α3 |S |2(α1 +α2 +α3 ) T1 ε 1+ε 2mod 2 |V |αε 3 . 2 mod 2 1

2

(3.28)

Since ε1 + ε2 + ε ≡ j mod 2, this can be written as

α +α |1 + τ2 |αε 3 |S |2(α1 +α2 +α3 ) T1 ε 1+ε 2mod 2 |V |αε 3 .

(3.29)

−3n − ν + ν1 + ν2 , α1 + α2 = −n − ν, 2 ε1 + ε2 ≡ j + ε ≡ δ mod 2,

(3.30)

α +α2 +α3

|1 + τ2 |j 1

1

2

Now, one has α1 + α2 + α3 =

so that, besides the conditions (3.26), it suffices to assume moreover that 3n + ν − ν1 − ν2 = j + 1, j + 3, . . . , 2

3n + ν − ν1 − ν2 = 1, 3, . . . ,

(3.31)

and that n + ν = δ + 1, δ + 3, . . . . These conditions are clearly invariant under the exchange of pairs (ε2 , ν1 ) and (ε1 , ν2 ). They are not fully necessary: the reason for this is that, in our desingularization procedure, we have started with giving the pair (1 , 2 ) special consideration, while we might just as well started from giving the pair (2 , 3 ) or (3 , 1 ) special consideration. This takes us to the assumptions in Proposition 3.1, not forgetting that in the one-dimensional case, the desingularization process stops at (3.18). The rest of the proof is trivial. 2 We shall also need the following result, in the same spirit as Proposition 3.1, though of course its proof presents no difficulty.

968


Proposition 3.3. Set, assuming −ρ = δ + 1, δ + 3, . . . and ρ = δ, δ + 2, . . . , 1

c(ρ, δ) = (−i)δ π − 2 −ρ

( ρ+1+δ 2 ) ( −ρ+δ 2 )

,

(3.32)

so that one should have, in one dimension, ρ −ρ−1 F |s|δ (σ ) = c(ρ, δ)|σ |δ

(3.33)

(of course, we are using here the usual Fourier transformation, with integral kernel e−2iπsσ : there is no symplectic Fourier transformation on an odd-dimensional space). Recalling (2.22), consider the integral kernel

−n+ν . kν,δ (x, ξ∗ ; y, η∗ ) = (−1)δ c(n − 1 − ν, δ) x1 − y1 + x∗ , η∗ − y∗ , ξ∗ δ

(3.34)

When −n < Re ν < 1 − n, this is the integral kernel of an operator θν,δ well defined, in the weak sense, from the space of C ∞ vectors of the representation πν,δ to the dual of that space (which contains the space of C ∞ vectors of the representation π−ν,δ ). As an operator-valued function of ν, θν,δ extends as a holomorphic function in C \ P , where the set P consists of the values ν such that −n + ν = δ, δ + 2, . . . or n − ν = δ + 1, δ + 3, . . . . The operator θν,δ is an intertwiner from the representation πν,δ to the representation π−ν,δ . When ν ∈ iR, it coincides with the one introduced in another way in Definition 2.2. The latter way to define the operator θiλ,δ has the advantages, especially in the version (2.19), that on one hand it continues to be meaningful after ν ∈ C has been substituted for iλ, on the other hand that it extends to a (tempered) distribution setting: but this requires that the homogeneous functions, or distributions, under consideration, should have a well-defined meaning as distributions in R2n , not only as functions, or distributions, in R2n \ {0}. 4. Hyperplane waves and rays We decompose here symbols as integral superpositions of homogeneous hyperplane waves, also of homogeneous rays, by which we mean homogeneous measures carried by straight lines through the origin of R2n . With the help of such decompositions, we shall transform, in this section, the triple product studied in Section 3 in a way crucial towards the proof of the main theorem. Consider the transformation G, a rescaled version of the symplectic Fourier transformation (also a unitary involution of L2 (R2n )) defined as h(Y )e−4iπ[X,Y ] dy: (4.1) (Gh)(X) = 2n R2n

part of our interest in this transformation [11, p. 120] is that, for every S ∈ S (R2n ), the distribution GS is the Weyl symbol of the operator u → Op(S)u, ˇ where u(x) ˇ = u(−x). If a symbol h = h(x; ξ ) depends only on ξ1 , say h(x; ξ ) = φ(ξ1 ), it is immediate that (Gh)(x; ξ ) = ˆ 2φ(−2x 1 )δ(x∗ )δ(ξ ): in other words, Gh is the measure carried by the line {te1 : t ∈ R}, with


969

ˆ density 2φ(−2t) dt. More generally, if S ∈ R2n \ {0}, setting S = ge1 with g ∈ Sp(n, R), the Gtransform of the hyperplane wave X → φ([X, S]) is the measure carried by the line {tS: t ∈ R}, ˆ with density 2φ(−2t) dt. In particular, for any ρ ∈ C, −ρ = δ + 1, δ + 3, . . . , we shall denote as μS (ρ, δ) the measure ρ carried by the line {tS: t ∈ R}, with density |t|δ dt. Recalling the definition (3.32) of c(ρ, δ), we have, provided that n + ν = δ + 1, δ + 3, . . . and −n − ν = δ, δ + 2, . . . ,

−n−ν

= (−1)δ 2ν c(−n − ν, δ)μS (n − 1 + ν, δ). G X → [X, S] δ

(4.2)

Note that the measure μS (ρ, δ) is a homogeneous distribution of type (ρ + 1 − 2n, δ) (do not forget that, in R2n−1 , the Dirac mass at the origin is homogeneous of degree 1 − 2n). Let us first decompose functions in S(R2n ) into homogeneous hyperplane waves. Start from the continuation of (2.4), to wit 1 hν,δ (X) = 4π

∞ |t|δn−1+ν h(tX) dt,

(4.3)

−∞

where the integral converges for every X = 0 provided that Re ν > −n. In this case, the function hν,δ is, as we now show, a C ∞ vector of the representation πν,δ . With X∗ = (x; 1, ξ∗ ), one has for every N the inequality |h(tX∗ )| C(1 + |t|)−N (1 + |x| + |ξ∗ |)−N for some constant C: then, with the norm defined in (3.2), one has X∗ → h(tX∗ )ν C(1 + |t|)−N , from which one obtains, since Re(n − 1 + ν) > −1, that the function hν,δ lies in the Hilbert space Hν defined in association with this norm. That it is a C ∞ vector of the representation πν,δ follows from the fact that this representation corresponds, under the transformation (4.3) from h to hν,δ , to the phase space representation of Sp(n, R) in S(R2n ). In the case when, moreover, Re ν < 1 − n, one may write

∞ hν,δ (X) = 2

n

|t|δn−1+ν

−∞

=

e−4iπt[X,S] (Gh)(S) dS

dt R2n

2−ν c(n − 1 + ν, δ) 4π

[X, S] −n−ν (Gh)(S) dS, δ

(4.4)

R2n

which leads to the decomposition of h into homogeneous hyperplane waves if coupled with the equation h=

δ=0,1 Re ν=a

hν,δ

dν , i

(4.5)

in which −n < a < 1 − n. From (2.3), however, the line of integration we are particularly interested in is the pure imaginary line, for which this decomposition is just the spectral decomposition of h relative to the (self-adjoint) operator E in L2 (R2n ). Starting from (4.4) and moving the set of values of ν, we certainly reach, for fixed S, poles of the distribution-valued function ν → |[X, S]|δ−n−ν , at points ν = −n + δ + 1, ν = −n + δ + 3, . . . , but these poles are

970


simple, and disappear after multiplication by the factor c(n − 1 + ν, δ), as seen from (3.32). This makes it possible to continue the decomposition of h into homogeneous hyperplane waves up to the spectral line. Starting from Gh in place of h and noting that (Gh)−ν,δ = Ghν,δ , one obtains also, if Re ν < n, hν,δ = =

2ν c(n − 1 − ν, δ) 4π 1 4π

−n+ν

dS h(S)G X → [X, S] δ

R2n

h(S)μS (n − 1 − ν, δ) dS,

(4.6)

(−1)δ c(ρ, δ)c(−ρ − 1, δ) = 1:

(4.7)

R2n

after one has used the equation

this leads to a decomposition of h into rays if coupled with the equation dν h= h−ν,δ , i

(4.8)

δ=0,1 Re ν=a

in which, starting from a value of a between −n and 1 − n, we can actually take a = 0 when so desired. The following lemma will enable us to deal with multipliers of the species which occurs consistently in the present work. Lemma 4.1. Let S ∈ R2n \ {0}. If ε, δ = 0 or 1 and α, ν ∈ C satisfy the condition − 12 < Re α < 1 α ∞ ∞ 2 + Re ν, the multiplication by the function X∗ → |[S, X∗ ]|ε sends the space C (πν,δ ) of C vectors of the representation πν,δ to the space L2 (M0 ). Proof. It is no loss of generality to assume that S = en+1 , i.e., [S, X∗ ] = x1 . Given f ∈ C ∞ (πν,δ ) extending to R2n \ {0} as a function f of type (−n − ν, δ), the function k(x; ξ ) = |x1 |αε |ξ1 |ν−α ε+δ mod 2 f (x; ξ )

(4.9)

is of type (−n, 0). Since the corresponding representation π0,0 preserves the Hilbert space L2 (M0 ), it suffices, in view of Remark 2.2, to check that the restriction of the function k, to M0 lies in the space L2loc (M0 ), which leads to the two conditions indicated. 2 2 ;ε We now come back to a study of the bilinear operator (f1 , f2 ) → Jεν11 ,ε ,ν2 ;ν (f1 , f2 ), or of the associated triple product obtained when testing this distribution against f ∈ C ∞ (π−ν,δ ). Recall from the end of Section 2 that such expressions can also use as arguments objects with the proper type defined in R2n \ {0} rather than their restrictions to M0 , the distinction being purely notational. We shall eventually assume, but not at one stroke, that

f1 = (h1 )ν1 ,δ1 ,

f2 = (h2 )ν2 ,δ2 ,

for a triple of functions h1 , h2 , h ∈ S(R2n ).

f = h−ν,δ

(4.10)


971

Lemma 4.2. Assume that h2 ∈ S(R2n ) and that all hypotheses of Proposition 3.1 are valid. Moreover, assume that Re ν2 < n and that 1 Re ν1 > − , 2

Re(ν − ν1 + ν2 ) = n,

1 Re ν < . 2

(4.11)

If f1 ∈ C ∞ (πν1 ,δ1 ), one has in the weak sense, i.e., when integrated against f (X∗ ) dm(X∗ ) for some f ∈ C ∞ (π−ν,δ ), 2 ;ε Jεν11 ,ε ,ν2 ;ν f1 , (h2 )ν2 ,δ2 (X∗ ) =

(−1)ε2 1 n−2+ν−ν 1 +ν2 4π c( , ε2 ) 2

× [X∗ , S]

−n−ν−ν1 +ν2 2 ε1

h2 (S) dS R2n

−n+ν+ν1 +ν2 2 θ n−ν+ν1 −ν2 ,ε Y∗ → [S, Y∗ ] ε f1 (Y∗ ) (X∗ ). 2

2

(4.12)

Proof. First, we observe, as a consequence of Lemma 4.1, that, under the conditions (4.11), the −n+ν+ν1 +ν2 2

multiplication by the function Y∗ → |[S, Y∗ ]|ε

sends the space C ∞ (πν1 ,δ1 ) to the space −n−ν−ν1 +ν2

L2 (M0 ) and that the multiplication by the function X∗ → |[X∗ , S]|ε1 2 sends the space L2 (M0 ) to the space of distributions C −∞ (πν,δ ), the topological dual of C ∞ (π−ν,δ ) (i.e., the linear space of continuous linear forms on that space). On the other hand, the first condition (4.11) gives the intertwining operator θ n−ν+ν1 −ν2 ,ε a meaning as a unitary operator in L2 (M0 ), 2 2 so that the right-hand side of the equation to be proved is meaningful. If one makes there the integral kernel of the operator θ n−ν+ν1 −ν2 ,ε explicit, as 2

−n−ν+ν1 −ν2 2

1 +ν2 (−1)ε2 c( n−2+ν−ν , ε2 )|[Y∗ , X∗ ]|ε2 2 −n−ν−ν1 +ν2 2

|s|ε1

, then if one sets S = sZ∗ , so that

−n+ν+ν1 +ν2 2

|s|ε

2

2 dS = |s|n−1+ν ds dm(Z∗ ), δ2

(4.13)

and if one uses the equation 1 (h2 )ν2 ,δ2 (X) = 4π

∞

2 |s|n−1+ν h2 (sX) ds, δ

(4.14)

−∞

one transforms the right-hand side of (4.12) into the left-hand side. However, the operator on the left-hand side has been defined with the help of the desingularization of its integral kernel as done in Section 3, while on the right-hand side, the claimed unitarity of the intertwining operator into consideration is a consequence of Definition 2.2: to identify the two ways to introduce it, one must use again the connection between (2.21) and (2.22). 2 Let us rewrite (4.12), as tested against f , with 1 f (X∗ ) = h−ν,δ (X∗ ) = 4π

∞ |t|δn−1−ν h(tX∗ ) dt. −∞

(4.15)

972


One has ε1 ,ε2 ;ε Jν1 ,ν2 ;ν f1 , (h2 )ν2 ,δ2 , h−ν,δ =

1 (−1)ε2 (4π)2 c( n−2+ν−ν1 +ν2 , ε2 ) 2

−n+ν+ν1 +ν2

−n−ν−ν1 +ν2

2 2 h2 (S) F Y → [S, Y ] ε f1 (Y ) , T → [T , S] ε h(T ) dS: × 1

R2n

(4.16) note that the two pairs of brackets , do not denote the same pairings: on the left-hand side, it corresponds to the duality between C −∞ (πν,δ ) and C ∞ (π−ν,δ ); within the integrand on the right-hand side, it corresponds to the one between S (R2n ) and S(R2n ). To prove this, we start from the right-hand side, expressing the intertwining operator there as a Fourier transformation. The function

−n+ν+ν1 +ν2

−n−ν−ν1 +ν2 2 2 F Y → [S, Y ] ε f1 (Y ) (T ) T → [T , S] ε 1

(4.17)

is of type (recalling (2.33))

−n − ν − ν1 + ν2 n − ν − ν1 − ν2 , ε1 + (−2n, 0) + , ε + (n + ν1 , δ1 ) 2 2 = (−n − ν, δ).

(4.18)

Set T = tX∗ , so that dT = |t|2n−1 dt dm(X∗ ): then, the right-hand side of (4.16) transforms into the left-hand side in view of (4.18) and (4.15). As a last step, we now use the decomposition 2−ν1 c(n − 1 + ν1 , δ1 ) (h1 )ν1 ,δ1 (Y ) = 4π

−n−ν1

(Gh1 )(R) [Y, R] δ dR 1

(4.19)

R2n

of f1 = (h1 )ν1 ,δ1 , as provided by (4.4). Proposition 4.3. Assume that all hypotheses from Proposition 3.1 are satisfied and that, moreover, −n + ν + ν1 + ν2 = ε, ε + 2, . . . , 2 2 − n − ν + ν1 + ν2 = ε2 + 1, ε2 + 3, . . . 2

ν + ν1 = δ2 , δ2 + 2, . . . ,

(4.20)

and Re ν1 > −n, Then,

Re ν2 < n,

Re ν < n.

(4.21)


2 ;ε Jεν11 ,ε ,ν2 ;ν (h1 )ν1 ,δ1 , (h2 )ν2 ,δ2 , h−ν,δ

973

=

−n+ν+ν1 +ν2 , ε) (−1)ε2 2−ν1 c( 2 3 n−2+ν−ν +ν 1 2 (4π) c( , ε2 ) 2

×

n−2−ν+ν1 +ν2 2

|r|j

1

R2n ×R2n

n−2−ν−ν1 −ν2 2

|s|ε

−n−ν−ν1 +ν2

2 (Gh1 )(R)h2 (S) [R, S] ε dR dS

h(rR + sS) dr ds,

(4.22)

R2

where the last integral must be understood in the distribution sense: recall that j was defined in (2.34). Proof. First, write the equation, of immediate verification, −n+ν+ν1 +ν2 1 2 (t1 , t∗ ; τ1 , τ∗ ) | − η1 |ε F (y; η) → | − y1 |−n−ν δ1 n−2−ν−ν −ν 1 2 −n + ν + ν1 + ν2 δ1 1 2 = (−1) c(−n − ν1 , δ1 )c |τ1 |n−1+ν δ(t∗ )δ(τ∗ ). , ε |t1 |ε δ1 2 (4.23)

Next, under the generic condition [R, S] = 0, one can find g ∈ Sp(n, R) such that S = ge1 ,

R = [R, S]gen+1 :

(4.24)

it follows that

−n−ν−ν1 +ν2

−n+ν+ν1 +ν2

2 2

[Y, R] −n−ν1 , T → [T , S]

h(T ) F Y → [S, Y ] ε δ1 ε1

−n−ν1

−n + ν + ν1 + ν2 δ1 , ε [R, S] δ = (−1) c(−n − ν1 , δ1 )c 1 2 −n−ν−ν1 +ν2 1 −ν2 n−2−ν−ν 2 1 × |t1 |ε |τ1 |n−1+ν δ(t∗ )δ(τ∗ ), |τ1 |ε1 2 (h ◦ g)(t1 , t∗ ; τ1 , τ∗ ) . δ1

(4.25)

Since (h ◦ g)(t1 , 0; τ1 , 0) = h t1 S + τ1

R , [R, S]

(4.26)

we set τ1 = [R, S]r and, for clarity, t1 = s, getting

−n−ν−ν1 +ν2

−n+ν+ν1 +ν2

2 2

[Y, R] −n−ν1 , T → [T , S]

h(T ) F Y → [S, Y ] ε δ1 ε1

−n−ν−ν1 +ν2

−n + ν + ν1 + ν2 2 , ε [R, S] ε = (−1)δ1 c(−n − ν1 , δ1 )c 1 2 n−2−ν+ν1 +ν2 n−2−ν−ν1 −ν2 2 2 × |r|j |s|ε h(rR + sS) dr ds R2

as a result.

(4.27)

974


Then, using (4.16) and (4.19) together with Eq. (4.7) (−1)δ1 c(n − 1 + ν1 , δ1 )c(−n − ν1 , δ1 ) = 1,

(4.28)

we obtain (4.22) under the conditions which made Lemma 4.2, and (4.16) as a consequence, valid. Analytic continuation is possible, the hypotheses from Proposition 3.1 giving a meaning to the left-hand side. The conditions (4.21) make it possible to extract (h1 )ν1 ,δ1 , (h2 )ν2 ,δ2 and h−ν,δ 1 from h1 , h2 , h; the first condition (4.20) gives a meaning to |s|−1−ν−ν as a distribution (the δ2 factor depending on r is already locally summable from the previous condition), and the other two inequalities (4.20) make up half the conditions needed in order that the ratio

−n+ν+ν1 +ν2 ,ε) 2 n−2+ν−ν1 +ν2 c( ,ε2 ) 2

c(

be well defined and nonzero while, as it turns out, the other two conditions necessary for that have already been taken care of by the assumptions of Proposition 3.1. 2 5. Some one-dimensional preparation Let us briefly recall the spectral decomposition of the one-dimensional Euler operator in L2 (R), with the notation of Section 2. Given a function hiλ,δ on R2 , homogeneous of degree −1 − iλ and with a given parity specified by the index δ = 0 or 1, we set

hiλ,δ (s) = hiλ,δ (s, 1)

(5.1)

so that hiλ,δ (x, ξ ) = |ξ |δ−1−iλ hiλ,δ

x . ξ

(5.2)

Then, every function h ∈ L2 (R2 ) can be decomposed as h=

∞

(5.3)

hiλ,δ dλ

δ=0,1 −∞

with 1 hiλ,δ (x, ξ ) = 2π

∞ t iλ hδ (tx, tξ ) dt,

(5.4)

0

where hδ denotes the even, or odd, part of h, according to whether δ = 0 or 1. Note that we denote here as hiλ,δ the function denoted as hλ,δ in [11, p. 34]. Using the equations (in which signed powers such as |s|αδ have been defined in (2.2)) d −ν−2 |x|−1−ν = −(1 + ν)|x|1−δ dx δ

and

d log|x| = x −1 , dx

(5.5)

one obtains the well-known fact, already used in Section 3, that the function ν → |x|δ−1−ν , a locally summable function if Re ν < 0, extends as a distribution-valued holomorphic function of ν for ν = δ, δ + 2, . . . .


975

1 2 If |x|−1−ν and |ξ |−1−ν make sense as distributions as just defined, the symbol h(x, ξ ) = δ1 δ2

1 2 |x|−1−ν # |ξ |−1−ν makes sense as a tempered distribution on R2 : in other words, the composiδ1 δ2 tion of the two operators, the first of which is the convolution by the inverse Fourier transform 2 1 , and the second is the multiplication by |x|−1−ν , is well defined as an operator from of |ξ |−1−ν δ2 δ1 S(R) to S (R). To see this, one may use as an intermediary space the space OM [10, p. 101] of C ∞ functions on the line each derivative of which is bounded by some polynomial. Under the lift from hiλ,δ to hiλ,δ provided by (5.2), the distribution associated to the function

|s|

−1−ν1 +ν2 −iλ 2

is given as (x, ξ ) → |x|

−1−ν1 +ν2 −iλ 2

and the distribution associated to the function s

−1−ν1 +ν2 −iλ 2

−1−ν1 +ν2 −iλ 2

(x, ξ ) → x

−1+ν1 −ν2 −iλ 2

|ξ |δ

(5.6)

is given as

−1+ν1 −ν2 −iλ 2

|ξ |1−δ

.

(5.7)

Both distributions make sense if −1±(ν12−ν2 )−iλ = −1, −2, . . . , which is the case whenever λ ∈ R if one assumes that |Re(ν1 − ν2 )| < 1. We may then recall Lemma 5.1 from [11] as follows: Lemma 5.1. Let ν1 , ν2 ∈ C and δ1 , δ2 = 0 or 1: assume that ν1 = δ1 , ν2 = δ2 and that |Re(ν1 ± ν2 )| < 1 which implies that |Re ν1 | < 1, |Re ν2 | < 1. Let δ = 0 or 1 be such that 1 2 δ ≡ δ1 + δ2 mod 2. Set h1 (x, ξ ) = |x|−1−ν , h2 (x, ξ ) = |ξ |−1−ν and h = h1 # h2 , a tempered δ1 δ2 2 2 distribution in R . It admits the weak decomposition in S (R ) given as ∞ h=

hiλ,δ dλ

(5.8)

−∞

with hiλ,δ (x, ξ ) = 2

ν1 +ν2 −iλ−5 2

π

ν1 +ν2 −iλ 2

( −ν12+δ1 )( −ν22+δ2 )

( ν1 +δ21 +1 )( ν2 +δ22 +1 )

2 +iλ ( 1+ν1 −ν )( 1+ν1 +ν24−iλ+2δ1 )( 1−ν1 +ν42 +iλ+2δ ) 4 × i δ2 −δ 1−ν +ν ( 1 4 2 −iλ )( 1−ν1 −ν24+iλ+2δ1 )( 1+ν1 −ν42 −iλ+2δ ) × |x|

−1−ν1 +ν2 −iλ 2

+ i −δ2 −δ+1

−1+ν1 −ν2 −iλ 2

|ξ |δ

2 +iλ ( 3+ν1 −ν )( 3+ν1 +ν24−iλ−2δ1 )( 3−ν1 +ν42 +iλ−2δ ) 4

2 −iλ ( 3−ν1 +ν )( 3−ν1 −ν24+iλ−2δ1 )( 3+ν1 −ν42 −iλ−2δ ) 4 −1+ν1 −ν2 −iλ −1−ν1 +ν2 −iλ 2 2 . |ξ |1−δ × x

(5.9)

976


Note that the integrand, as a distribution-valued function of λ, has no singularity on the real line. Also, as a consequence of Stirling’s formula, the coefficient is bounded, for large |λ|, by some power of |λ|: since our claim is that the integral decomposition (5.8) is valid in a weak sense in S (R2 ), we may ensure convergence by means of the equation |x|

−1−ν1 +ν2 −iλ 2

−1+ν1 −ν2 −iλ 2

|ξ |δ

−N N −1−ν1 +ν2 −iλ −1+ν12−ν2 −iλ 2 1 + 4π 2 E 2 |x| , = 1 + λ2 |ξ |δ

(5.10)

∂ ∂ + ξ ∂ξ , and of a similar one involving the second term on the right-hand in which 2iπE = 1 + x ∂x side of (5.9). 1 2 We now need to consider the case of two symbols |x|−n−ν and |ξ |−n−ν , in which n = δ1 δ2 1, 2, . . . is given, the same in both functions. The reason is that, even though the proof of the main theorem depends on the decomposition of symbols into homogeneous hyperplane waves, which are essentially one-dimensional objects, the spectral decomposition of the Euler operator in L2 (R2n ) demands that we consider decompositions of the same species as (5.3) in which, however, the degrees of homogeneity of the functions in the decomposition lie on the complex line with real part −n rather than −1. Let Q and P be the basic infinitesimal operators of Heisenberg’s representation, where Q is 1 d the operator of multiplication by the variable x on the real line, and P = 2iπ dx . Then, in the one-dimensional Weyl calculus, one has the commutation relations

∂h 1 Op , Q, Op(h) = − 2iπ ∂ξ

1 ∂h P , Op(h) = Op . 2iπ ∂x

(5.11)

1 ∂h Also, P Op(h) = Op(ξ h + 4iπ ∂x ). If h1 (resp. h2 ) is a tempered distribution depending only on x (resp. ξ ), and if one sets A1 = Op(h1 ), A2 = Op(h2 ), one has (using the facts that A1 commutes 1 with Q, A2 commutes with P and the Heisenberg relation [P , Q] = 2iπ )

[P , A1 ][Q, A2 ] = P [Q, A1 A2 ] − [Q, A1 A2 P ] −

1 A1 A2 : 2iπ

(5.12)

it follows that if h = h1 # h2 , the symbol of the operator [P , Op(h1 )][Q, Op(h2 )] is the function

1 ∂ ξ+ 4iπ ∂x

1 ∂h 1 ∂ 1 ∂h 1 1 ∂ 2h − + ξh − − h= . 2iπ ∂ξ 2iπ ∂ξ 4iπ ∂x 2iπ 4π 2 ∂x∂ξ

(5.13)

In other words, under the present assumptions, ∂h1 ∂h2 ∂ 2h # = . ∂x ∂ξ ∂x∂ξ

(5.14)

Introduce, for k = 0, 1, . . . and a ∈ C, the Pochhammer symbols (a)k = a(a + 1) . . . (a + k − 1), and extend the definition of |s|αδ beyond the case when δ = 0 or 1, setting |s|αp = |s|αp mod 2 . With the same assumptions about ν1 , ν2 , δ1 , δ2 as in Lemma 3.1, one has for n = 1, 2, . . . (using (5.5)) the equation


977

−n−ν2 1 (1 + ν1 )n−1 (1 + ν2 )n−1 |x|−n−ν n−1−δ1 # |ξ |n−1−δ2

∞ = −∞

×2

1 + ν1 − ν2 + iλ 2

ν1 +ν2 −iλ−5 2

π

ν1 +ν2 −iλ 2

n−1

1 − ν1 + ν2 + iλ 2

n−1

( −ν12+δ1 )( −ν22+δ2 )

( ν1 +δ21 +1 )( ν2 +δ22 +1 )

2 +iλ ( 1+ν1 −ν )( 1+ν1 +ν24−iλ+2δ1 )( 1−ν1 +ν42 +iλ+2δ ) 4 × i δ2 −δ 1−ν +ν ( 1 4 2 −iλ )( 1−ν1 −ν24+iλ+2δ1 )( 1+ν1 −ν42 −iλ+2δ ) 1−2n−ν1 +ν2 −iλ 2

× |x|n−1

+ i −δ2 −δ+1

1−2n+ν1 −ν2 −iλ

|ξ |n−1+δ2

2 +iλ ( 3+ν1 −ν )( 3+ν1 +ν24−iλ−2δ1 )( 3−ν1 +ν42 +iλ−2δ ) 4

2 −iλ ( 3−ν1 +ν )( 3−ν1 −ν24+iλ−2δ1 )( 3+ν1 −ν42 −iλ−2δ ) 4 1−2n−ν1 +ν2 −iλ 1−2n+ν1 −ν2 −iλ 2 dλ. |ξ |n−δ 2 × |x|n

(5.15)

Note that the degree of homogeneity of each of the two terms under the integral sign is 1 − 2n − iλ, not −n − iλ as we would wish it to be: we must thus perform a deformation of contour. We substitute z ∈ C for iλ and we must move z from the pure imaginary line to the line with real part 1 − n. There is no convergence problem at infinity in the process, in view of (5.10). We must then chase for possible poles, setting μ = ν1 −ν22 +z and μ = ν1 −ν22 −z . The only singularities can arise from the factors depending on x or ξ , or from the first and third Gamma functions in the numerator of each of the two major coefficients. We make a group of each of the expressions

1 1 μ 2 −n−μ + |x|n−1 , 4 2 n−1 1 3 μ −n−μ +μ + |x|n2 , 4 2 n−1 1 1 δ μ 2 −n+μ −μ + − |ξ |n−1+δ , 4 2 2 n−1 1 3 δ μ 2 −n+μ −μ − − |ξ |n−δ . 4 2 2 n−1

1 +μ 2 1 2 1 2 1 2

(5.16)

We now show that each of the four functions under consideration remains a holomorphic function of z in a neighbourhood of the closed strip 1 − n Re z 0. First we show that the Gamma factor and the distribution (in x or ξ ) on any of the four lines have disjoint sets of singularities as functions of z. This is a consequence of the fact, noted just after (5.5), that |x|−α δ a well-defined distribution in x provided that α = δ + 1, δ + 3, . . . . For, as a consequence, the singularities of the factor depending on x or ξ on the four lines are reached when μ ∈ 12 + 2N, resp. μ ∈ 32 + 2N, resp. μ ∈ −δ − 12 − 2N, resp. μ ∈ δ − 32 + 2N, while the singularities of the corresponding Gamma factors are reached when μ ∈ − 12 − 2N, resp. μ ∈ − 32 − 2N, resp. μ ∈ −δ + 12 + 2N, resp. μ ∈ −δ + 32 + 2N.

978


Since the two sets of singularities under consideration are disjoint, what remains to be proved is that each of the eight expressions

1 2 1 2 1 2

1 μ + , 4 2 n−1 3 μ +μ + , 4 2 n−1 1 δ − μ + − 4 2 n−1 3 δ − μ − − 4 2 n−1

1 +μ 2

μ , 2 μ , 2

1 +μ 2 1 +μ 2

1

n−1

1 − μ 2 1 − μ 2

−n−μ

2 |x|n−1 1

|x|n2

−n−μ

, ,

n−1

n−1

n−1

1

−n+μ

1

−n+μ

2 |ξ |n−1+δ ,

2 |ξ |n−δ

(5.17)

is regular for z lying in the strip 1 − n Re z 0. So far as the distribution on the right of d n−1 d n−1 ) , or ( dξ ) -derivative of the each line is concerned, we write it as (−1)n−1 times the ( dx 1

1

1

+μ

− 1 +μ

2 , resp. |ξ |1−δ . Now, the condition Re z 0, distribution |x|− 2 −μ , resp. x− 2 −μ , resp. |ξ |δ2 together with the assumption |Re(ν1 − ν2 )| < 1, implies that Re μ < 12 and Re μ > − 12 , which gives the four distributions under consideration a meaning as locally summable functions. So far as the Gamma factors are concerned, every other term in the product

3 1 + μ ... n − + μ or = 2 2 n−1 1 3 1 1 −μ −μ − μ ... n − − μ = 2 2 2 2 n−1 1 +μ 2

1 +μ 2

(5.18)

will help in killing the relevant poles of the corresponding Gamma factor. Indeed, with p = 1, 2, . . . , each of the two expressions ( 12 + μ)2p−1 ( 14 + μ2 ) and ( 12 + μ)2p−2 ( 14 + μ2 ) is the product of a polynomial in μ by the function (p + 14 + μ2 ), while each of the two expressions ( 12 + μ)2p−1 ( 34 + μ2 ) and ( 12 + μ)2p−2 ( 34 + μ2 ) is the product of a polynomial in μ by the

μ μ 1 1 2 ). The last two expressions to be analyzed are ( 2 − μ )n−1 ( 4 − 2 ) and ( 12 − μ )n−1 ( 34 − μ2 ). We use this time the inequality Re μ < n2 and observe that each of the two expressions ( 12 − μ )2p−1 ( 14 − μ2 ) and ( 12 − μ )2p−2 ( 14 − μ2 ) is the product of a polynomial by (p + 14 − μ2 ), while each of the two expressions ( 12 − μ )2p−1 ( 34 − μ2 ) and ( 12 − μ )2p−2 ( 34 − μ2 ) is the product of a polynomial by (p − 14 − μ2 ).

function (p −

1 4

+

Performing the change of contour which was the aim of the lengthy preparation just made, we finally obtain the following. Lemma 5.2. Let ν1 , ν2 ∈ C and δ1 , δ2 = 0 or 1: assume that ν1 = δ1 , ν2 = δ2 and that |Re(ν1 ± ν2 )| < 1. Let n = 1, 2, . . . , and let δ, δ1 , δ2 be the numbers, all equal to 0 or 1, characterized by the congruences mod 2 δ ≡ δ1 + δ2 ,

δ1 ≡ n − 1 − δ1 ,

δ2 ≡ n − 1 − δ2 .

(5.19)


979

1 2 Set h1 (x, ξ ) = |x|−n−ν , h2 (x, ξ ) = |ξ |−n−ν and let h = h1 # h2 , a tempered distribution in R2 . δ1 δ2 2 It admits the weak decomposition in S (R ) given as

∞ h=

(n)

(5.20)

hiλ,δ dλ

−∞

with (n) −1 hiλ,δ (x, ξ ) = (1 + ν1 )−1 n−1 (1 + ν2 )n−1

×2

ν1 +ν2 −iλ+n−6 2

π

2 − n + ν1 − ν2 + iλ 2

n−1+ν1 +ν2 −iλ 2

n−1

2 − n − ν1 + ν2 + iλ 2

n−1

−ν +δ −ν +δ ( 12 1 )( 22 2 ) ν +δ +1 ν +δ +1 ( 1 21 )( 2 22 )

n+ν +ν −iλ+2δ1 2−n+ν1 −ν2 +iλ 2 +iλ+2δ )( 1 24 )( 2−n−ν1 +ν ) δ2 −δ ( 4 4 × i 2−n−ν1 −ν2 +iλ+2δ1 n−ν1 +ν2 −iλ n+ν1 −ν2 −iλ+2δ ( )( )( ) 4 4 4 −n−ν1 +ν2 −iλ 2

× |x|n−1 +i

−n+ν1 −ν2 −iλ

|ξ |n−1−δ2

n+2+ν1 +ν2 −iλ−2δ1 4−n+ν1 −ν2 +iλ 2 +iλ−2δ )( )( 4−n−ν1 +ν ) 4 4 4 4−n−ν −ν +iλ−2δ 1 2 2 −iλ−2δ 1 ( n+2−ν14+ν2 −iλ )( )( n+2+ν1 −ν ) 4 4

−δ2 −δ+1 (

−n−ν1 +ν2 −iλ 2

× |x|n

−n+ν1 −ν2 −iλ 2

|ξ |n−δ

(5.21)

,

where we recall our convention that |s|αp = |s|αp with p = 0 or 1 and p ≡ p mod 2. In the proof of Lemma 5.2, we have avoided moving ν1 and ν2 , which would have complicated the pole chasing even more. It is, however, necessary to check that analytic continuation with respect to ν1 and ν2 is possible up to some point, in the sense of the following lemma. −1−ν1

Lemma 5.3. Set ν1 = n − 1 + ν1 , ν2 = n − 1 + ν2 , so that |x|δ1 2 . |ξ |−n−ν δ2

(n) hiλ,δ

−1−ν2

1 = |x|−n−ν and |ξ |δ2 δ1

=

To obtain the term from the decomposition (5.20) of h1 # h2 (same notation as in Lemma 5.2), it suffices to perform the substitutions ν1 → ν1 , ν2 → ν2 and iλ → ν = iλ + n − 1 on the right-hand side of (5.9). Proof. The proof, based on the duplication formula and on the formula of complements for the Gamma function, is perfectly ugly, though one can take solace in the fact that it offers a means of verification. Starting from the right-hand side of (5.9) and making the substitution (ν1 , ν2 , iλ) → (ν1 , ν2 , iλ + n − 1), we want to show that we just obtain the right-hand side of (5.21). We shall limit ourselves to the case when n is odd. One has (1 + ν1 )−1 n−1

1−n−ν +δ

2−n−ν −δ

1 1 1 1 ( )( ) (1 − n − ν1 ) 2 2 = 21−n = , −ν +δ 1−ν −δ 1 1 (−ν1 ) 1 1 ( )( )

2

2

(5.22)

980


so that (1 + ν1 )−1 n−1

−ν1 +δ1 1−n−ν1 +δ1 2−n−ν1 −δ1 ) 1−n ( )( ) 2 2 2 2 1+ν +δ 1+ν +δ 1−ν −δ ( 21 1 ) ( 21 1 )( 21 1 )

(

21−n times the corresponding coefficient

−ν1 +δ1 ) 2 1+ν1 +δ1 ( ) 2

(

=2

1−n−ν1 +δ1 ) 2 , n+ν +δ ( 21 1 )

1−n (

(5.23)

arising after the shift ν1 → ν1 from a factor

in (5.9). The same goes so far as the comparable coefficient depending on ν2 is concerned. The powers of 2 and π , as well as the Gamma factors in the middle of the coefficients we are interested in, transform in an immediately satisfactory way. The remaining headache arises from the coefficient, obtained from (5.9) and the required shift, B :=

2 +iλ )( n−ν1 +ν42 +iλ+2δ ) ( n+ν1 −ν 4

2 −iλ+2δ ( 2−n−ν14+ν2 −iλ )( 2−n+ν1 −ν ) 4

:

(5.24)

2 −iλ−2δ multiplying by ( 4−n−ν14+ν2 −iλ )( 4−n+ν1 −ν ) up and down, using the formula of com4 plements upstairs and the duplication formula downstairs, we obtain

n + ν1 − ν2 + iλ n − ν1 + ν2 + iλ + 2δ −1 sin π B = n+iλ sin π 2 4 4 2 − n + ν1 − ν2 − iλ −1 2 − n − ν1 + ν2 − iλ × . 2 2 π

(5.25)

This must be compared to the similar coefficient from (5.21), which must be accompanied, as a factor, by the product of the two remaining Pochhammer symbols. This is A :=

2 −iλ ( n−ν1 +ν ) 2

2 −iλ ) ( n+ν1 −ν 2

( 2−n−ν12+ν2 −iλ ) ( 2−n+ν12−ν2 −iλ ) ×

2 +iλ+2δ ( 2−n+ν14−ν2 +iλ )( 2−n−ν1 +ν ) 4 2 −iλ ( n−ν1 +ν )( n+ν1 −ν42 −iλ+2δ ) 4

:

(5.26)

if we multiply the product of fractions on the second line, up and down, by 2 −iλ−2δ ( 2+n−ν14+ν2 −iλ )( 2+n+ν1 −ν ), if we apply again the formula of complements upstairs 4 and the duplication formula downstairs, it becomes

2 − n + ν1 − ν2 + iλ 2 − n − ν1 + ν2 + iλ + 2δ −1 sin π 4 4 22−n+iλ −1 n + ν1 − ν2 − iλ n − ν1 + ν2 − iλ × . 2 2 π

sin π

(5.27)

It follows that A = 22n−2 B, which completes our verification, in the case when n is odd, so far as the coefficient of the first term on the right-hand side of (5.9) or (5.21) is concerned. We shall not write down everything in the case when (still with n odd) the coefficient of the


981

second term is concerned. The trick is, this time, to multiply the fraction B which takes the 2 −iλ+2δ ); next, the fraction on the place of B, up and down, by ( 2−n−ν14+ν2 −iλ )( 2−n+ν1 −ν 4 second line of the expression A which takes the place of A is to be multiplied, up and down, by 2 −iλ )( n+ν1 −ν42 −iλ+2δ ): again, we find that A = 22n−2 B . The lemma is thus proved ( n−ν1 +ν 4 in the case when n is odd. The proof is of course similar in the case when it is even: only, one should not forget that, in this case, δ1 = 1 − δ1 and δ2 = 1 − δ2 . Also, the right-hand side of (5.9) will yield, after transformation, the two terms on the right-hand side of (5.21) in reverse order. 2 Making all Gamma factors apparent has been necessary for the discussion of the change of complex contour. Using the shorthand provided by (3.32), i.e., making the substitution ( ρ+1+δ 2 ) ( −ρ+δ 2 )

1

= i δ π ρ+ 2 c(ρ, δ),

(5.28)

one obtains the following. Proposition 5.4. Under the assumptions of Lemma 5.2, one has h(n) iλ,δ (x, ξ ) = C0 (ν1 , ν2 , iλ; δ1 , δ2 , δ)|x|

−n−ν1 +ν2 −iλ 2

+ C1 (ν1 , ν2 , iλ; δ1 , δ2 , δ)x

−n+ν1 −ν2 −iλ 2

|ξ |δ

−n−ν1 +ν2 −iλ 2

−n+ν1 −ν2 −iλ 2

|ξ |1−δ

,

(5.29)

with C0 (ν1 , ν2 , iλ; δ1 , δ2 , δ)

n − 2 + ν1 − ν2 + iλ π (−1) c(−n − ν1 , δ1 )c(−n − ν2 , δ2 )c =2 ,0 2 n − 2 − ν1 + ν2 + iλ n − 2 + ν1 + ν2 − iλ , δ1 c ,δ (5.30) ×c 2 2 ν1 +ν2 −iλ+n−6 2

−1

δ

and C1 (ν1 , ν2 , iλ; δ1 , δ2 , δ)

n − 2 + ν1 − ν2 + iλ ,1 π (−1) c(−n − ν1 , δ1 )c(−n − ν2 , δ2 )c =2 2 n − 2 − ν1 + ν2 + iλ n − 2 + ν1 + ν2 − iλ , 1 − δ1 c ,1 − δ . (5.31) ×c 2 2 ν1 +ν2 −iλ+n−6 2

−1

δ

In view of the proof of the main theorem in next section, and as a final topic in this very 1 2 computational section, we compute the G-transform (4.1) of the symbol |x1 |−n−ν # |ξ1 |−n+ν , δ1 δ2 2n considered as a distribution in R : we still set x = (x1 , x∗ ), ξ = (ξ1 , ξ∗ ). The change ν2 → −ν2 is needed for the application in next section: at the same time, we change the variable of integration λ to −λ below so as to decompose the result as an integral superposition of distributions

982

T. Kobayashi et al. / Journal of Functional Analysis 257 (2009) 948–991 (n)

(n)

of type (−n − iλ, δ); we denote as k−iλ,δ the function obtained from hiλ,δ after these two sign changes. Proposition 5.5. Assume that ν1 = δ1 , −ν2 = δ2 and |Re(ν1 ± ν2 )| < 1. One has the weak decomposition in S (R2n ), given by the equation 1 2 (x, ξ ) = # |η1 |−n+ν G Y → |y1 |−n−ν δ1 δ2

∞

(n) Gk−iλ,δ (x, ξ ) dλ

(5.32)

−∞

with n−2−ν1 −ν2 −iλ n−2+ν1 +ν2 −iλ (n) 2 2 Gk−iλ,δ (x, ξ ) = B0 (ν1 , ν2 , iλ; δ1 , δ2 , δ)|x1 |δ |ξ1 | δ(x∗ )δ(ξ∗ ) n−2−ν1 −ν2 −iλ 2

+ B1 (ν1 , ν2 , iλ; δ1 , δ2 , δ)|x1 |1−δ

ξ1

n−2+ν1 +ν2 −iλ 2

δ(x∗ )δ(ξ∗ ), (5.33)

where B0 (ν1 , ν2 , iλ; δ1 , δ2 , δ) =2

ν1 −ν2 −iλ+n−6 2

π −1 c(−n − ν1 , δ1 )c(−n + ν2 , δ2 )c

n − 2 + ν1 − ν2 + iλ , δ1 2

(5.34)

and B1 (ν1 , ν2 , iλ; δ1 , δ2 , δ) = −2

ν1 −ν2 −iλ+n−6 2

π

−1

n − 2 + ν1 − ν2 + iλ , 1 − δ1 . c(−n − ν1 , δ1 )c(−n + ν2 , δ2 )c 2 (5.35)

Proof. This is a consequence of the preceding proposition, together with the equation G Y → |y1 |αω1 |ξ1 |βω2 (x, ξ ) = 2−n−α−β (−1)ω2 c(α, ω1 )c(β, ω2 )|x1 |ω−1−β |ξ1 |ω−1−α δ(x∗ )δ(ξ∗ ). 2 1

(5.36)

A simplification occurs from the use of Eqs. (4.7) −n − ν1 − ν2 + iλ n − 2 + ν1 + ν2 − iλ ,0 c , 0 = 1, 2 2 n − 2 + ν1 + ν2 − iλ −n − ν1 − ν2 + iλ c ,δ c , δ = (−1)δ , 2 2 −n − ν1 − ν2 + iλ n − 2 + ν1 + ν2 − iλ ,1 c , 1 = −1, c 2 2 n − 2 + ν1 + ν2 − iλ −n − ν1 − ν2 + iλ c ,1 − δ c , 1 − δ = (−1)1−δ . 2 2

c

2

(5.37)


983

6. Another composition of Weyl symbols Theorem 6.1. Given δ1 , δ2 and δ = 0 or 1 with δ ≡ δ1 + δ2 mod 2, and j = 0 or 1, define ε1 , ε2 , ε by means of (2.34), and set, for real λ1 , λ2 , λ, (j )

aδ1 ,δ2 ;δ (iλ1 , iλ2 ; iλ) =2

n−6+i(λ1 +λ2 −λ) 2

×

i ε−ε1 −ε2 π

3(1−n)−2+i(λ1 +λ2 −λ) 2

( n+i(λ1 −λ22 +λ)+2ε1 )

2 +λ)+2ε2 ( n+i(−λ1 +λ ) 2

( n−i(λ1 +λ22 +λ)+2ε )

2 +λ)+2ε1 2 +λ)+2ε2 2 +λ)+2ε ( 2−n−i(λ1 −λ ) ( 2−n−i(−λ1 +λ ) ( 2−n+i(λ1 +λ ) 2 2 2

.

(6.1)

Given two symbols h1 and h2 in the space S(R2n ), one has, in the weak sense in S (R2n ), ∞ h1 # h2 =

(h1 # h2 )iλ dλ,

(6.2)

−∞

with (h1 # h2 )iλ =

∞ ∞

(j )

δ1 =0,1 δ2 =0,1 −∞ −∞ j =0,1

aδ1 ,δ2 ;δ (iλ1 , iλ2 ; iλ)

2 ;ε × Jεiλ1 ,ε (h1 )iλ1 ,δ1 , (h2 )iλ2 ,δ2 dλ1 dλ2 , 1 ,iλ2 ;iλ

(6.3)

2 ;ε where Jεiλ1 ,ε is the bilinear operator from C ∞ (πiλ1 ,δ1 ) × C ∞ (πiλ2 ,δ2 ) to C −∞ (πiλ,δ ) for1 ,iλ2 ;iλ mally introduced in (2.38) and discussed in Section 3.

Proof. One has h1 # h2 = G(h1 # Gh2 ), as it follows from the interpretation of the transformation G of symbols recalled in the beginning of Section 4. Next, we decompose h1 into hyperplane waves with the help of (4.4), and h2 into rays with the help of (4.6), recalling that one can move the line of integration up to the spectral line and writing h1 =

∞

(h1 )iλ1 ,δ1 dλ1 ,

∞

Gh2 =

δ1 =0,1 −∞

(Gh2 )−iλ2 ,δ2 dλ2 ,

(6.4)

δ2 =0,1 −∞

with (h1 )iλ1 ,δ1 (X) =

2−iλ1 c(n − 1 + iλ1 , δ1 ) 4π

−n−iλ1

(Gh1 )(R) [X, R] δ dR, 1

R2n

(Gh2 )−iλ2 ,δ2 (X) =

2iλ2 c(n − 1 − iλ2 , δ2 ) 4π

R2n

−n+iλ2

h2 (S) [X, S] δ dS: 2

(6.5)

984


1 recall that the product c(n − 1 + ν1 , δ1 )|[X, R]|−n−ν , can be continued analytically with respect δ1 to ν1 , as a distribution in X. Then,

(h1 # h2 )(X) =

∞ ∞

δ1 =0,1 δ2 =0,1 −∞ −∞

δ1 ,δ2 Fiλ (X) dλ1 dλ2 1 ,iλ2

(6.6)

with ,δ2 Fνδ11,ν (X) = 2

2−ν1 +ν2 c(n − 1 + ν1 , δ1 )c(n − 1 − ν2 , δ2 ) (4π)2

−n−ν1

−n+ν2

× dR dS, (6.7) (Gh1 )(R)h2 (S) G [X, R] δ # [X, S] δ 1

2

R2n ×R2n

the two signed powers under the sharp product of which appears under the integral sign being regarded as functions of X. Actually, so as to obtain the last equation, we have changed the order of the bilinear operation # and of the integration with respect to dR dS. Though not completely trivial, the justification is fully similar to that, based on the consideration of the domains of powers of the harmonic oscillator, which occurred, in the one-dimensional case, in [12, p. 209]: we shall not reproduce it here. Generically, one has [R, S] = 0 and, as noticed in (4.24), there exists g ∈ Sp(n, R) such that g −1 S = e1 ,

g −1 R = [R, S]en+1

(6.8)

in terms of the canonical basis of Rn × Rn . Then, using the covariance of the Weyl calculus, and the fact that the transformation G commutes with symplectic changes of coordinates, we obtain ,δ2 Fνδ11,ν (X) = 2

×

2−ν1 +ν2 c(n − 1 + ν1 , δ1 )c(n − 1 − ν2 , δ2 ) (4π)2

−n−ν1

(Gh1 )(R)h2 (S) [S, R]

G Y → |y1 |−n−ν1 # |η1 |−n+ν2 g −1 X dR dS. δ1

δ1

δ2

R2n ×R2n

(6.9) ,δ2 The function Fνδ11,ν 2 can then be made explicit, starting from (6.9), with the help of Proposition 5.5. Rewrite the result of this proposition, tested against h ∈ S(R2n ), as

1 2 ,h # |η1 |−n+ν G Y → |y1 |−n−ν δ1 δ2

∞ =

dλ

−∞

n−2−ν1 −ν2 −iλ n−2+ν1 +ν2 −iλ 2 2 h(se1 + ren+1 ) B0 (ν1 , ν2 , iλ; δ1 , δ2 , δ)|r| |s|δ

R2

+ B1 (ν1 , ν2 , iλ; δ1 , δ2 , δ)r1 Then,

n−2+ν1 +ν2 −iλ 2

n−2−ν1 −ν2 −iλ 2

|s|1−δ

dr ds.

(6.10)


985

−1 1 2 G Y → |y1 |−n−ν ◦ g ,h # |η1 |−n+ν δ1 δ2

∞ =

dλ

−∞

n+ν1 +ν2 −iλ n−2+ν1 +ν2 −iλ n−2−ν12−ν2 −iλ

2 2 h(sS + rR) B0 (ν1 , ν2 , iλ; δ1 , δ2 , δ) [R, S]

|r| |s|δ

R2

n+ν1 +ν2 −iλ n−2+ν1 +ν2 −iλ n−2−ν12−ν2 −iλ 2 2 dr ds, + B1 (ν1 , ν2 , iλ; δ1 , δ2 , δ) [R, S] r |s|1−δ

(6.11)

as seen after one has used (6.8) and the change of variable r → [R, S]r, and δ1 ,δ2 Fiλ 1 ,iλ2

∞ =

δ1 ,δ2 Fiλ dλ 1 ,iλ2 ;iλ

(6.12)

−∞

with i(−λ1 +λ2 ) δ1 ,δ2 δ1 2 , h = (−1) c(n − 1 + iλ1 , δ1 )c(n − 1 − iλ2 , δ2 ) Fiλ ,iλ ;iλ 1 2 (4π)2 × (Gh1 )(R)h2 (S) dR dS h(rR + sS)

R2n ×R2n

× B0 (iλ1 , iλ2 , iλ; δ1 , δ2 , δ)

R2

−n+i(−λ21 +λ2 −λ) n−2+i(λ1 +λ2 −λ) n−2+i(−λ21 −λ2 −λ)

2 × [R, S] δ1 |r| |s|δ + B1 (iλ1 , iλ2 , iλ; δ1 , δ2 , δ)

−n+i(−λ1 +λ2 +λ) n−2+i(λ1 +λ2 −λ) n−2+i(−λ21 −λ2 −λ)

2 dr ds. r |s|1−δ × [R, S] 1−δ1 2

(6.13)

Finally, making the coefficients B0 and B1 explicit with the help of Proposition 5.5 and using (4.7) again, n−2+i(−λ1 +λ2 −λ) 2

(−1)δ2 2 1 δ1 ,δ2 Fiλ1 ,iλ2 ;iλ , h = 4π (4π)4

h(rR + sS)

(Gh1 )(R)h2 (S) dR dS R2n ×R2n

n − 2 + i(λ1 − λ2 + λ) , δ1 × c 2

R2

−n+i(−λ21 +λ2 −λ) n−2+i(λ1 +λ2 −λ) n−2+i(−λ21 −λ2 −λ)

2 × [R, S] δ1 |r| |s|δ n − 2 + i(λ1 − λ2 + λ) −c , 1 − δ1 2

−n+i(−λ1 +λ2 +λ) n−2+i(λ1 +λ2 −λ) n−2+i(−λ21 −λ2 −λ)

2 × [R, S] 1−δ1 2 r |s|1−δ dr ds. (6.14)

986


δ1 ,δ2 The distribution Fiλ ∈ S (R2n ) is of type (−n − iλ, δ). Now, given any element S of 1 ,iλ2 ;iλ −∞ C (πiλ,δ ) extended as a distribution in R2n of type (−n − iλ, δ) with the same name, and any function h ∈ S(R2n ), one has the equation

S, hS (R2n )×S (R2n ) = 4πS, h−iλ,δ C −∞ (πiλ,δ )×C ∞ (π−iλ,δ )

(6.15)

linking the two kinds of pairings. Starting from the case when S is a function, one obtains (6.15) from the equation S(tX∗ ) = |t|δ−n−iλ S(X∗ ) and (2.4) or, if preferred, from a polarization δ1 ,δ2 , h−iλ,δ , the of (2.13). The left-hand side of (6.14) can thus also be regarded as being Fiλ 1 ,iλ2 ;iλ −∞ ∞ pairing now denoting that between C (πiλ,δ ) and C (π−iλ,δ ). The comparison with (4.22) is now easy. δ1 ,δ2 ;δ 2 ;ε With another look at (2.34), one sees that Jεν11 ,ε ,ν2 ;ν coincides with Jν1 ,ν2 ;ν when j = 0, and with

1 ,1−δ2 ;1−δ J1−δ when j = 1. Then, the first or second term on the right-hand side of (6.14) is a ν1 ,ν2 ;ν multiple of the right-hand side of (4.22) taken with j = 0 or 1, as it follows from a comparison of the exponents and subscripts in (4.22) and in each of the two terms of (6.14) of the signed powers of [R, S], r and s. The coefficient by which one must multiply the expression on right-hand side of (4.22) to obtain the corresponding term in right-hand side of (6.14) is

n−2+i(−λ1 +λ2 +λ) , ε2 ) c( 1 n−2+i(λ1 +λ2 −λ) n − 2 + i(λ1 − λ2 + λ) 2 2 c . 2 , ε1 −n+i(λ1 +λ2 +λ) 4π 2 c( , ε)

(6.16)

2

Expanding, we can write this as

2

n−6+i(λ1 +λ2 −λ) 2

×

i ε−ε1 −ε2 π

3(1−n)−2+i(λ1 +λ2 −λ) 2

( n+i(λ1 −λ22 +λ)+2ε1 )

2 +λ)+2ε2 ( n+i(−λ1 +λ ) 2

( n−i(λ1 +λ22 +λ)+2ε )

2 +λ)+2ε1 2 +λ)+2ε2 2 +λ)+2ε ( 2−n−i(λ1 −λ ) ( 2−n−i(−λ1 +λ ) ( 2−n+i(λ1 +λ ) 2 2 2

This concludes the proof of Theorem 6.1.

.

(6.17)

2

As an example, let us consider the harmonic oscillator L = Op(π) with (x, ξ ) = |x|2 + |ξ |2 , and sharp products of fractional powers of . Proposition 6.2. Let ν1 , ν2 ∈ C satisfy the conditions −n < Re ν1 < n, −n < Re ν2 < n. Then, −n−ν1 −n−ν2 the decomposition into homogeneous components hiλ of the symbol h = 2 # 2 is given by the equation n−2+ν1 +ν2 −iλ −n−iλ 1 2 hiλ = (2π) 2 4

×

2 −iλ 2 +iλ 2 +iλ 2 −iλ ( n+ν1 +ν )( n+ν1 −ν )( n−ν1 +ν )( n−ν1 −ν ) 4 4 4 4

n+ν2 n−iλ 1 ( n+ν 2 )( 2 )( 2 )

.

(6.18)


987

Proof. It is identical to that of the one-dimensional case, as treated in [12, p. 214]. Only, one starts this time from the equation L −2πs n 2 −2 1 − s Op e = 1−s 1+s

(6.19)

(same reference as in the one-dimensional case), leading rapidly to the equation

h=

(2π)

ν1 +ν2 +2n 2

n+ν2 1 ( n+ν 2 )( 2 )

∞∞

n+ν1 −2 2

s1

n+ν2 −2 2

s2

e

s +s2 1 s2

1 −2π 1+s

0 0

ds1 ds2 , (1 + s1 s2 )n

(6.20)

then ν1 +ν2 +n−2−iλ ( n+iλ −n−iλ 1 2 ) 2 2 hiλ = (2π) n+ν 2 2 ( 2 1 )( n+ν ) 2

∞∞ ×

n+ν1 −2 2

s1

n+ν2 −2 2

s2

(s1 + s2 )

−n−iλ 2

(1 + s1 s2 )

−n+iλ 2

ds1 ds2 ,

(6.21)

0 0

from which it is easy to conclude. 1 2 Let us observe that, if not dealing with differential operators (i.e., when −n−ν and −n−ν are 2 2 not both non-negative integers), Moyal’s expansion (1.11) would lead in this example to a sum of terms with increasing singularities at 0, without significance, even asymptotic, as a distribution in R2n : however, let us hasten to say that microlocal analysis does not attach much significance to points of the phase space. 2 As a comment, let us express our conviction that the new composition formula has at best limited interest so far as applications of pseudodifferential analysis to partial differential equations are concerned. This is not to mean that symplectic covariance does not play any role in PDE: only, its role is essentially subordinate to that of the covariance under translations. It would be more correct to say that, in the more technical classes of symbols used in pseudodifferential analysis, it is rather the notion of uniformity under actions of conjugates of the group of translations under local families of symplectic transformations that is important. Here, our tilt is entirely towards the symplectic action, to the point that we have completely forgotten about the action of translations. On the other hand, automorphic pseudodifferential analysis calls for the present point of view, as experienced in the one-dimensional case: automorphic symbols are much too singular to be even remotely reminiscent of symbols in any of the classes developed for PDE applications. This does not imply that, to obtain the sharp composition of two automorphic symbols, it suffices to apply the present formula. Rather, the specific formula developed in this case, which has many special features inherent in the theory of modular forms, is based on the same principles (coupling symplectic covariance with the decomposition of automorphic symbols into their homogeneous components of a definite parity) as the ones which made the formula discussed here a natural one.

988


R2n ) 7. Irreducibility of the decomposition of L2 (R We prove here the irreducibility of most unitary representations appearing in the spectral decomposition of Proposition 2.1. In the last decades, general irreducibility results such as Kostant’s irreducibility theorem for spherical (minimal) principal series representations [6] and Vogan–Wallach’s irreducibility theorem for generic parameters [14] have been developed. Also, many specific cases have been studied in detail by R. Howe, E.-T. Tan, S.-T. Lee, S. Sahi, etc. by algebraic and combinatorial methods. However, to the best of our knowledge, neither the general theory nor the known special results contain Theorem 7.3 below, the proof of which is based on the extension of the idea of branching laws to non-compact subgroups [5] and on properties of the Weyl calculus in Rn−1 . Lemma 7.1. Let Mvect 0 = {S = (s1 , s∗ ; 0, σ∗ )} denote the linear space of translations of the affine hyperplane M0 . Given S ∈ M0 , define the linear automorphism TS of R2n by the equation TS X = X + [S, X]e1 + [e1 , X]S.

(7.1)

2n For every S ∈ Mvect 0 , TS is a symplectic transformation of R preserving M0 . The group of all such symplectic transformations is generated by the group N of transformations TS , S ∈ Mvect 0 , together with the group M of transformations (x1 , x∗ ; ξ1 , ξ∗ ) → (x1 , y∗ ; ξ1 , η∗ ), where the map (x∗ ; ξ∗ ) → (y∗ ; η∗ ) is a symplectic transformation in the 2n − 2 variables involved; the latter normalizes the first within Sp(n, R).

Proof. That [TS X, TS Y ] = [X, Y ] for every pair X, Y is an immediate consequence of the relations [e1 , e1 ] = [e1 , S] = [S, S] = 0. That the group MN generates the stabilizer of M0 is a consequence of the observation following (2.28). 2 Eq. (2.8) reduces when g ∈ MN to πν,δ (g)f (X) = f g −1 X ,

X ∈ M0 .

(7.2)

If one sets S∗∗ = (s∗ ; σ∗ ), X∗∗ = (x∗ ; ξ∗ ), the transformation T−S expresses itself when considered on M0 as T−S (x1 , x∗ ; 1, ξ∗ ) = x1 − 2s1 + [S∗∗ , X∗∗ ], x∗ − s∗ ; 1, ξ∗ − σ∗ :

(7.3)

it follows in particular that, given (iλ, δ) ∈ iR × {0, 1}, all transformations πiλ,δ (g) with g ∈ MN, when regarded as unitary transformations of L2 (M0 ), commute with the differential operator 1 ∂ 2iπ ∂x1 . Let us first decompose the restriction of the representation πiλ,δ to MN: from what has just been said, it can be analyzed when coupled with the spectral decomposition of the operator 1 ∂ 2iπ ∂x1 , in other words when fixing the first variable t in the partial Fourier transform F1 f of f ∈ L2 (M0 ), as already done in Section 2. From (7.2), one has if n 2 the identity F1 πiλ,δ (TS )f (t, x∗ ; ξ∗ ) = e−2iπt (2s1 −[S∗∗ ,X∗∗ ]) (F1 f )(t, x∗ − s∗ ; ξ∗ − σ∗ ),

(7.4)


989

a group of transformations in which we may regard t = 0 as a parameter by specializing to s1 = 0, (t) getting a projective representation πiλ,δ of R2n−2 , actually independent of (iλ, δ), as a result; the same is true when considering transformations F1 (πiλ,δ (g))F1−1 with g ∈ M. Lemma 7.2. Assume that n 2. For fixed t = 0, the linear space of bounded operators in (t) L2 (R2n−2 ) which commute with all transformations F1 (πiλ,δ (g))F1−1 with g ∈ MN is generated by the identity and the transformation F1 Σt F1−1 characterized by the equation e−2iπt[X∗∗ ,Y∗∗ ] (F1 f )(t, Y∗∗ ) dY∗∗ . (7.5) (F1 Σt f )(t, X∗∗ ) = |t|n−1 R2n−2

Proof. First assume that t = 2. Looking at (7.4), one sees that the linear space of infinitesimal operators of the representation of N under consideration is generated by the following operators, 1 ∂ where j, k 2: (i) the operators ξj + 4iπ ∂xj , where ξj denotes the operator of multiplication 1 ∂ by ξj ; (ii) the operators xk − 4iπ ∂ξk . From (1.11), these are just the operators h → ξj # h and h → xk # h. Taking advantage of the Weyl calculus in Rn−1 , set

(2) (2) (g)Op(h) = Op F1 πiλ,δ (g) F1−1 h ,

g ∈ MN,

(7.6)

defining in this way a unitary representation (2) of MN in the space of Hilbert–Schmidt operators in L2 (Rn−1 ). From what has just been seen, the image (2) (N ) consists of the automorphisms A → exp 2iπ η, Q − y, P A (7.7) (where the first factor was defined in the introduction). On the other hand, in view of (1.12), the image under (2) of M consists of the maps A → U AU −1 with U in the image of the metaplectic representation. Since the Heisenberg representation in L2 (Rn−1 ) is irreducible, while that of the metaplectic representation decomposes into its restrictions to spaces of functions with a given parity, it follows that the commutant of the representation (2) of MN is the linear space generated by the identity together with the automorphism A → ACh, where Ch is the parity map u → u, ˇ of the space of Hilbert–Schmidt operators in L2 (Rn−1 ). Going back to symbols and using what immediately follows (4.1), one obtains the case t = 2 of Lemma 7.2, from which one obtains the general case by a simple rescaling of coordinates of S. 2 Consider now any bounded operator K in the commutant of the representation πiλ,δ . Restricting the representation to MN, it follows from Lemma 7.2 that the operator F1 KF1−1 is a linear combination, with coefficients depending on t (the variable used in the definition of the partial Fourier transform), of the operators I and F1 Σt F1−1 . Introduce the group A of symplectic transformations of R2n defined as ga : (x, ξ ) → ax, a −1 ξ , a > 0. (7.8) From (2.8), one has πiλ,δ (ga )f (x1 , x∗ ; 1, ξ∗ ) = a −n−iλ f a −2 x1 , a −2 x∗ ; 1, ξ∗ .

(7.9)

990


Then, the operator K must also commute with the Euler operator j 1 xj ∂x∂ j , and the operator F1 KF1−1 must commute with the operator −t ∂t∂ + j 2 xj ∂x∂ j : after a change of variables in (7.5), it follows that the above-referred coefficients depend only on sign t. Theorem 7.3. Given any n 1, and any pair (iλ, δ) ∈ iR × {0, 1} such that (iλ, δ) = (0, 1) and (iλ, δ) = (0, 0), the representation πiλ,δ is irreducible; if (iλ, δ) = (0, 1), it decomposes as the direct sum of two irreducible representations, and such is the case if (iλ, δ) = (0, 0) and n 2. Proof. We may assume that n 2, since the one-dimensional case is classical [2]. From the considerations that precede in this section, any operator commuting with the representation πiλ,δ must lie in the algebra generated by the following two involutions: (i) the transformation Σ defined by (F1 f )(t, X∗∗ ) = |t|n−1

e−2iπt[X∗∗ ,Y∗∗ ] (F1 f )(t, Y∗∗ ) dY∗∗ ;

(7.10)

R2n−2 1 (ii) the transformation Ψ = sign( 2iπ

∂ ∂x1 )

defined by

F1 (Ψf ) (t, X∗∗ ) = (sign t)(F1 f )(t, X∗∗ ).

(7.11)

Looking at (2.26), one may note that Σ = θ0,0 and that the composition ΣΨ = Ψ Σ coincides with the intertwining operator θ0,1 . Now, θ0,1 is a non-trivial (i.e., distinct from a scalar) intertwining operator of the representation π0,1 with itself, and θ0,0 is an intertwining operator of the representation π0,0 with itself, non-trivial as soon as n 2. What remains to be seen, fixing n 2, is that the operator θ0,1 cannot commute with the representation πiλ,δ unless (iλ, δ) = (0, 1) and that the operator θ0,0 cannot commute with the representation πiλ,δ unless (iλ, δ) = (0, 0), finally that Ψ can never (if n 2) commute with a representation πiλ,δ . Given (iλ, δ), set Θj = θiλ,δ θ0,j

(7.12)

(F1 Θj f )(t, X∗∗ ) = |t|−iλ j −δ (F1 f )(t, X∗∗ ).

(7.13)

so that, from (2.27),

If θ0,j happens to be an intertwining operator from the representation πiλ,δ to itself, the operator Θj is an intertwining operator from πiλ,δ to π−iλ,δ . This operator, in its realization on L2 (M0 ), has an integral kernel which, evaluated at some pair ((x1 , X∗∗ ), (y1 , Y∗∗ )), is the product of some distribution in x1 − y1 by δ(X∗∗ − Y∗∗ ): as n 2, it is obvious that such an integral kernel, unless it is that of a scalar operator, cannot satisfy the covariance property that would make it an intertwining operator between two representations of the species under consideration. The same applies to the operator Ψ . 2


991

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

M. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970) 145–150. V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. 48 (1947) 568–640. I.N. Bernstein, S.I. Gelfand, Meromorphy of the function P λ , Funktsional. Anal. i Prilozhen. 3 (1969) 84–85. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math. (2) 79 (1964) 109–203; Ann. of Math. (2) 79 (1964) 205–326. T. Kobayashi, Branching problems of unitary representations, in: Proc. of ICM 2002, Beijing, vol. 2, 2002, pp. 615– 627. B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969) 627–642. P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions, Ann. of Math. Stud., vol. 87, Princeton Univ. Press, 1976. S.D. Miller, W. Schmid, The Rankin–Selberg method for automorphic distributions, in: Representation Theory and Automorphic Forms, in: Progr. Math., vol. 255, Birkhäuser Boston, Boston, MA, 2008, pp. 111–150. A.I. Osak, Trilinear Lorentz invariant forms, Comm. Math. Phys. 29 (1973) 189–217. L. Schwartz, Théorie des distributions, vols. 1, 2, Hermann, Paris, 1959. A. Unterberger, Quantization and Non-holomorphic Modular Forms, Lecture Notes in Math., vol. 1742, SpringerVerlag, Berlin, Heidelberg, 2000. A. Unterberger, Automorphic Pseudodifferential Analysis and Higher-Level Weyl Calculi, Progr. Math., vol. 209, Birkhäuser, Basel, Boston, Berlin, 2002. A. Unterberger, Quantization and Arithmetic, Pseudodifferential Operators, vol. 1, Birkhäuser, 2008. D.A. Vogan Jr., N.R. Wallach, Intertwining operators for real reductive groups, Adv. Math. 82 (1990) 203–243. A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964) 143–211.


Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups ✩ Giuseppe Da Prato b , Michael Röckner c,d , Feng-Yu Wang a,e,∗ a School of Math. Sci. and Lab. Math Com. Sys., Beijing Normal University, 100875, China b Scuola Normale Superiore di Pisa, Italy c Faculty of Mathematics, University of Bielefeld, Germany d Department of Mathematics and Statistics, Purdue University, W. Lafayette, 47906 IN, USA e Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, Swansea, UK

Received 3 January 2009; accepted 7 January 2009 Available online 21 January 2009 Communicated by Paul Malliavin

Abstract We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261–303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic differential equations; Harnack inequality; Monotone coefficients; Yosida approximation; Kolmogorov operators

✩ Supported in part by “Equazioni di Kolmogorov” from the Italian “Ministero della Ricerca Scientifica e Tecnologica”, WIMCS, Creative Research Group Fund of the National Natural Science Foundation of China (No. 10721091), the 973Project, the DFG through SFB-701 and IRTG 1132, by NSF-Grant 0603742 as well as by the BIBOS-Research Center. * Corresponding author at: School of Math. Sci. and Lab. Math Com. Sys. Beijing Normal University, 100875, China. E-mail address: [email protected] (F.-Y. Wang).

0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.007

G. Da Prato et al. / Journal of Functional Analysis 257 (2009) 992–1017

993

1. Introduction, framework and main results In this paper we continue our study of stochastic equations in Hilbert spaces with singular drift through its associated Kolmogorov equations started in [6]. The main aim is to prove a Harnack inequality for its transition semigroup in the sense of [16] (see also [1,14,17] for further development) and exploit its consequences. See also [12] for an improvement of the main results in [14] concerning generalized Mehler semigroups. To describe our results more precisely, let us first recall the framework from [6]. Consider the stochastic equation dX(t) = AX(t) + F X(t) dt + σ dW (t), (1.1) X(0) = x ∈ H. Here H is a real separable Hilbert space with inner product ·,· and norm | · |, W = W (t), t 0, is a cylindrical Brownian motion on H defined on a stochastic basis (Ω, F , (Ft )t0 , P) and the coefficients satisfy the following hypotheses: (H1) (A, D(A)) is the generator of a C0 -semigroup, Tt = etA , t 0, on H and for some ω ∈ R Ax, x ω|x|2 ,

∀x ∈ D(A).

(1.2)

(H2) σ ∈ L(H ) (the space of all bounded linear operators on H ) such that σ is positive definite, self-adjoint and ∞ (i) 0 (1 + t −α )Tt σ 2HS dt < ∞ for some α > 0, where · HS denotes the norm on the space of all Hilbert–Schmidt operators on H . (ii) σ −1 ∈ L(H ). (H3) F : D(F ) ⊂ H → 2H is an m-dissipative map, i.e., u − v, x − y 0,

∀x, y ∈ D(F ), u ∈ F (x), v ∈ F (y),

(“dissipativity”) and Range(I − F ) :=

x − F (x) = H. x∈D(F )

Furthermore, F0 (x) ∈ F (x), x ∈ D(F ), is such that F0 (x) = min |y|. y∈F (x)

Here we recall that for F as in (H3) we have that F (x) is closed, nonempty and convex. The corresponding Kolmogorov operator is then given as follows: Let EA (H ) denote the linear span of all real parts of functions of the form ϕ = eih,· , h ∈ D(A∗ ), where A∗ denotes the adjoint operator of A, and define for any x ∈ D(F ), L0 ϕ(x) =

1 2 2 Tr σ D ϕ(x) + x, A∗ Dϕ(x) + F0 (x), Dϕ(x) , 2

Additionally, we assume:

ϕ ∈ EA (H ).

(1.3)

994


(H4) There exists a probability measure μ on H (equipped with its Borel σ -algebra B(H )) such that (i) μ(D(F )) = 1, (ii) H (1 + |x|2 )(1 + |F0 (x)|)μ(dx) < ∞, (iii) H L0 ϕdμ = 0 for all ϕ ∈ EA (H ). Remark 1.1. (i) A measure for which the last equality in (H4) (makes sense and) holds is called infinitesimally invariant for (L0 , EA (H )). (ii) Since ω in (1.2) is an arbitrary real number we can relax (H3) by allowing that for some c ∈ (0, ∞) u − v, x − y c|x − y|2 ,

∀x, y ∈ D(F ), u ∈ F (x), v ∈ F (y).

We simply replace F by F − c and A by A + c to reduce this case to (H3). (iii) At this point we would like to stress that under the above assumptions (H1)–(H4) (and (H5) below) because F0 is merely measurable and σ is not Hilbert–Schmidt, it is unknown whether (1.1) has a strong solution. (iv) Similarly as in [6] (see [6, Remark 4.4] in particular) we expect that (H2)(ii) can be relaxed to the condition that σ = (−A)−γ for some γ ∈ [0, 1/2]. However, some of the approximation arguments below become more involved. So, for simplicity we assume (H2)(ii). The following are the main results of [6] which we shall use below. Theorem 1.2. (Cf. [5, Theorem 2.3 and Corollary 2.5].) Assume (H1), (H2)(i), (H3) and (H4). Then for any measure μ as in (H 4) the operator (L0 , EA (H )) is dissipative on L1 (H, μ), hence μ closable. Its closure (Lμ , D(Lμ )) generates a C0 -semigroup Pt , t 0, on L1 (H, μ) which μ μ is Markovian, i.e., Pt 1 = 1 and Pt f 0 for all nonnegative f ∈ L1 (H, μ) and all t > 0. μ Furthermore, μ is Pt -invariant, i.e.,

μ Pt f dμ = f dμ, ∀f ∈ L1 (H, μ). H

H

Below Bb (H ), Cb (H ) denote the bounded Borel-measurable, continuous functions respectively from H into R and · denotes the usual norm on L(H ). Theorem 1.3. (Cf. [5, Proposition 5.7].) Assume (H1)–(H4) hold. Then for any measure μ as in (H4) and H0 := supp μ (:= largest closed set of H whose complement is a μ-zero set) there μ μ μ exists a semigroup pt (x, dy), x ∈ H0 , t > 0, of kernels such that pt f is a μ-version of Pt f for all f ∈ Bb (H ), t > 0, where as usual

μ μ pt f (x) = f (y)pt (x, dy), x ∈ H0 . H

Furthermore, for all f ∈ Bb (H ), t > 0, x, y ∈ H0 , |ω|t μ pt f (x) − ptμ f (y) √e f 0 σ −1 |x − y| t ∧1

(1.4)


and for all f ∈ Lipb (H ) (:= all bounded Lipschitz functions on H ) μ pt f (x) − ptμ f (y) e|ω|t f Lip |x − y|, ∀t > 0, x, y ∈ H0 ,

995

(1.5)

and μ

lim pt f (x) = f (x),

t→0

∀x ∈ H0 .

(1.6) μ

(Here f 0 , f Lip denote the supremum, Lipschitz norm of f respectively.) Finally, μ is pt invariant. Remark 1.4. (i) Both results above have been proved in [6] on L2 (H, μ) rather than on L1 (H, μ), but the proofs for L1 (H, μ) are entirely analogous. (ii) In [6] we assume ω in (H1) to be negative, getting a stronger estimate than (1.4) (cf. [6, (5.11)]). But the same proof as in [6] leads to (1.4) for arbitrary ω ∈ R (cf. the proof of [6, μ Proposition 4.3] for t ∈ [0, 1]). Then by virtue of the semigroup property and since pt is Markov we get (1.4) for all t > 0. (iii) Theorem 1.3 holds in more general situations since (H2)(ii) can be relaxed (cf. [6, Remark 4.4] and [4, Proposition 8.3.3]). μ μ (iv) (1.4) above implies that pt , t > 0, is strongly Feller, i.e., pt (Bb (H )) ⊂ C(H0 ) (= all continuous functions on H0 ). We shall prove below that under the additional condition (H5) we μ even have pt (Lp (H, μ)) ⊂ C(H0 ) for all p > 1 and that μ in (H4) is unique. However, so far we have not been able to prove that for this unique μ we have supp μ = H , though we conjecture that this is true. For the results on Harnack inequalities, in this paper we need one more condition. (H5) (i) (1 + ω − A, D(A)) satisfies the weak sector condition (cf. e.g. [10]), i.e., there exists a constant K > 0 such that 1/2 1/2 (1 + ω − A)y, y , (1 + ω − A)x, y K (1 + ω − A)x, x ∀x, y ∈ D(A).

(1.7)

(ii) There exists a sequence of A-invariant finite dimensional subspaces Hn ⊂ D(A) such that ∞ n=1 Hn is dense in H . We note that if A is self-adjoint, then (H2) implies that A has a discrete spectrum which in turn implies that (H5)(ii) holds. Remark 1.5. Let (A, D(A)) satisfy (H1). Then the following is well known: (i) (H5)(i) is equivalent to the fact that the semigroup generated by (1 + ω − A, D(A)) on the complexification HC of H is a holomorphic contraction semigroup on HC (cf. e.g. [10, Chapter I, Corollary 2.21]). (ii) (H5)(i) is equivalent to (1 + ω − A, D(A)) being variational. Indeed, let (E, D(E)) be the coercive closed form generated by (1 + ω − A, D(A)) (cf. [10, Chapter I, Section 2]) and

D(E)) be its symmetric part. Then define (E, V := D(E)

with inner product E and V ∗ to be its dual.

(1.8)

996


Then V ⊂H ⊂V∗

(1.9)

and 1 + ω − A : D(A) → H has a natural unique continuous extension from V to V ∗ satisfying all the required properties (cf. [10, Chapter I, Section 2, in particular Remark 2.5]). μ

Now we can formulate the main result of this paper, namely the Harnack inequality for pt , t > 0. μ

Theorem 1.6. Suppose (H1)–(H5) hold and let μ be any measure as in (H4) and pt (x, dy) as in Theorem 1.3 above. Let p ∈ (1, ∞). Then for all f ∈ Bb (H ), f 0, 2 μ p μ p pt f (x) pt f (y) exp σ −1

pω|x − y|2 , (p − 1)(1 − e−2ωt )

t > 0, x, y ∈ H0 . (1.10)

As consequences in the situation of Theorem 1.6 (i.e. assuming (H1)–(H5)) we obtain: Corollary 1.7. For all t > 0 and p ∈ (1, ∞) μ pt Lp (H, μ) ⊂ C(H0 ). Corollary 1.8. μ in (H4) is unique. μ

Because of this result below we write pt (x, dy) instead of pt (x, dy). Finally, we have Corollary 1.9. (i) For every x ∈ H0 , pt (x, dy) has a density ρt (x, y) with respect to μ and ρt (x, ·)p/(p−1) p

1 , 2 −1 2 pω|x−y| μ(dy) exp −σ −2ωt H (1−e )

x ∈ H0 , p ∈ (1, ∞).

(1.11)

(ii) If μ(eλ|·| ) < ∞ for some λ > 2(ω ∧ 0)2 σ −1 2 , then pt is hyperbounded, i.e. pt L2 (H,μ)→L4 (H,μ) < ∞ for some t > 0. 2

Corollary 1.10. For simplicity, let σ = I and instead of (H1) assume that more strongly (A, D(A)) is self-adjoint satisfying (1.2). We furthermore assume that |F0 | ∈ L2 (H, μ). (i) There exists M ∈ B(H0 ), M ⊂ D(F ), μ(M) = 1 such that for every x ∈ M Eq. (1.1) has a pointwise unique continuous strong solution (in the mild sense see (4.11) below), such that X(t) ∈ M for all t 0 P-a.s.


997

(ii) Suppose there exists Φ ∈ C([0, ∞)) positive and strictly increasing such that lims→∞ s −1 Φ(s) = ∞ and

∞ Ψ (s) :=

dr < ∞, Φ(r)

∀s > 0.

(1.12)

s

If there exists a constant c > 0 such that F0 (x) − F0 (y), x − y c − Φ |x − y|2 ,

∀x, y ∈ D(F ),

(1.13)

then pt is ultrabounded with λ(1 + Ψ −1 (t/4)) pt L2 (H,μ)→L∞ (H,μ) exp , (1 − ε −ωt/2 )2

t > 0,

holding for some constant λ > 0. Remark 1.11. We emphasize that since the nonlinear part F0 of our Kolmogorov operator is in general not continuous, it was quite surprising for us that in this infinite dimensional case nevertheless the generated semigroup Pt maps L1 -functions to continuous ones as stated in Corollary 1.7. The proof that Corollary 1.9 follows from Theorem 1.6 is completely standard. So, we will omit the proofs and instead refer to [14,17]. Corollary 1.7 is new and follows whenever a semigroup pt satisfies the Harnack inequality (see Proposition 4.1 below). μ Corollary 1.8 is new. Since (1.10) implies irreducibility of pt and Corollary 1.7 implies that it is strongly Feller, a well known theorem due to Doob immediately implies that μ is the unique μ μ invariant measure for pt , t > 0. pt , however, depends on μ, so Corollary 1.8 is a stronger statement. Corollary 1.10 is also new. Theorem 1.6 as well as Corollaries 1.7, 1.8 and 1.10 will be proved in Section 4. In Section 3 we first prove Theorem 1.6 in case F0 is Lipschitz, and in Section 2 we prepare the tools that allow us to reduce the general case to the Lipschitz case. In Section 5 we prove two results (see Theorems 5.2 and 5.4) on the existence of a measure satisfying (H4) under some additional conditions and present an application to an example where F0 is not continuous. For a discussion of a number of other explicit examples satisfying our conditions see [6, Section 9]. 2. Reduction to regular F0 Let F be as in (H3). As in [6] we may consider the Yosida approximation of F , i.e., for any α > 0 we set 1 Jα (x) − x , α

x ∈ H,

Jα (x) := (I − αF )−1 (x),

α > 0,

Fα (x) := where for x ∈ H

(2.1)

998


and I (x) := x. Then each Fα is single valued, dissipative and it is well known that lim Fα (x) = F0 (x), ∀x ∈ D(F ), Fα (x) F0 (x), ∀x ∈ D(F ).

α→0

(2.2) (2.3)

Moreover, Fα is Lipschitz continuous, so F0 is B(H )-measurable. Since Fα is not differentiable in general, as in [6] we introduce a further regularization by setting

(2.4) Fα,β (x) := eβB Fα eβB x + y N 1 B −1 (e2βB −1) (dy), α, β > 0, 2

H

where B : D(B) ⊂ H → H is a self-adjoint, negative definite linear operator such that B −1 is of trace class and as usual for a trace class operator Q the measure NQ is just the standard centered Gaussian measure with covariance given by Q. Fα,β is dissipative, of class C ∞ , has bounded derivatives of all the orders and Fα,β → Fα pointwise as β → 0. Furthermore, for α > 0 |Fα,β (x)| cα := sup : x ∈ H, β ∈ (0, 1] < ∞. (2.5) 1 + |x| We refer to [8, Theorem 9.19] for details. Now we consider the following regularized stochastic equation

dXα,β (t) = AXα,β (t) + Fα,β Xα,β (t) dt + σ dW (t), Xα,β (0) = x ∈ H.

(2.6)

It is well known that (2.6) has a unique mild solution Xα,β (t, x), t 0. Its associated transition semigroup is given by α,β

Pt

f (x) = E f Xα,β (t, x) ,

t > 0, x ∈ H,

for any f ∈ Bb (H ). Here E denotes expectation with respect to P. Proposition 2.1. Assume (H1)–(H4). Then there exists a Kσ -set K ⊂ H such that μ(K) = 1 and for all f ∈ Bb (H ), T > 0 there exist subsequences (αn ), (βn ) → 0 such that for all x ∈ K lim

lim P•αn ,βm f (x) = p•μ f (x)

n→∞ m→∞

weakly in L2 (0, T ; dt).

(2.7)

Proof. This follows immediately from the proof of [6, Proposition 5.7]. (A closer look at the proof even shows that (2.7) holds for all x ∈ H0 = supp μ.) 2 As we shall see in Section 4, the proof of Theorem 1.6 follows from Proposition 2.1 if we can α,β prove the corresponding Harnack inequality for each Pt . Hence in the next section we confine ourselves to the case when F0 is dissipative and Lipschitz.


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3. The Lipschitz case In this section we assume that (H1)–(H3) and (H5) hold and that F0 in (H3) is in addition Lipschitz continuous. The aim of this section is to prove Theorem 1.6 for such special F0 (see Proposition 3.1 below). We shall do this by finite dimensional (Galerkin) approximations, since for the approximating finite dimensional processes we can apply the usual coupling argument. We first note that since F0 is Lipschitz (1.1) has a unique mild solution X(t, x), t 0, for every initial condition x ∈ H (cf.[8]) and we denote the corresponding transition semigroup by Pt , t > 0, i.e. Pt f (x) := E f X(t, x) ,

t > 0, x ∈ X,

where f ∈ Bb (H ). Now we need to consider an appropriate Galerkin approximation. To this end let ek ∈ D(A), k ∈ N, be orthonormal such that Hn = linear span{e1 , . . . , en }, n ∈ N. Hence {ek : k ∈ N} is an orthonormal basis of (H, ·,·). Let πn : H → Hn be the orthogonal projection with respect to (H, ·,·). So, we can define An := πn A|Hn = A|Hn by (H5)(ii)

(3.1)

and, furthermore Fn := πn F0|Hn ,

σn := πn σ|Hn .

Obviously, σn : Hn → Hn is a self-adjoint, positive definite linear operator on Hn . Furthermore, σn is bijective, since it is one-to-one. To see the latter, one simply picks an orthonormal basis {e1σ , . . . , enσ } of Hn with respect to the inner product ·,·σ defined by x, yσ := σ x, y. Then if x ∈ Hn is such that σn x = πn σ x = 0, it follows that

x, eiσ

σ

= σ x, eiσ = 0,

∀1 i n.

But x = ni=1 x, eiσ σ eiσ , hence x = 0. Now fix n ∈ N and on Hn consider the stochastic equation

dXn (t) = An Xn (t) + Fn Xn (t) dt + σn dWn (t), Xn (0) = x ∈ Hn ,

(3.2)

where Wn (t) = πn W (t) = ni=1 ek , W (t)ek . (3.2) has a unique strong solution Xn (t, x), t 0, for every initial condition x ∈ Hn which is pathwise continuous P-a.s. Consider the associated transition semigroup defined as before by Ptn f (x) = E f Xn (t, x) , where f ∈ Bb (Hn ).

t > 0, x ∈ Hn ,

(3.3)

1000


Below we shall prove the following: Proposition 3.1. Assume that (H1)–(H5) hold. Then: (i) For all f ∈ Cb (H ) and all t > 0, lim Ptn f (x) = Pt f (x),

n→∞

∀x ∈ Hn0 , n0 ∈ N.

(ii) For all nonnegative f ∈ Bb (H ) and all n ∈ N , p ∈ (1, ∞) 2 n p pω|x − y|2 Pt f (x) Ptn f p (y) exp σ −1 , (p − 1)(1 − e−2ωt )

t > 0, x, y ∈ Hn .

(3.4)

Proof. (i) Define

t WA,σ (t) :=

t 0.

e(t−s)A σ dW (s), 0

Note that by (H2)(i) we have that WA,σ (t), t 0, is well defined and pathwise continuous. For x ∈ Hn0 , n0 ∈ N fixed, let Z(t), t 0, be the unique variational solution (with triple V ⊂ H ⊂ V ∗ as in Remark 1.5(ii), see e.g. [13]) to dZ(t) = AZ(t) + F0 Z(t) + WA,σ (t) dt, (3.5) Z(0) = x, which then automatically satisfies 2 E sup Z(t) < +∞.

(3.6)

t∈[0,T ]

Then we have (see [8]) that Z(t) + WA,σ (t), t 0, is a mild solution to (1.1) (with F0 Lipschitz), hence by uniqueness X(t, x) = Z(t) + WA,σ (t),

t 0.

(3.7)

Clearly, since 2 E sup WA,σ (t) < +∞,

(3.8)

t∈[0,T ]

we have πn WA,σ (t) → WA,σ (t)

as n → ∞ in L2 (Ω, F , P), ∀t 0.

We set Xn (t) := Xn (t, x) (= solution of (3.2)). Defining

t WAn ,σn (t) =

e(t−s)An σn dWn (t), 0

t 0,


1001

and Zn (t) := Xn (t) − WAn ,σn (t),

n ∈ N, t 0,

it is enough to show that Zn (t) → Z(t)

as n → ∞ in L2 (Ω, F , P), ∀t 0,

Xn (t) → X(t)

as n → ∞ in L2 (Ω, F , P), ∀t 0,

(3.9)

because then by (3.7)

and the assertion follows by Lebesgue’s dominated convergence theorem. To show (3.9) we first note that by the same argument as above dZn (t) = An Zn (t) + Fn Zn (t) + WAn ,σn (t) dt and thus (in the variational sense), since A = An on Hn by (3.1) d Z(t) − Zn (t) = A Z(t) − Zn (t) + F0 X(t) − Fn Xn (t) dt. Applying Itô’s formula we obtain that for some constant c > 0 2 1 Z(t) − Zn (t) 2

t

2 (ω + 1/2)Z(s) − Zn (s)

0

2 2 + F0 X(s) − F0 Xn (s) + (1 − πn )F0 X(s) ds

t

c

Z(s) − Zn (s)2 ds + c

0

t

WA,σ (s) − WA ,σ (s)2 ds n n

0

t +

(1 − πn )F0 X(s) 2 ds.

0

Now (3.9) follows by the linear growth of F0 , (3.6)–(3.8) and Gronwall’s lemma, if we can show that

T

2 EWA,σ (s) − WAn ,σn (s) ds → 0 as n → ∞.

(3.10)

0

To this end we first note that a straightforward application of Duhamel’s formula yields that etA |Hn = etAn

∀t 0.

1002


Therefore

s WA,σ (s) − WAn ,σn (s) =

e(t−r)A (σ − πn σ πn ) dW (r), 0

and thus EWA,σ (s) − WA

n ,σn

2 (s) =

s

2 (t−r)A e (σ − πn σ πn ) dr

0 ∞

rA e (σ − πn σ πn )ei 2 dr. = s

i=1 0

Since for any i ∈ N, r ∈ [0, s], the integrands converge to 0, Lebesgue’s dominated convergence theorem implies (3.10). (ii) Fix T > 0, n ∈ N and x, y ∈ Hn . Let ξ T ∈ C 1 ([0, ∞)) be defined by ξ T (t) :=

2ωe−ωt |x − y| , 1 − e−2ωT

t 0.

Consider for Xn (t) = Xn (t, x), t 0, see the proof of (i), the stochastic equation ⎧ ⎪ T (t) Xn (t) − Yn (t) 1 ⎪ dY Y (t) = A Y (t) + F (t) + ξ ⎪ n n n n X (t) =Yn (t) dt ⎨ n |Xn (t) − Yn (t)| n ⎪ + σn dWn (t), ⎪ ⎪ ⎩ Yn (0) = y.

(3.11)

Since z→

Xn (t) − z 1X (t) =z |Xn (t) − z| n

is dissipative on Hn for all t 0 (cf. [17]), (3.11) has a unique strong solution Yn (t) = Yn (t, y), t 0, which is pathwise continuous P-a.s. Define the first coupling time (3.12) τn := inf t 0: Xn (t) = Yn (t) . Writing the equation for Xn (t) − Yn (t), t 0, applying the chain rule to φ (z) := z ∈ (− 2 , ∞), > 0, and letting → 0 subsequently, we obtain d Xn (t) − Yn (t) ωXn (t) − Yn (t) − ξ T (t)1Xn (t) =Yn (t) dt

√ z + 2,

t 0,

which yields d e−ωt Xn (t) − Yn (t) −e−ωt ξ T (t)1Xn (t) =Yn (t) dt,

t 0.

(3.13)


1003

In particular, t → e−ωt |Xn (t) − Yn (t)| is decreasing, hence Xn (T ) = Yn (T ) for all T τn . But by (3.13) if T τn then Xn (T ) − Yn (T )e−ωT |x − y| − |x − y|

T 0

2ωe−2ωt dt = 0. 1 − e−2ωT

So, in any case Xn (T ) = Yn (T ),

P-a.s.

(3.14)

Let

T ∧τn

R := exp − 0

1 − 2

T ∧τn

0

ξ T (t) Xn (t) − Yn (t), σ −1 dWn (t) |Xn (t) − Yn (t)|

(ξ T (t))2 |σ −1 (Xn (t) − Yn (t))|2 dt . |Xn (t) − Yn (t)|2

By (3.14) and Girsanov’s theorem for p > 1, p p p n = E Rf Xn (T ) PT f (y) = E f Yn (T ) p−1 PTn f p (x) E R p/(p−1) .

(3.15)

Let

p Mp = exp − p−1 p2 − 2(p − 1)2

T ∧τn

0 T ∧τn

0

ξ T (t) Xn (t) − Yn (t), σ −1 dWn (t) |Xn (t) − Yn (t)| (ξ T (t))2 |σ −1 (Xn (t) − Yn (t))|2 dt . |Xn (t) − Yn (t)|2

We have EMp = 1 and hence,

ER

p/(p−1)

p = E Mp exp 2(p − 1)2

p sup exp 2(p − 1)2 Ω 2 exp σ −1

T ∧τn

0

(ξ T (t))2 |σ −1 (Xn (t) − Yn (t))|2 dt |Xn (t) − Yn (t)|2

T ∧τn

T 2 −1 2 ξ (t) σ dt

0

pω|x − y|2 . (p − 1)2 (1 − e−2ωT )

Combining this with (3.15) we get the assertion (with T replacing t).

2

1004


4. Proof and consequences of Theorem 1.6 On the basis of Propositions 3.1 and 2.1 we can now easily prove Theorem 1.6. Proof of Theorem 1.6. Let f ∈ Lipb (H ), f 0. By Proposition 3.1(i) it then follows that (3.4) holds with Pt f replacing Ptn f provided F is Lipschitz. Using that n∈N Hn is dense in H and that Pt f (x) is continuous on x (cf. [8]) we obtain (3.4) for all x, y ∈ H . In particular, this is true α ,β for Pt n n f from Proposition 2.1. Now fix t > 0 and k ∈ N, let 1 χk (s) := 1[t,t+1/k] (s), k αn ,βm

Using (3.4) for Pt

μ pt f (x) =

1 lim k→∞ k

s 0.

f , (1.6), Proposition 2.1 and Jensen’s inequality, we obtain for x, y ∈ K

t+1/k

psμ f (x) ds t

= lim lim

t+1 lim χk (s)Psαn ,βm f (x) dx

k→∞ n→∞ m→∞

0

lim lim

t+1 2 α ,β p 1/p n m lim χk (s) Ps f (y) exp σ −1

lim lim

t+1 2 lim χk (s)Psαn ,βm f p (y) exp σ −1

k→∞ n→∞ m→∞

0

k→∞ n→∞ m→∞

0

2 μ p 1/p = pt f (y) exp σ −1

ω|x − y|2 ds (p − 1)(1 − e−2ωs )

pω|x − y|2 ds (p − 1)(1 − e−2ωs )

1/p

ω|x − y|2 , (p − 1)(1 − e−2ωt )

where we note that we have to choose the sequences (αn ), (βn ) such that (2.7) holds both for f and f p instead of f . Since K is dense in H0 , (1.10) follows for f ∈ Cb (H ), for all x, y ∈ H0 , μ since pt f is continuous on H0 by (1.4). Let now f ∈ Bb (H ), f 0. Let fn ∈ Cb (H ), n ∈ N, such that fn → f in Lp (H, μ) as μ n → ∞, p ∈ (1, ∞) fixed. Then, since μ is invariant for pt , t > 0, selecting a subsequence if necessary, it follows that there exists K1 ∈ B(H ), μ(K1 ) = 1, such that μ

μ

pt fn (x) → pt f (x)

as n → ∞,

∀x ∈ K1 . μ

Taking this limit in (1.10) we obtain (1.10) for all x, y ∈ K1 . Taking into account that pt is continuous and that K1 is dense in H0 = supp μ, (1.10) follows for all x, y ∈ H0 . 2 Corollary 1.7 immediately follows from Theorem 1.6 and the following general result:


1005

Proposition 4.1. Let E be a topological space and P a Markov operator on Bb (E). Assume that for any p > 1 there exists a continuous function ηp on E × E such that ηp (x, x) = 0 for all x ∈ E and 1/p η (x,y) P |f |(x) P |f |p (y) e p

∀x, y ∈ E, f ∈ Bb (E).

(4.1)

Then P is strong Feller, i.e. maps Bb (E) into Cb (E). Furthermore, for any σ -finite measure μ on (E, B(E)) such that

|Pf | dμ C |f | dμ, ∀f ∈ Bb (E), (4.2) E

E

for some C > 0, P uniquely extends to Lp (E, μ) with P Lp (E, μ) ⊂ C(E) for any p > 1. Proof. Since P is linear, we only need to consider f 0. Let f ∈ Bb (E) be nonnegative. By (4.1) and the property of ηp we have 1/p lim sup Pf (x) Pf p (y) ,

p > 1.

x→y

Letting p ↓ 1 we obtain lim supx→y Pf (x) Pf (y). Similarly, using f 1/p to replace f and replacing x with y, we obtain 1/p p Pf (y) Pf (x) epηp (y,x) ,

∀x, y ∈ E, p > 1.

First letting x → y then p → 1, we obtain lim infx→y Pf (x) Pf (y). So Pf ∈ Cb (E). Next, for any nonnegative f ∈ Lp (E, μ), let fn = f ∧ n, n 1. By (4.2) and fn → f in Lp (E, μ) we have P |fn − fm |p → 0 in L1 (E, μ) as n, m → ∞. In particular, there exists y ∈ E such that lim P |fn − fm |p (y) = 0.

n,m→∞

(4.3)

Moreover, by (4.1), for BN := {x ∈ E: ηp (x, y) < N} p p sup Pfn (x) − Pfm (x) sup P |fn − fm |(x) P |fn − fm |p (y) epN . x∈BN

x∈BN

Since by the strong Feller property Pfn ∈ Cb (E) for any n 1 and noting that Cb (BN ) is complete under the uniform norm, we conclude from (4.3) that Pf is continuous on BN for any N 1, and hence, Pf ∈ C(H ). 2 Proof of Corollary 1.8. Let μ1 , μ2 be probability measures on (H, B(H )) satisfying (H4). Define μ := 12 μ1 + 12 μ2 . Then μ satisfies (H4) and μi = ρi μ, i = 1, 2, for some B(H )-measurable ρi : H → [0, 2]. Let i ∈ {1, 2}. Since ρi is bounded, by (H4)(iii) and Theorem 1.2 it follows that

Lμ u dμi = 0, ∀u ∈ D(Lμ ). H

1006


Hence d dt

Lμ etLμ u dμi = 0,

etLμ u dμi = H

∀u ∈ D(Lμ ),

H

i.e.

μ pt u dμi

H

=

u dμi

∀u ∈ EA (H ).

H μ

Since EA (H ) is dense in L1 (H, μi ), μi is (pt )-invariant. But as mentioned before, by Theμ orem 1.6 it follows that (pt ) is irreducible on H0 (see [9]) and it is strong Feller on H0 by Corollary 1.7. So, since μi (H0 ) = 1, μi = μ. 2 Proof of Corollary 1.10. Let A˜ := A − ωI,

˜ := D(A), D(A)

F˜0 := F0 + ωI. By (H2), A˜ has discrete spectrum. Let ek ∈ H , −λk ∈ (−∞, 0], be the corresponding orthonormal eigenvectors, eigenvalues respectively. For k ∈ N define ϕk (x) := ek , x,

x ∈ H.

We note that by a simple approximation (1.5) also holds for any Lipschitz function on H and thus (cf. the proof of [6, Proposition 5.7(iii)]) also (1.6) holds for such functions, i.e. in particular, for all k ∈ N [0, ∞) t → pt ϕk (x)

is continuous for all x ∈ H0 .

(4.4)

Since any compactly supported smooth function on RN is the Fourier transform of a Schwartz test function, by approximation it easily follows that setting F Cb∞ {ek } := g e1 , ·, . . . , eN , · : N ∈ N, g ∈ Cb∞ RN , we have F Cb∞ ({ek }) ⊂ D(Lμ ) and for ϕ ∈ F Cb∞ ({ek }) Lμ ϕ(x) =

1 2 Tr D ϕ(x) + x, ADϕ(x) + F0 (x), Dϕ(x) , 2

x ∈ H.

Then by approximation it is easy to show that ϕk , ϕk2 ∈ D(Lμ )

and Lμ ϕk = −λk ϕk + ek , F˜0 ,

Lμ ϕk2 = −2λk ϕk2 + 2ϕk ek , F˜0 + 2,

∀k ∈ N.

(4.5)


1007

Since we assume that |F0 | is in L2 (H, μ), by [3, Theorem 1.1] we are in the situation of [15, Chapter II]. So, we conclude that by [15, Chapter II, Theorem 1.9] there exists a normal (that is Px [X(0) = x] = 1) Markov process (Ω, F , (Ft )t0 , (X(t))t0 , (Px )x∈H0 ) with state space H0 and M ∈ B(H0 ), μ(M) = 1, such that X(t) ∈ M for all t 0 Px -a.s. for all x ∈ M and which has continuous sample paths Px -a.s for all x ∈ M and for which by the proof of [6, Proposition 8.2] and (4.4), (4.5) we have that for all k ∈ N

βkx (t) := ϕk

X(t) − ϕk (x) −

t

Lμ ϕk X(s) ds,

t 0,

0

Mkx (t) := ϕk2

X(t) − ϕk2 (x) −

t

Lμ ϕk2 X(s) ds,

t 0,

(4.6)

0

are continuous local (Ft )-martingales with βkx (0) = Mk (0) = 0 under Px for all x ∈ M. Fix x ∈ M. Below Ex denotes expectation with respect to Px . Since for T > 0

T

2 Ex 1 + X(s) 1 + F0 X(s) dsμ(dx)

H 0

1 + |x|2 1 + F0 (x) μ(dx) < ∞,

=T H

making M smaller if necessary, by (H4)(ii) we may assume that

T Ex

2 1 + X(s) 1 + F0 X(s) ds < ∞.

(4.7)

0

By standard Markov process theory we have for their covariation processes under Px ,

βkx , βkx t

t =

Dϕk X(s) , Dϕk X(s) ds = tδk,k ,

t 0.

(4.8)

0

Indeed, an elementary calculation shows that for all k ∈ N, t 0,

t βkx (t)2

−

Dϕk X(s) 2 ds

0

t = Mkx (t) − 2ϕk (x)βkx (t) − 0

x βk (t) − βkx (s) Lμ ϕk X(s) ds,

(4.9)

1008


where all three summands on the right-hand side are martingales. Since we have a similar formula for finite linear combinations of ϕk s replacing a single ϕk , by polarization we get (4.8). Note that by (4.5) and (4.7) all integrals in (4.6), (4.9) are well defined. Hence, by (4.8) βkx , k ∈ N, are independent standard (Ft )-Brownian motions under Px . Now it follows by [11, Theorem 13] that, with W x = (W x (t))t0 , being the cylindrical Wiener process on H given by W x = (βkx ek )k∈N , we have for every t 0,

t X(t) = e x + tA

e

(t−s)A

F0 X(s) ds +

0

t e(t−s)A dW x (s),

P-a.s.,

(4.10)

0

that is, the tuple (Ω, F , (Ft )t0 , Px , W x , X) is a solution to ⎧

t

t ⎪ ⎪ ⎨ Y (t) = etA Y (0) + e(t−s)A F Y (s) ds + e(t−s)A dW (s), 0 ⎪ 0 0 ⎪ ⎩ law Y (0) = δx (:= Dirac measure in x),

P-a.s., ∀t 0,

(4.11)

in the sense of [11, p. 4]. We note that the zero set in (4.10) is indeed independent of t, since all terms are continuous in t Px -a.s. because of (H2)(ii) and (4.7). Claim. We have X-pathwise uniqueness for Eq. (4.11) (in the sense of [11, p. 98]). For any given cylindrical (Ft )-Wiener process W on a stochastic basis (Ω , F , (Ft )t0 , P ) let Y = Y (t), Z = Z(t), t 0, be two solutions of (4.11) such that law(Z) = law(Y ) = law(X) and Y (0) = Z(0) P -a.s. Then by (4.7)

E

T

F0 Y (s) ds = E

T

0

F0 Z(s) ds = Ex

0

T

F0 X(s) ds < ∞

(4.12)

0

(which, in particular implies by (4.11) and by (H2)(i) that both Y and Z have P -a.s. continuous sample paths). Hence applying [11, Theorem 13] again (but this time using the dual implication) we obtain for all k ∈ N

ek , Y (t) − Z(t) = −λk

t

ek , Y (s) − Z(s) ds

0

t +

ek , F˜0 Y (s) − F˜0 Z(s) ds,

0

Therefore, by the chain rule for all k ∈ N,

t 0, P -a.s.


2 ek , Y (t) − Z(t) = −2λk

t

1009

2 ek , Y (s) − Z(s) ds

0

t +2

ek , Y (s) − Z(s) ek , F˜0 Y (s) − F˜0 Z(s) ds,

t 0, P -a.s.

0

Dropping the first term on the right-hand side and summing up over k ∈ N (which is justified by (4.11) and the continuity of Y and Z), we obtain from (H3) that Y (t) − Z(t)2 2

t

Y (s) − Z(s), F˜0 Y (s) − F˜0 Z(s) ds

0

t 2ω

Y (s) − Z(s)2 ds,

t 0, P -a.s.

0

Hence, by Gronwall’s lemma Y = Z P -a.s. and the Claim is proved. By the Claim we can apply [11, Theorem 10, (1) ⇔ (3)] and then [11, Theorem 1] to conclude that Eq. (4.11) has a strong solution (see [11, Definition 1]) and that there is one strong solution with the same law as X, which hence by (4.7) has continuous sample paths a.s. Now all conditions in [11, Theorem 13.2] are fulfilled and, therefore, we deduce from it that on any stochastic basis (Ω, F , (Ft )t0 , P) with (Ft )-cylindrical Wiener process W on H and for x, y ∈ M there exist pathwise unique continuous strong solutions X(t, x), X(t, y), t 0, to (4.11) such that P ◦ X(·, x)−1 = Px ◦ X −1 and P ◦ X(·, y)−1 = Py ◦ X −1 , in particular, X(0, x) = x and X(0, y) = y and P ◦ X(t, x)−1 (dz) = pt (x, dz),

t 0,

P ◦ X(t, y)−1 (dz) = pt (y, dz),

t 0.

(4.13)

In particular, we have proved (i). To prove (ii), below for brevity we set X := X(·, x), X := X(·, y). Then proceeding as in the proof of the Claim, by (1.13) and noting that s −1 Φ(s) → ∞ as s → ∞, we obtain 2 2 d X(t) − X (t) a − Φ0 X(t) − X (t) dt for some constant a > 0, only depending on ω and Φ, where Φ0 = 12 Φ. Now we consider two cases.

(4.14)

1010


Case 1. |x − y|2 Φ0−1 (2a). Define f (t) := |X(t) − X (t)|2 , t 0, and suppose there exists t0 ∈ (0, ∞) such that f (t0 ) > Φ0−1 (a). Then we can choose δ ∈ [0, t0 ] maximal such that f (t) > Φ0−1 (a),

∀t ∈ (t0 − δ, t0 ].

Hence, because by (4.14) f is decreasing on every interval where it is larger than Φ0−1 (a), we obtain that f (t0 − δ) f (t0 ) > Φ0−1 (a). Suppose t0 − δ > 0. Then f (t0 − δ) Φ0−1 (a) by the continuity of f and the maximality of δ. So, we must have t0 − δ = 0, hence f (t0 ) f (t0 − δ) = f (0) = |x − y|2 Φ0−1 (2a). So, X(t) − X (t)2 Φ −1 (2a), 0

∀t > 0.

Case 2. |x − y|2 > Φ0−1 (2a). Define t0 = inf{t 0: |X(t) − X (t)|2 Φ0−1 (2a)}. Then by Case 1, starting at t = t0 rather than t = 0 we know that X(t) − X (t)2 Φ −1 (2a), 0

∀t t0 .

(4.15)

Furthermore, it follows from (4.14) that 2 2 1 d X(t) − X (t) − Φ0 X(t) − X (t) dt, 2

∀t t0 .

This implies 2 1 Ψ X(t) − X (t) 2

2 |x−y|

|X(t)−X (t)|2

t dr , Φ0 (r) 4

∀t t0 .

Therefore, X(t) − X (t)2 Ψ −1 (t/4),

∀t t0 .

(4.16)


1011

Combining Case 1, (4.15) and (4.16) we conclude that X(t) − X (t)2 Ψ −1 (t/4) + Φ −1 (2a),

∀t > 0.

0

(4.17)

Combining (4.17) with Theorem 1.6 for all f ∈ Bb (H ) we obtain 2 λ(1 + Ψ −1 (t/8)) , pt/2 |f | X(t/2) pt/2 f 2 X (t/2) exp (1 − ε −ωt/2 )2

∀t > 0

for some constant λ > 0. By Jensen’s inequality and approximation it follows that for all f ∈ L2 (H, μ) 2 2 pt |f |(x) E pt/2 |f | X(t/2) λ(1 + Ψ −1 (t/8)) 2 , pt f (y) exp (1 − ε −ωt/2 )2

∀t > 0, ∀x, y ∈ M.

(4.18)

But since H0 = supp μ, M is dense in H0 , hence by the continuity of pt f (cf. Corollary 1.7) (4.18) holds for all x ∈ H0 , y ∈ M. Since μ(M) = 1 this completes the proof by integrating both sides with respect to μ(dy). 2 Remark 4.2. We would like to mention that by using [2] instead of [15] we can drop the assumption that |F0 | ∈ L2 (H, μ). So, by (4.9) and the proof above we can derive (4.8) avoiding to assume the usually energy condition

t

F0 X(s) 2 ds < ∞,

Px -a.s.

0

Details will be included in a forthcoming paper. We would like to thank Tobias Kuna at this point from whom we learnt identity (4.9) by private communication. 5. Existence of measures satisfying (H4) To prove existence of invariant measures we need to strengthen some of our assumptions. So, let us introduce the following conditions. (H1) (A, D(A)) is self-adjoint satisfying (1.2). (H6) There exists η ∈ (ω, ∞) such that F0 (x) − F0 (y), x − y −η|x − y|2 ,

∀x, y ∈ D(F ).

Remark 5.1. (i) Clearly, (H1) implies (H1) and (H5). (H1) and (H2)(i) imply that (A, D(A)) and thus also (1 + ω − A, D(A)) has a discrete spectrum. Let λi ∈ (0, ∞), i ∈ N, be the eigenvalues of the latter operator. Then by (H2) ∞ i=1

λ−1 i < ∞.

(5.1)

1012


(ii) If we assume (5.1), i.e. that (1 + ω − A)−1 is trace class, then all what follows holds with (H2) replaced by (H2)(i). So, σ −1 ∈ L(H ) is not needed in this case. Let Fα , α < 0, be as in Section 2. Then e.g. by [5, Theorem 3.2] equation (1.1) with Fα replacing F0 has a unique mild solution Xα (t, x), t 0. Since there exist η˜ ∈ (ω, ∞) and α0 > 0 such that each Fα , α ∈ (0, α0 ), satisfies (H6) with η˜ replacing η, by [5, Section 3.4] Xα has a unique invariant measure μα on (H, B(H )) such that for each m ∈ N

|x|m μα (dx) < ∞.

sup α∈(0,α0 )

(5.2)

H

That these moments are indeed uniformly bounded in α, follows from the proof of [5, Proposition 3.18] and the fact that η˜ ∈ (ω, ∞). Let NQ denote the centered Gaussian measure on (H, B(H )) with covariance operator Q defined by

∞ Qx :=

x ∈ H,

etA σ etA x dt, 0

which by (H2)(ii) is trace class. Let W 1,2 (H, NQ ) be defined as usual, that is as the completion of EA (H ) with respect to the norm !

ϕW 1,2 :=

2 ϕ + |Dϕ|2 dNQ

"1/2 ϕ ∈ EA (H ),

,

H

where D denotes first Fréchet derivative. By [7] we know that W 1,2 (H, NQ ) ⊂ L2 (H, NQ ),

compactly.

(5.3)

Theorem 5.2. Assume that (H1) , (H2), (H3) and (H6) hold and let μα , α ∈ (0, α0 ) be as above. Suppose that there exists a lower semi-continuous function G : H → [0, ∞] such that

{G < ∞} ⊂ D(F ),

|F0 | G on D(F )

and

G2 dμα < ∞.

sup α∈(0,α0 )

(5.4)

H

Then {μα : α ∈ (0, α0 )} is tight and any limit point μ satisfies (H4) and hence by Corollary 1.8 all of these limit points coincide. Furthermore, for all m ∈ N

F0 (x)2 + |x|m μ(dx) < ∞

(5.5)

H

and there exists ρ : H → [0, ∞), B(H )-measurable, such that μ = ρNQ and

√ ρ ∈ W 1,2 (H, μ).


1013

Proof. We recall that by [3, Theorem 1.1] for each α ∈ (0, α0 ) μα = ρα NQ ;

√ ρα ∈ W 1,2 (H, NQ )

and as is easily seen from its proof, that

1 √ 2 |D ρα | dNQ |Fα |2 dμα . 4 H

(5.6)

(5.7)

H

But by (2.3) and (5.4) the right-hand side of (5.7) is uniformly bounded in α. Hence by (5.3) there exists a zero sequence {αn } such that √ √ ραn → ρ for some

in L2 (H, NQ ) as n → ∞,

√ ρ ∈ W 1,2 (H, NQ ) and therefore, in particular, ραn → ρ

in L1 (H, NQ ) as n → ∞.

(5.8)

Define μ := ρNQ and ρn := ραn , n ∈ N. Since G is lower semi-continuous and μαn → μ as n → ∞ weakly, (5.2) and (5.4) imply

2 G (x) + |x|m μ(dx) < ∞, ∀m ∈ N. (5.9) H

Hence by (5.4) both (H4)(i) and (H4)(ii) follow. So, it remains to prove (H4)(iii). Since σ is independent of α, to show (5.9) it is enough to prove that for all ϕ ∈ Cb (H ), h ∈ D(A),

Fαhn (x)ϕ(x)μαn (dx) = F0h (x)ϕ(x) μ(dx), (5.10) lim n→∞

H

H

where Fαh := h, Fα , α ∈ [0, α0 ). We have

F h ϕ dμα − F h ϕ dμ n αn 0 H

ϕ∞

H

h F − F h ρn dNQ + αn 0

H

h F ϕ |ρn − ρ| dNQ . 0

H

But by (2.3) and (5.4) we have

h F − F h ρn dNQ αn 0 H

h F − F h ρn dNQ αn 0

{|G|M}

+

2|h| sup M α∈(0,α0 )

G2 dμα . H

(5.11)

1014


Hence first letting n → ∞ then M → ∞ by (2.2), (5.4) and (5.8) Lebesgue’s generalized dominated convergence theorem implies that the first term on the right-hand side of (5.11) converges to 0. Furthermore, for every δ ∈ (0, 1)

F0h F h ϕ dμα − F h ϕ dμ n 0 0 1 + δ|F h | ϕ(ρn − ρ) dNQ 0 H

H

H

!

+ δϕ∞

h 2 F dμα + n 0

H

" h 2 F dμ . 0

(5.12)

H

Since by (2.3) and (5.4)

sup α∈(0,α0 )

h 2 F dμα < ∞, 0

H

(H4)(iii) follows from (5.12) by letting first n → ∞ and then δ → 0, since for fixed δ > 0 the first term in the right-hand side converges to zero by (5.8). 2 Example 5.3. Let H = L2 (0, 1), Ax = x, x ∈ D(A) := H 2 (0, 1) ∩ H01 (0, 1). Let f : R → R be decreasing such that for some c3 > 0, m ∈ N, f (s) c3 1 + |s|m , ∀s ∈ R. (5.13) Let si ∈ R, i ∈ N, be the set of all arguments where f is not continuous and define [f (si + ), f (si − )], if s = si for some i ∈ N, f¯(s) = f (s), else. Define F : D(F ) ⊂ H → 2H ,

x → f¯ ◦ x,

where D(F ) = {x ∈ H : f¯ ◦ x ⊂ H }. Then F is m-dissipative. Let F0 be defined as in Section 2. Since A ω for some ω < 0, it is easy to check that all conditions (H1) , (H2), (H3), (H6) with η = 0 hold for any σ ∈ L(H ) such that σ −1 ∈ L(H ). Define 1/2 1 2m if x ∈ L2m (0, 1), 0 |x(ξ )| dξ G(x) := +∞ if x ∈ / L2m (0, 1). Then {G < ∞} ⊂ D(F ) and |F0 | = |F0 |L2 (0,1) G on D(F ). Furthermore, by [6, (9.3)]

G2 dμα < ∞.

sup α∈(0,α0 )

H

(5.14)


1015

Note that from [6, Hypothesis 9.5] only the first inequality, which clearly holds by (5.13) in our case, was used to prove [6, (9.3)]. Hence all assumptions of Theorem 5.2 above hold and we obtain the existence of the desired unique probability measure μ satisfying (H4) in this case. We emphasize that no continuity properties of f and F0 are required. In particular, then all results stated in Section 1 except for Corollary 1.10(ii) hold in this case. If moreover there exists an increasing positive convex function Φ on [0, ∞) satisfying (1.12) such that f (s) − f (t) (s − t) c − Φ |s − t|2 , s, t ∈ R, then by Jensen’s inequality (1.13) holds. Hence, by Corollary 1.10 one obtains an explicit upper bound for pt L2 (H,μ)→L∞ (H,μ) . A natural and simple choice of Φ is Φ(s) = s m for m > 1. One can extend these results to the case, where (0, 1) above is replaced by a bounded open 1 set in Rd , d = 2 or 3 for σ = (−)γ , γ ∈ ( d−2 4 , 2 ), based on Remark 1.1(iv). Before to conclude we want to present a condition in the general case (i.e for any Hilbert space H as above) that implies (5.4), hence by Theorem 5.2 ensures the existence of a probability measure satisfying (H4) so that all results of Section 1 apply also to this case. As will become clear from the arguments below, such condition is satisfied if the eigenvalues of A grow fast enough in comparison with |F 0 |. To this end we first note that by (5.1) for i ∈ N we can find qi −1 qi ∈ (0, λi ), qi ↑ ∞ such that ∞ i=1 qi < ∞ and λi → 0 as i → ∞. Define Θ : H → [0, ∞] by Θ(x) :=

∞ λi i=1

qi

x, ei 2 ,

x ∈ H,

(5.15)

where {ei }i∈N is an eigenbasis of (1 + ω − A, D(A)) such that ei has eigenvalue λi . Then Θ has compact level sets and | · |2 Θ. Below we set Hn := lin span {e1 , . . . , en }, A˜ := A − (1 + ω)I,

πn := projection onto Hn , ˜ := D(A), D(A)

F˜0 := F0 + (1 + ω)I.

(5.16) (5.17)

We note that obviously Hn ⊂ {Θ < +∞} for all n ∈ N. Theorem 5.4. Assume that (H1) , (H2), (H3) and (H6) hold and let μα , α ∈ (0, α0 ), be as above. Suppose that {Θ < +∞} ⊂ D(F ) and that for some C ∈ (0, ∞), m ∈ N F0 (x) C 1 + |x|m + Θ 1/2 (x) , ∀x ∈ D(F ). (5.18) Then

Θ dμα < ∞

sup α∈(0,α0 )

and (5.4) holds, so Theorem 5.2 applies.

H

(5.19)

1016


Proof. Consider the Kolmogorov operator Lα corresponding to Xα (t, x), t 0, x ∈ H, which for ϕ ∈ F Cb2 ({en }), i.e., ϕ = g(e1 , ·, . . . , eN , ·) for some N ∈ N, g ∈ Cb2 (RN ), is given by Lα ϕ(x) :=

1 2 2 Tr σ D ϕ(x) + x, ADϕ(x) + Fα (x), Dϕ(x) , 2

x ∈ H,

(5.20)

where D 2 denotes the second Fréchet derivative. Then, an easy application of Itô’s formula shows that the L1 (H, μα )-generator of (Ptα ) (given as before by Ptα f (x) = E[f (Xα (t, x))]) is given on F Cb2 ({en }) by Lα . In particular,

Lα ϕ dμα = 0, ∀ϕ ∈ F Cb2 {en } . H

By a simple approximation argument and (5.2) we get for α ∈ (0, α0 ) and ϕn (x) :=

n

qi−1 x, ei 2 ,

x ∈ H, n ∈ N,

i=1

that also

Lα ϕn dμα = 0.

(5.21)

H

But for all x ∈ H , with F˜α defined as F˜0 in (5.17), we have Lα ϕn (x) = −2

n λi i=1

+

n

qi

x, ei 2 + 2

n

qi−1 F˜α (x), ei x, ei

i=1

qi−1 σn ei , σn ej

i,j =1

−2Θ(πn x) + 2

n

2 qi−1 F˜α (x), ei

i=1

+

n

1/2 n

1/2

qi−1 x, ei 2

i=1

qi−1 |σn ei |2

i=1 ∞ −2Θ(πn x) + c1 1 + |x|m+1 + Θ 1/2 (x)|x| + σ 2 qi−1 ,

(5.22)

i=1

for some constant c1 independent of n and α. Here we used (2.3) and (5.18). Now (5.21), (5.2) and (5.22) immediately imply that for some constant c˜1 ! "

∞ m+2 sup Θ(x)μα (dx) sup c˜1 1 + |x| μα (dx) + σ 2 qi−1 < ∞. α∈(0,α0 )

H

α∈(0,α0 )

H

i=1

So, (5.19) is proved, which by (5.18) implies (5.4) and the proof is complete.

2


1017

Acknowledgments The second named author would like to thank UCSD, in particular, his host Bruce Driver, for a very pleasant stay in La Jolla where a part of this work was done. The authors would like to thank Ouyang for his comments leading to a better constant involved in the Harnack inequality. References [1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math. 130 (3) (2006) 223–233. [2] L. Beznea, N. Boboc, M. Röckner, Markov processes associated with Lp -resolvents and applications to stochastic differential equations in Hilbert spaces, J. Evol. Equ. 6 (4) (2006) 745–772. [3] V. Bogachev, G. Da Prato, M. Röckner, Regularity of invariant measures for a class of perturbed Ornstein– Uhlenbeck operators, Nonlinear Differential Equations Appl. 3 (1996) 261–268. [4] S. Cerrai, Second Order PDE’s in Finite and Infinite Dimensions. A Probabilistic Approach, Lecture Notes in Math., vol. 1762, Springer-Verlag, 2001. [5] G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Birkhäuser, 2004. [6] G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261–303. [7] G. Da Prato, P. Malliavin, D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris 315 (1992) 1287–1291. [8] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. [9] W. Liu, Doctor-Thesis, Bielefeld University, 2008. [10] Z.M. Ma, M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, 1992. [11] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.) 426 (2004). [12] S.-X. Ouyang, Doctor-Thesis, Bielefeld University, 2008. [13] C. Prevot, M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., Springer, 2007. [14] M. Röckner, F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal. 203 (1) (2003) 237–261. [15] W. Stannat, (Nonsymmetric) Dirichlet operators on L1 : Existence, uniqueness and associated Markov processes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1) (1999) 99–140. [16] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424. [17] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (4) (2007) 1333–1350.


Stable invariant manifolds for parabolic dynamics ✩ Luis Barreira ∗ , Claudia Valls Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal Received 11 December 2008; accepted 14 January 2009 Available online 31 January 2009 Communicated by Paul Malliavin

Abstract We consider nonautonomous equations v = A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t) = t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v = A(t)v + f (t, v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations. © 2009 Elsevier Inc. All rights reserved. Keywords: Invariant manifolds; Parabolic dynamics; Stability theory

1. Introduction We consider linear equations v = A(t)v

(1)

in a Banach space, which exhibit stable and unstable behaviors with respect to growth rates of the form ecρ(t) for an arbitrary function ρ(t). This corresponds to the existence of what we ✩

Partially supported by FCT through CAMGSD, Lisbon.


E-mail addresses: [email protected] (L. Barreira), [email protected] (C. Valls). URL: http://www.math.ist.utl.pt/~barreira/ (L. Barreira). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.014

L. Barreira, C. Valls / Journal of Functional Analysis 257 (2009) 1018–1029

1019

call a ρ-nonuniform exponential dichotomy. This includes linear dynamics that may have all Lyapunov exponents zero or all Lyapunov exponents infinite, of course besides the usual case when ρ(t) = t. In such a situation, one is not able, at least without further modifications, to apply the existing stability theory, using for example Lyapunov’s so-called regularity theory (see for example [1] for details), which can only be applied successfully when all Lyapunov exponents are finite. On the other hand, we show in [3] that for ρ in a large class of rate functions, any linear equation as in (1) in a finite-dimensional space, and with two blocks with asymptotic rates ecρ(t) respectively with c negative and positive, has a ρ-nonuniform exponential dichotomy. Moreover, we show in [5] that a ρ-nonuniform exponential dichotomy is robust under sufficiently small linear perturbations. This shows that there are plenty exponential dichotomies with the generalized behavior given by a function ρ. Therefore, it is of interest to develop a corresponding theory. Our main aim is to show that the exponential behavior exhibited by a linear nonuniform exponential dichotomy persists under sufficiently small nonlinear perturbations. Namely, we establish the existence of stable invariant manifolds for the equation v = A(t)v + f (t, v),

(2)

provided that there are constants c, q > 0 with q sufficiently large such that f (t, u) − f (t, v) cu − v uq + vq .

(3)

We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations. We first consider contractions, and then dichotomies with an elaboration of the proof for contractions. The strategy of proof is essentially based on work in [2], although that paper does not consider exponential contractions. In the theory of differential equations, the notion of exponential dichotomy plays a central role in the study of invariant manifolds. In particular, the existence of an exponential dichotomy for Eq. (1) ensures the existence of stable and unstable invariant manifolds under sufficiently small nonlinear perturbations. The theory of exponential dichotomies and its applications are widely developed. Moreover, there exist large classes of linear equations with exponential dichotomies. We refer to the books [6–9,15] for details and further references. On the other hand, the notion of exponential dichotomy is too stringent for the dynamics and it is of interest to look for more general types of hyperbolic behavior. It is in this context that we consider the more general notion of nonuniform exponential dichotomy. Our work can also be considered a contribution to the theory of nonuniformly hyperbolic dynamics. We refer to [1] for an introduction to the theory, which goes back to the works of Oseledets [11] and Pesin [12,13]. Invariant manifolds were first obtained for nonuniformly hyperbolic trajectories by Pesin in [12]. The first related results in Hilbert spaces were established by Ruelle in [14]. The case of transformations in Banach spaces under some compactness assumptions was considered by Mañé in [10]. 2. Stability of exponential contractions Let X be a Banach space and let A : R+ 0 → B(X) be a continuous function, where B(X) is the set of bounded linear operators in X. We consider the linear equation (1). Given an increasing

1020


+ differentiable function ρ : R+ 0 → R0 such that

lim

t→+∞

log t = 0, ρ(t)

(4)

we say that Eq. (1) admits a ρ-nonuniform exponential contraction if there exist constants λ < 0, a 0 and D 1 such that for every t s 0 we have T (t, s) Deλ(ρ(t)−ρ(s))+aρ(s) ,

(5)

where T (t, s) is the linear evolution operator associated to Eq. (1). Let also f : R+ 0 × X → X be a continuous function with f (t, 0) = 0 for every t 0. We consider the nonlinear equation (2), and we study the persistence of the stability of an exponential contraction. Let B(δ) ⊂ X be the open ball of radius δ centered at zero, and set α = (1 + 2/q)a.

(6)

Theorem 1. Assume that Eq. (1) admits a ρ-nonuniform exponential contraction, and that there exist constants c, q > 0 such that (3) holds for every t 0 and u, v ∈ X, with qλ + a < 0. Then for any D > D and any sufficiently small δ > 0, provided that s 0 is sufficiently large, each solution of Eq. (2) with v(s) = ξ ∈ B(δe−αρ(s) ) satisfies v(t) D eλ(ρ(t)−ρ(s))+aρ(s) ξ ,

t s.

Proof. We consider the space B = v : [s, +∞) → X: v is continuous and v δe−αρ(s) , with the norm v =

1 sup v(t)e−λ(ρ(t)−ρ(s))−aρ(s) : t s . 2D

We can easily verify that B is a complete metric space. Set t (J v)(t) = T (t, s)ξ +

U (t, τ )f τ, v(τ ) dτ

s

for each v ∈ B and t s. Clearly, J v is continuous and (J v)(s) = ξ . By the assumptions in the theorem, for each v, w ∈ B and τ s we have f τ, v(τ ) − f τ, w(τ ) cv(τ ) − w(τ ) v(τ )q + w(τ )q c2q+2 δ q D 1+q eλ(q+1)(ρ(τ )−ρ(s))+a(q+1)ρ(s)−αqρ(s) v − w .


1021

Therefore, using (6) we obtain (J v)(t) − (J w)(t)

t

T (t, τ ) · f τ, v(τ ) − f τ, w(τ ) dτ

s

c2q+2 δ q D 2+q eλ(ρ(t)−ρ(s)) v − w

t e(qλ+a)(ρ(τ )−ρ(s)) dτ.

(7)

s

It follows from (4) that ρ(t) → +∞ as t → +∞, and thus tρ (t) → +∞ as t → +∞. Therefore, for each c > 0 there exists t0 > 0 such that tρ (t) > c for every t > t0 . For s > t0 we obtain t c

t e

(qλ+a)(ρ(τ )−ρ(s))

τρ (τ )e(qλ+a)(ρ(τ )−ρ(s)) dτ

dτ

s

s

t τ e(qλ+a)(ρ(τ )−ρ(s)) ∞ 1 − e(qλ+a)(ρ(τ )−ρ(s)) dτ, qλ + a qλ + a s s

since qλ + a < 0. By (4) we have log τ e(qλ+a)ρ(τ ) = log τ + (qλ + a)ρ(τ )

log τ + qλ + a → −∞ = ρ(τ ) ρ(τ ) as τ → +∞, and thus,

1 c+ qλ + a

t e(qλ+a)(ρ(τ )−ρ(s)) dτ s

s . |qλ + a|

Taking c sufficiently large, we have t e(qλ+a)(ρ(τ )−ρ(s)) dτ s.

(8)

s

By (7), setting K = supst0 (se−aρ(s) ) < +∞, we obtain q+2 q 2+q (J v)(t) − (J w)(t) c2 δ D K eλ(ρ(t)−ρ(s))+aρ(s) v − w , |qλ + a|

and thus, J v − J w θ v − w ,

(9)

1022


with θ = c2q+1 δ q D 1+q K/|qλ + a|. Taking δ sufficiently small so that θ < 1/2, the operator J becomes a contraction. Furthermore, by (9) and (5), 1 J v J 0 + θ v ξ + θ v 2 1 1 δe−αρ(s) + δe−αρ(s) = δe−αρ(s) , 2 2

(10)

and J (B) ⊂ B. Therefore, there exists a unique function v ∈ B such that J v = v. It follows from (10) that v ξ /(2(1 − θ )). This completes the proof of the theorem. 2 3. Stable invariant manifolds We establish in this section the existence of stable invariant manifolds under sufficiently small perturbations of nonuniform exponential dichotomies. We first introduce the notion of exponential dichotomy. We say that Eq. (1) admits a ρ-nonuniform exponential dichotomy if there exist projections P (t) for t 0 satisfying P (t)T (t, s) = T (t, s)P (s),

t, s 0,

and there exist constants λ < 0 μ,

a 0 and D 1

such that for every t s 0 we have T (t, s)P (s) Deλ(ρ(t)−ρ(s))+aρ(s) , T (t, s)−1 Q(t) De−μ(ρ(t)−ρ(s))+aρ(t) ,

(11)

where Q(t) = Id −P (t). For each t 0 we consider the linear subspaces E(t) = P (t)(X)

and F (t) = Q(t)(X).

Now we assume that Eq. (1) admits a ρ-nonuniform exponential dichotomy. For each t 0, let Bt (δ) ⊂ E(t) be the open ball of radius δ centered at zero, and set β = a(1 + 3/q). We consider the set Zβ = Zβ (δ) = (s, ξ ): s 0, ξ ∈ Bs δe−βρ(s) ,

(12)


1023

and we denote by Xβ the space of continuous functions φ : Zβ → X such that for every s 0 we have φ(s, 0) = 0, φ(s, ξ ) ∈ F (s), and φ(s, ξ ) − φ(s, ξ¯ ) ξ − ξ¯ for every ξ, ξ¯ ∈ e−βρ(s) . With the norm φ = sup φ(s, ξ )/ξ : s 0 and ξ ∈ Bs δe−βρ(s) \ {0} , Xβ becomes a complete metric space. Given φ ∈ Xβ we consider its graph W=

s, ξ, φ(s, ξ ) : (s, ξ ) ∈ Zβ .

(13)

For each (s, ξ, η) ∈ R+ 0 × E(s) × F (s) we also consider the semiflow Ψτ (s, ξ, η) = s + τ, x(s + τ ), y(s + τ ) ,

τ 0,

where t x(t) = T (t, s)ξ +

T (t, τ )P (τ )f τ, x(τ ), y(τ ) dτ,

s

t y(t) = T (t, s)η +

T (t, τ )Q(τ )f τ, x(τ ), y(τ ) dτ.

s

The following is our stable manifold theorem. Theorem 2. Assume that Eq. (1) admits a ρ-nonuniform exponential dichotomy, and that there exist constants c, q > 0 such that (3) holds for every t 0 and u, v ∈ X, with qλ + a < 0 and λ + a < μ. Then there exist δ, R, D > 0, and a unique function φ ∈ Xβ such that: 1. the set W is forward invariant under the semiflow Ψτ , that is, Ψτ s, ξ, φ(s, ξ ) ∈ W

whenever (s, ξ ) ∈ Zβ+a (δ/R);

2. for every s 0, ξ, ξ¯ ∈ Bs (δe−(β+a)ρ(s) ), and τ 0 we have Ψτ s, ξ, φ(s, ξ ) − Ψτ s, ξ¯ , φ(s, ξ¯ ) D eλ(ρ(τ )−ρ(s))+aρ(s) ξ − ξ¯ . Proof. We consider the space X of continuous functions φ : Z → X, where Z = (s, ξ ): s 0, ξ ∈ E(s) ,

1024


such that φ|Zβ ∈ Xβ , and φ(s, ξ ) = φ s, δe−βρ(s) ξ/ξ whenever ξ ∈ / Bs (δe−βρ(s) ). We note that there is a one-to-one correspondence between functions in Xβ and X. In particular, X is a Banach space with the norm X φ → φ|Zβ . Moreover, for each φ ∈ X and s 0 we have (see Lemma 4.3 in [4]) φ(s, x) − φ(s, y) 2x − y for every x, y ∈ E(s). (14) Now we prove some auxiliary results. Lemma 1. There exists R > 0 such that for every δ > 0 sufficiently small: 1. for each φ ∈ X, given (s, ξ ) ∈ Zβ there is a unique continuous function x = xφ : [s, +∞) × Bs (δe−βρ(s) ) → X such that for every t s, t x(t) = T (t, s)ξ +

T (t, τ )P (τ )f τ, x(τ, ξ ), φ τ, x(τ, ξ ) dτ ;

(15)

s

xφ (t, ξ ) Reλ(ρ(t)−ρ(s))+aρ(s) ξ ,

2.

t s.

Proof. This is an elaboration of the proof of Theorem 1. We consider the space Bs of continuous functions x : [s, +∞) × Bs δe−βρ(s) → X such that x δe−βρ(s) , with the norm x =

1 sup x(t, ξ )e−λ(ρ(t)−ρ(s))−aρ(s) : t s, (s, ξ ) ∈ Zβ (δ) . 2D

We can easily verify that Bs is a complete metric space. We define an operator Js by t (Js x)(t, ξ ) = U (t, s)ξ +

U (t, τ )f τ, x(τ, ξ ), φ τ, x(τ, ξ ) dτ

s

for each x ∈ Bs . Clearly, Js x is continuous, and (Js x)(s, ξ ) = ξ . By the assumptions in the theorem, for each x, y ∈ Bs , φ ∈ X, and τ s we have f τ, x(τ, ξ ), φ τ, x(τ, ξ ) − f τ, y(τ, ξ ), φ τ, y(τ, ξ ) 2c6q+1 δ q D 1+q eλ(q+1)(ρ(τ )−ρ(s))+a(q+1)ρ(s)−βqρ(s) x − y . Therefore, proceeding as in the proof of Theorem 1 (see also (8)) we obtain (Js x) − (Js y) θ x − y ,

(16)


1025

with θ = 2c6q+1 δ q D 2+q K/|qλ + a|, and K = supst0 (se−aρ(s) ). Taking δ sufficiently small so that θ < 1/2, the operator Js becomes a contraction. Moreover, by (16) and (11), proceeding again as in the proof of Theorem 1 we obtain Js x δe−βρ(s) ,

that is,

Js (Bs ) ⊂ Bs ,

and the unique fixed point of Js satisfies x ξ /(2(1 − θ )).

2

Lemma 2. There exists K > 0 such that for every δ > 0 sufficiently small, φ, ψ ∈ X, s 0 sufficiently large, ξ, ξ¯ ∈ Bs (δe−βρ(s) ), and t s we have xφ (t, ξ ) − xφ (t, ξ¯ ) + xφ (t, ξ ) − xψ (t, ξ ) K eλ(ρ(t)−ρ(s))+aρ(s) ξ − ξ¯ + ξ · φ − ψ .

(17)

Proof. For every τ s, setting η = 2c3q+1 R q δ q ,

(18)

and using the definition of β in (12) and Lemma 1, we have f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) − f τ, xφ (τ, ξ¯ ), φ τ, xφ (τ, ξ¯ ) ηe−3aρ(s) eqλ(ρ(t)−ρ(s)) xφ (τ, ξ ) − xφ (τ, ξ¯ ),

(19)

f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) − f τ, xψ (τ, ξ ), ψ τ, xψ (τ, ξ ) ηe−3aρ(s) eqλ(ρ(t)−ρ(s)) xφ (τ, ξ ) · φ − ψ + 3xφ (τ, ξ ) − xψ (τ, ξ ) .

(20)

and

Setting γ (t) = xφ (τ, ξ ) − xφ (τ, ξ¯ ) + xφ (τ, ξ ) − xψ (τ, ξ ), we obtain γ (t) U (t, s) · ξ − ξ¯ + 3ηe−3aρ(s)

t

U (t, τ )eqλ(ρ(τ )−ρ(s)) γ (τ ) dτ

s

+ ηe−3aρ(s)

t

U (t, τ )eqλ(ρ(τ )−ρ(s)) xφ (τ ) · φ − ψ dτ,

s

where U (t, s) = T (t, s)P (s). Using (11) we conclude that

1026


e−λ(ρ(t)−ρ(s)) γ (t) Deaρ(s) ξ − ξ¯ + 3Dηe

−2aρ(s)

t

e(qλ+a)(ρ(τ )−ρ(s)) e−λ(ρ(τ )−ρ(s)) γ (τ ) dτ

s

+ ηDRe

−aρ(s)

t ξ · φ − ψ

e(qλ+a)(ρ(τ )−ρ(s)) dτ. s

It was shown in the proof of Theorem 1 that there exists t0 > 0 such that (8) holds for any t > t0 . Therefore, for s > t0 we have e−λ(ρ(t)−ρ(s)) γ (t) Deaρ(s) ξ − ξ¯ + ηDRKξ · φ − ψ + 3Dηe

−2aρ(s)

t

e(qλ+a)(ρ(τ )−ρ(s)) e−λ(ρ(τ )−ρ(s)) γ (τ ) dτ.

s

Applying Gronwall’s lemma to the function e−λ(ρ(t)−ρ(s)) γ (t), we obtain γ (t) K eλ(ρ(t)−ρ(s))+aρ(s) ξ − ξ¯ + ξ · φ − ψ , for some constant K > 0.

2

The next step is to show the existence of a function φ ∈ X satisfying

φ t, x(t) = T (t, s)φ s, x(s) +

t

T (t, τ )Q(τ )f τ, x(τ ), φ τ, x(τ ) dτ

(21)

s

with x = xφ . First we transform this problem into an equivalent one. Lemma 3. Given δ > 0 sufficiently small, s 0 sufficiently large, and φ ∈ X the following properties hold: 1. if (21) holds for every (s, ξ ) ∈ Zβ and t s, then ∞ φ(s, ξ ) = −

T (τ, s)−1 Q(τ )f τ, xφ (τ ), φ τ, xφ (τ ) dτ

(22)

s

for every (s, ξ ) ∈ Zβ ; 2. if (22) holds for every (s, ξ ) ∈ Zβ , then (21) holds for every (s, ξ ) ∈ Zβ+a (δ/R) and t s. Proof. By Lemma 1 we have f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) c3q+1 xφ (τ, ξ )q+1 c3q+1 R q+1 e(q+1)λ(ρ(τ )−ρ(s))+qaρ(s) ξ q+1 ,


1027

and hence, ∞ T (τ, s)−1 Q(τ )f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) dτ s

∞ c3

q+1

DR

q+1

ξ

q+1 (q+1)aρ(s)

e((q+1)λ−μ+a)(ρ(τ )−ρ(s)) dτ.

e

s

Since (q + 1)λ − μ + a < 0, arguing as in the proof of Theorem 1 (see (8)) we find that the integral is well defined. Now we assume that (21) holds for every (s, ξ ) ∈ Zβ and t s. This is equivalent to φ(s, ξ ) = V (t, s)−1 φ t, xφ (t, ξ ) t −

V (τ, s)−1 f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) dτ,

(23)

s

where V (t, s) = T (t, s)Q(s). Since V (t, s)−1 φ t, xφ (t, ξ ) 2DRξ e2aρ(s) e(λ−μ+b)(ρ(t)−ρ(s)) , letting t → +∞ in (23) we conclude that (22) holds for every (s, ξ ) ∈ Zβ and t s. The second property follows by repeating arguments in the proof of Lemma 4.7 in [4]. 2 Lemma 4. Given δ > 0 sufficiently small, there is a unique function φ ∈ X such that (22) holds for every (s, ξ ) ∈ Zβ . Proof. We consider the operator Φ defined for each φ ∈ X by ∞ (Φφ)(s, ξ ) = −

V (τ, s)−1 f τ, xφ (τ, ξ ), φ τ, xφ τ, x(τ, ξ ) dτ

s

when (s, ξ ) ∈ Zβ , and by (Φφ)(s, ξ ) = (Φφ) s, δe−βρ(s) ξ/ξ otherwise. When ξ = 0 we have xφ (t, ξ ) = 0, and hence (Φφ)(t, 0) = 0. By (19) and Lemmas 1 and 2 we obtain a(τ ) := f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) − f τ, xφ (τ, ξ¯ ), φ τ, xφ (τ, ξ¯ ) ηK e−2aρ(s) e(q+1)λ(ρ(τ )−ρ(s)) ξ − ξ¯ , and using (11),

1028


(Φφ)(s, ξ ) − (Φφ)(s, ξ¯ )

∞ V (τ, s)−1 a(τ ) dτ s −aρ(s)

ηK e

ξ − ξ¯

∞ e((q+1)λ−μ+a)(ρ(τ )−ρ(s)) dτ. s

Since (q + 1)λ − μ + a < 0, arguing as in the proof of Theorem 1 (see (8)) we find that there exists t0 > 0 such that ∞ e((q+1)λ−μ+a)(ρ(τ )−ρ(s)) dτ s s

for every s t0 . Hence, we obtain (Φφ)(s, ξ ) − (Φφ)(s, ξ¯ ) ηK Kξ − ξ¯ whenever (s, ξ ), (s, ξ¯ ) ∈ Zβ , and proceeding as in the proof of Lemma 4.3 in [4] yields (Φφ)(s, ξ ) − (Φφ)(s, ξ¯ ) 2ηK Kξ − ξ¯ for arbitrary ξ and ξ¯ . Therefore, taking δ sufficiently small (see (18)) the operator Φ : X → X is well defined. On the other hand, by (20) and Lemmas 1 and 2 we have b(τ ) := f τ, xφ (τ, ξ ), φ τ, xφ (τ, ξ ) − f τ, xψ (τ, ξ ), ψ τ, xψ (τ, ξ ) η(R + 3K )e−2aρ(s) e(q+1)λ(ρ(τ )−ρ(s)) ξ · φ − ψ, and using (11), (Φφ)(s, ξ ) − (Φψ)(s, ξ )

∞ V (τ, s)−1 b(τ ) dτ s

η(R + 3K )e

−2aρ(s)

∞ ξ φ − ψ

e((q+1)λ−μ+a)(ρ(τ )−ρ(s)) dτ. s

This implies that Φφ − Φψ η(R + 3K )K φ − ψ for some constant K > 0. Taking δ sufficiently small (see (18)) the operator Φ : X → X becomes a contraction. 2 We proceed with the proof of the theorem. By Lemma 1, for each φ ∈ X there is a unique function x = xφ satisfying (15), and thus it remains to solve (21) for φ setting x = xφ . This is


1029

an immediate consequence of Lemmas 3 and 4 for (s, ξ ) ∈ Zβ+a (δ/R) ⊂ Zβ . Furthermore, by Lemma 3, for every (s, ξ ) ∈ Zβ+a (δ/R) we have xφ (t) Reλ(ρ(t)−ρ(s))+aρ(s) ξ δe−βρ(s) ,

(24)

and (t, xφ (t)) ∈ Zβ for every t s. Thus, there exists a unique function φ ∈ X such that the graph set W in (13) obtained from φ|Zβ is invariant under the semiflow Ψτ for initial conditions (s, ξ ) ∈ Zβ+a (δ/R). Finally, it follows from (17) and (14) that for every t s, Ψt−s s, ξ, φ(s, ξ ) − Ψt−s s, ξ¯ , φ(s, ξ¯ ) 3xφ (t, ξ ) − xφ (t, ξ¯ ) 3K eλ(ρ(t)−ρ(s))+aρ(s) ξ − ξ¯ .

(25)

For each ξ, ξ¯ ∈ Bs (δe−(β+a)ρ(s) ), in view of (24) we can replace the function φ ∈ X in (25) by its restriction to Zβ . This completes the proof. 2 References [1] L. Barreira, Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Ser., vol. 23, Amer. Math. Soc., 2002. [2] L. Barreira, C. Valls, Stable manifolds for nonautonomous equations without exponential dichotomy, J. Differential Equations 221 (2006) 58–90. [3] L. Barreira, C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. 22 (2008) 509–528. [4] L. Barreira, C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math., vol. 1926, Springer, 2008. [5] L. Barreira, C. Valls, Robustness of general dichotomies, J. Funct. Anal. (2009), doi:10.1016/j.jfa.2008.11.018, in press. [6] C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., 1999. [7] W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer, 1978. [8] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1988. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer, 1981. [10] R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, in: J. Palis (Ed.), Geometric Dynamics, Rio de Janeiro, 1981, in: Lecture Notes in Math., vol. 1007, Springer, 1983, pp. 522–577. [11] V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968) 197–221. [12] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv. 10 (1976) 1261–1305. [13] Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys 32 (1977) 55– 114. [14] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982) 243–290. [15] G. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., vol. 143, Springer, 2002.


An asymptotic functional-integral solution for the Schrödinger equation with polynomial potential S. Albeverio a,b,c,d,e,f,g , S. Mazzucchi a,e,∗,1 a Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany b HCM, Germany c SFB611, BIBOS, Germany d IZKS, Austria e Dipartimento di Matematica, Università di Trento, 38050 Povo, Trento, TN, Italy f Cerfim (Locarno), Switzerland g Acc. Arch. (USI) (Mendrisio), Italy

Received 23 January 2009; accepted 5 February 2009 Available online 20 February 2009 Communicated by Paul Malliavin

Abstract A functional integral representation for the weak solution of the Schrödinger equation with a polynomially growing potential is proposed in terms of an analytically continued Wiener integral. The asymptotic expansion in powers of the coupling constant λ of the matrix elements of the Schrödinger group is studied and its Borel summability is proved. © 2009 Elsevier Inc. All rights reserved. Keywords: Feynman path integrals; Schrödinger equation; Analytic continuation of Wiener integrals; Polynomial potential; Asymptotic expansions

* Corresponding author at: Dipartimento di Matematica, Università di Trento, 38050 Povo, Trento, TN, Italy.

E-mail address: [email protected] (S. Mazzucchi). 1 F. Severi fellow.


S. Albeverio, S. Mazzucchi / Journal of Functional Analysis 257 (2009) 1030–1052

1031

1. Introduction The Schrödinger equation ⎧ ⎨

h¯ 2 ∂ ψ =− ψ + V ψ, 2m ⎩ ∂t ψ(0, x) = ψ0 (x) i h¯

(1)

with a polynomial potential V of the form V (x) = λ|x|2M and the asymptotic behaviour of its solution in some limiting situation (for instance when λ → 0 or h¯ → 0, or t → 0) is a largely studied topic [9,25,11,8]. Particularly interesting is the study of a possible functional integral representation, in the spirit of Feynman path integrals. During the last four decades, rigorous mathematical definitions of the heuristic Feynman path integrals have been given by means of different methods, and the properties of these rigorous integrals have been studied. Let us mention here only three of them, namely the one using the analytic continuation of Wiener integrals [10,12,16,17], the one provided by infinite-dimensional oscillatory integrals [1,3,20] and the one using white noise calculus [15] (see also the references given in [1,20] to other approaches). The main problem which is common to all the existing approaches is the restriction on the class of potentials V which can be handled. For most results one has to impose that V has at most quadratic growth at infinity. There are two exceptions to this restriction, the one of potentials which are Laplace transform of measures (like exponential potentials, see [1,15] and references therein) and quartic potentials [2,5–7,19]. A difficulty in the study of Eq. (1) with a polynomial potential is the non-regular behaviour of the solution. Indeed in [28] it has been shown that for superquadratic potentials the fundamental solution of (1) is nowhere of class C 1 . In [5,7,19] an infinite-dimensional oscillatory integral representation for the weak solution of the Schrödinger equation, i.e. the matrix element of the Schrödinger group, has been presented and studied in the case where the potential has precisely a quartic growth at infinity. The present paper generalizes partially these results to polynomial potentials with higher growth at infinity. For a dense set of vectors φ, ψ ∈ L2 (Rd ), we define an analytically continued Wiener integral Iti (φ, ψ) (Eq. (28)) which realizes rigorously the Feynman path integral i

representing the corresponding “matrix elements” of the Schrödinger group φ, e− h¯ H t ψ and prove that it solves the Schrödinger equation in a weak sense (Theorem 8). The relation between i the functional integral Iti (φ, ψ) and the matrix elements φ, e− h¯ H t ψ is investigated in details. In particular we prove that these quantities are asymptotically equivalent both as t → 0 and as i λ → 0. The asymptotic expansion in powers of λ of φ, e− h¯ H t ψ is studied and its Borel summability is proved. This result allows one to recover the matrix elements of the Schrödinger group from the asymptotic expansion in powers of λ of the functional integral Iti (φ, ψ), which in this sense can be recognized as an asymptotic weak solution of the Schrödinger equation. The paper is organized as follows. In Section 2 the analyticity properties of the spectrum i of the anharmonic oscillator Hamiltonian H and of the matrix elements φ, e− h¯ H t ψ of the Schrödinger group are studied. In Section 3 the Borel summability of the asymptotic expani sion in powers of the coupling constant λ (Dyson expansion) of the quantities φ, e− h¯ H t ψ is proved. Section 4 studies the definition and the properties of the functional integral Iti (φ, ψ), while Section 5 investigates its relations with the matrix elements of the Schrödinger group and their asymptotic equivalence.

1032


2. The Schrödinger equation with polynomial potential Let us consider the quantum anharmonic oscillator Hamiltonian with polynomial potential on L2 (Rd ), that is the operator defined on the vectors φ ∈ C0∞ (Rd ) by H φ(x) = −

h¯ 2 ψ(x) + λV2M (x)ψ(x), 2

x ∈ Rd ,

(2)

where λ ∈ R is a real positive “coupling” constant, V2M is a positive homogeneous 2M-order polynomial, and h¯ is the reduced Planck’s constant (the mass of the particle is set equal to 1 for simplicity). In the following, in order to simplify some notations, we shall put V2M (x) := |x|2M , but all our results are also valid in the more general case as they depend only on the positivity and the homogeneity properties of the potential V2M . H is essentially self-adjoint on C0∞ (Rd ) (see [23, Theorem X.28]). Its closure, denoted again by H , has the following domain: D(H ) = D() ∩ D |x|2M

2 2 d 2 4M 4 ˆ φ(x) dx < ∞, = φ∈L R : |x| |p| φ(p) dp < ∞ , where φˆ denotes the Fourier transform of φ. It is well known that H is a positive operator with a pure point spectrum {En } ⊂ R+ . Therefore −H generates an analytic contraction semigroup, P (z) : L2 (Rd ) → L2 (Rd ), P (z) = e−H z , with z being a complex parameter with positive real part. In the case where z is purely imaginary of the form z = hi¯ t, t ∈ R, one obtains a one pai

rameter group of unitary operators U (t) := e− h¯ H t , i.e. the Schrödinger group. Given a vector ψ0 ∈ L2 (Rd ), the vector ψ(t) := U (t)ψ0 belongs to D(H ) and it satisfies the Schrödinger equation: i h¯

∂ ψ(t) = H ψ(t). ∂t

(3)

The particular form of the potential allows one to prove a scaling property for the eigenvalues {En } of the operator H as well as their analyticity as a function of the coupling constant λ on a suitable region of a Riemann surface. The present lemma is taken from [25], which presents a detailed study of this problem, also in more general cases. Lemma 1. Let En (λ) denote the nth eigenvalue of the Hamiltonian (2). Then for λ, α > 0 one has En (λ) = α −1 En λα M+1 .

(4)

In particular 1

En (λ) = λ M+1 En (1).

(5)


1033

Proof. Let us consider, for any α ∈ R+ , the unitary operator V (α) : L2 (Rd ) → L2 (Rd ) given on vectors φ ∈ S(Rd ) by V (α)φ(x) = α 1/4 φ α 1/2 x ,

x ∈ Rd .

(6)

It simple to verify that V (α) leaves D(−) and D(|x|2M ) invariant and V (α)x 2M V (α)−1 = α M x 2M , V (α)V (α)−1 = α −1 . It follows that 2 h¯ V (α)H V (α)−1 = α −1 − + α M+1 λx 2M . 2 In particular, by taking α = λ−1/(M+1) one has

V λ

−1/(M+1)

2 −1/(M+1) −1 h¯ 1/(M+1) 2M − +x . HV λ =λ 2

(7)

As for any α ∈ R+ , the operator V (α)H V (α)−1 has the same spectrum of H , from Eq. (7) one easily deduces Eq. (5). 2 Remark 1. By analytic continuation, relation (5) allows to extends En (λ) to all complex values of λ belonging to a Riemann surface. In particular, the function En is many-sheeted and has an (M + 1)st order branch point at λ = 0. i

Let us consider now the matrix elements of the evolution operator U (t) = e− h¯ H t , i.e. the inner products φ, U (t)ψ, with φ, ψ ∈ L2 (Rd ). Let us consider also the function f : R+ → C of the coupling constant λ (present in H , hence in U (t)) defined by

f (λ) := φ, U (t)ψ ,

λ ∈ R, λ > 0.

(8)

˜ of the logarithm defined by Let us denote by Dθ1 ,θ2 the sector of the Riemann surface C Dθ1 ,θ2 := z ∈ C, z = ρeiφ : ρ > 0, φ ∈ (θ1 , θ2 ) . Let us consider the dense subset of L2 (Rd ) made of finite linear combinations of vectors of the form φ(x) = P (x)e−σ

2 |x|2

,

x ∈ Rd

(9)

with P being any polynomial with complex coefficients and σ 2 ∈ C a complex constant with positive real part (that these vectors are dense in L2 (Rd ) follows from the known fact that the finite linear combinations of Hermite functions are dense in the same space).

1034


¯ ψ ∈ S(Rd ) are of the Theorem 1. Let φ, ψ ∈ S(Rd ) ⊂ L2 (Rd ), such that the functions φ, form (9), with σ 2 ∈ C, σ 2 = |σ 2 |eiδ , δ ∈ R such that there exists an > 0 with

cos(δ + α) > ,

M −1 ∀α ∈ 0, π . M +1

(10)

M−1 .) Then the function f : R+ → C defined by (8) (This is the case for instance if δ = −π 2(M+1) ˜ of can be extended to an analytic function on the sector D−(M−1)π,0 of the Riemann surface C the logarithm.

Proof. Let us denote by {En (λ)}, resp. {en (λ)} the eigenvalues and resp. the eigenvectors of the Hamiltonian operator (2). Under the given assumptions on φ, ψ and by Eq. (5), for λ ∈ R+ the function f is given by f (λ) =

i

an (λ)bn (λ)e− h¯ En (λ)t =

i

an (λ)bn (λ)e− h¯ λ

1 M+1 En (1)t

with

an (λ) = φ, en (λ) ,

bn (λ) = en (λ), ψ .

On the other hand each coefficient an , bn can be extended to an analytic function of the variable λ on D−(M−1)π,0 . Indeed one has (for λ > 0) that en (λ) = V (λ−1/(M+1) )−1 en (1), where V (−λ1/(M+1) ) is the operator defined by (6). Without loss of generality, we can consider as an 2 2 instance a vector ψ of the form ψ(x) = x k e−σ x . In this case one has: −1

bn (λ) = V λ−1/(M+1) en (1), ψ = en (1), V λ−1/(M+1) ψ 1 1 k 2 − (M+1) |x|2 = λ− 4(M+1) en (1)(x)λ− 2(M+1) |x|k e−σ λ

(11) (12)

and the coefficient bn (λ) can be interpreted in terms of the inner product between the vector en (1) and the function 1

k

x → ψλ (x) := λ− 4(M+1) λ− 2(M+1) |x|k e−σ

1 2 λ− (M+1) |x|2

.

(13)

For λ ∈ D−(M−1)π,0 and for σ 2 satisfying the assumptions of the theorem, it is simple to verify that the function (13) belongs to L2 (Rd ) and its L2 -norm is uniformly bounded in D−(M−1)π,0 : ψλ
0)} and D2 = {Im(λ < 0)}. The function f1 , for λ ∈ R λ < 0 can be seen as the one-dimensional analogue of i φ, e− h¯ H t ψ, while function f2 as the one-dimensional analogue of the functional integral (28). It is possible to verify by means of a rotation of the integration contour in the complex 4 2 plane, that for λ ∈ R+ one has f1 (λ) = f2 (λ). For instance if λ = 1 one has eix +ix dx = 4 2 eiπ/4 e−ix −x dx.


1045

The extension of the equality between f1 and f2 on the negative real line is, on the other hand, not possible. Indeed it is possible to see that f1 and f2 are different branches on an analytic multivalued function defined on a Riemann sheet. A transformation of variable allows us to represent the two functions in a fashion allowing us to enlighten their analyticity properties and the nature of the singularity in λ = 0. Indeed when λ ∈ R+ , the following equality holds f2 (λ) = eiπ/4

e−iλx

4 −x 2

dx =

eiπ/4 λ1/4

e

−ix 4 −

x2 λ1/2

dx.

The right-hand side is an analytic function in the region λ ∈ C, λ = |λ|eiφ : |λ| > 0, −π < φ < π . It is continuous on the boundary of its analyticity domain but it is multivalued and it assumes different values approaching the negative real axis from above and from below: f2 |λ|eiπ =

eiλx +ix dx, 4 2 −iπ iπ/2 =e f2 |λ|e eiλx −ix dx. 4

2

By a rotation technique it is easy to verify that the latter integral is equal to eiπ/2

eiλx

4 −ix 2

dx = eiπ/4

e−iλx

4 −x 2

dx.

4 2 4 2 In other words we can say that the two integrals eiλx +ix dx and eiπ/4 e−iλx −x dx do not coincide on the negative real line: they are different branches of the same analytic but multivalued function. In a similar way, the functional integral (28) and the matrix elements of the Schrödinger group can be interpreted, as functions of the complex variable λ, as different branches of the same analytic but multivalued function. Remark 3. Despite the problems described so far, in the literature some particular cases have been handled by means of different techniques. In [19], equality (31) has been proved in the case where H is the inverse quartic oscillator Hamiltonian H0 ψ(x) = −

1 h¯ 2 ψ(x) + xΩ 2 xψ(x) − λ|x|4 ψ(x), 2 2

ψ ∈ C02 (Rd ), λ ∈ R+ ,

by means of an analytic continuation technique in the mass parameter. In [12] the pointwise solution of the heat equation (19) and its functional integral representation have been considered: − z Ht √ zλ t √ 2M e h¯ ψ (x) = e− h¯ 0 | h¯ zω(s)+x| ds ψ0 h¯ zω(t) + x W (dω). (34) Ct

1046


The right-hand side of (34) evaluated for z = i gives

iλ

e− h¯

t √ ¯ iω(s)+x|2M ds 0| h

ψ0

√ h¯ iω(t) + x W (dω).

(35)

Ct π

For suitable exponents 2M, namely for Re(ei(M+1) 2 ) < 0, the integral (35) is well defined and, as proved in [12] by means of a probabilistic argument, it represents the pointwise solution of Schrödinger equation. Let us consider now the integral (28) and let us assume that the hypothesis of Theorem 8 are satisfied. By considering two suitable sets of vectors in S1 , S2 ⊂ L2 (Rd ), with φ ∈ S2 and φ ∈ S1 , it is possible to interpret the integral Iti (φ, ψ) as the matrix element of an evolution operator in L2 (Rd ). Let us denote by S1 the subset of S(Rd ) made of the functions φ : Rd → C of the form φ(x) = P (x)e−

x2 2 (1−i)

,

x ∈ Rd ,

(36)

and by S2 the subset of S(Rd ) made of the functions φ : Rd → C of the form φ(x) = Q(x)e−

x2 2 (1+i)

(37)

,

where P and Q are polynomials with complex coefficients. As the Hermite functions form a complete orthonormal system in L2 (Rd ), it is simple to verify that both S1 and S2 are dense in L2 (Rd ). Moreover the functions φ ∈ S1 are such that: (1) the function z → φ(zx), x ∈ Rd , z ∈ D¯ 0,π/4 is analytic on D0,π/4 and continuous on D¯ 0,π/4 , π (2) the function x → φ(ei 4 x), x ∈ Rd is in L2 , while the functions φ ∈ S2 are such that: (1) the function z → φ(zx), x ∈ Rd , z ∈ D¯ −π/4,0 is analytic on D−π/4,0 and continuous on D¯ −π/4,0 , π (2) the function x → φ(e−i 4 x), x ∈ Rd belongs to L2 (Rd ). Let us denote by T : S1 → S2 the linear operator defined by π π T φ(x) = ei 8 d φ ei 4 x ,

φ ∈ S1 ,

and by T −1 : S2 → S1 its inverse, defined by π π T −1 φ(x) = e−i 8 d φ e−i 4 x ,

φ ∈ S2 .

By considering two vectors φ, ψ ∈ S1 one can easily verify that φ, T ψ = T φ, ψ.

(38)


1047

Analogously, by considering two vectors φ, ψ ∈ S2 , one can easily verify that

φ, T −1 ψ = T −1 φ, ψ .

(39)

This implies that, for φ ∈ S2 , ψ ∈ S1 , one has T −1 φ, T ψ = φ, ψ. Let HT : S2 → S2 be the operator defined by −iHT := T H T −1 . It is easy to verify that HT φ(x) = −

M+1 h¯ 2 φ(x) + λei 2 π |x|2M φ(x), 2

φ ∈ S2 .

M+1

Theorem 9. Let ψ ∈ S1 and φ ∈ S2 . Let us assume that Re(ei 2 π ) 0. Then the operator HT is the restriction to S2 of the generator A of a strongly continuous contraction semigroup 1 V (t) = e− h¯ At and the integral Iti (φ, ψ) given by (28) is equal to the matrix element T −1 φ, V (t)T ψ. Proof. Let V (t)t0 be the C0 -contraction semigroup defined by the Feynman–Kac type formula:

e−e

V (t)ψ(x) :=

i M+1 2 πλ t h¯ 0

√ | h¯ ω(s)+x|2M ds

ψ

√

h¯ ω(t) + x W (dω).

(40)

Ct

The operator-theoretic results for semigroups [13] of the form (40) have been investigated in t [21,18] (see also [16, Chapter 13.5]). In particular the generator A of the semigroup V (t) = e− h¯ A is given on smooth vectors ψ ∈ S(Rd ) by Aψ(x) = −

h¯ 2 ψ(x) + Q2M (x)ψ(x), 2

Q2M (x) := λei

M+1 2 π

|x|2M

(41)

with domain

h¯ 2 D(A) = ψ ∈ H 1 Rd : − ψ + Q2M ψ ∈ L2 Rd . 2

(42)

By a direct computation and by taking ψ ∈ S1 and φ ∈ S2 , one can easily verify that π

ei 4 d

π φ¯ ei 4 x

Rd

= T

e−

Ct −1

φ, V (t)T ψ

e

i(M+1) π 2 h¯

λ

t √ ¯ ω(s)+x|2M ds 0| h

ψ

√ i π π h¯ e 4 ω(t) + ei 4 x W (dω) dx

so that Iti (φ, ψ) = T −1 φ, V (t)T ψ.

2

Remark 4. Under the assumptions of Theorem 9, it is possible to give an alternative proof of Theorem 8, i.e. that the integral Iti (φ, ψ) is a weak solution of the Schrödinger equation in the sense that Eqs. (29) and (30) are satisfied.

1048


Indeed Eq. (29) follows by writing I0i (φ, ψ) as T −1 φ, T ψ and by Eqs. (38) and (39). t

Eq. (30) follows from the equality Iti (φ, ψ) := T −1 φ, e− h¯ A T ψ. Indeed i h¯

t t t d −1 T φ, e− h¯ A T ψ = T −1 φ, e− h¯ A (−iHT )T ψ = T −1 φ, e− h¯ HT T H ψ . dt it

Remark 5. To prove that Iti (φ, ψ) = φ, e− h¯ H ψ it would be sufficient to have that |Iti (φ, ψ)| t C φ ψ , or, in other words, that given ψ ∈ S1 , one has that the vector e− h¯ HT T ψ belongs to the domain of (T −1 )∗ , so that

∗ t t T −1 φ, e− h¯ HT T ψ = φ, T −1 e− h¯ HT T ψ .

If the inequality |Iti (φ, ψ)| C φ ψ holds true, then this would namely imply that there exists a bounded operator B(t) : L2 → L2 such that Iti (φ, ψ) = φ, B(t)ψ and B(0) = I . Iti (·,·) defined on S2 × S1 can be extended to L2 × L2 . In particular then Iti (U (t)φ, ψ) makes sense and by differentiating with respect to the time variable t we obtain i h¯

d i I U (t)φ, ψ = Iti U (t)φ, H ψ − Iti H U (t)φ, ψ = 0 dt t

so that Iti (U (t)φ, ψ) = I0i (U (0)φ, ψ) = φ, ψ for any t and this implies that B(t) = U (t). 5. The functional integral as asymptotic solution The present section is devoted to the proof that the functional integral (28) coincides asympi totically both as t → 0 and as λ → 0 with the matrix element φ, e h¯ H t ψ of the Schrödinger group. Theorem 10. Let φ, ψ ∈ L2 (Rd ) and M ∈ N satisfy the assumptions of Theorem 9. Then as i t → 0 the integral Iti (φ, ψ) and the matrix element φ, e h¯ H t ψ of the Schrödinger group admit the following asymptotic expansions Iti (φ, ψ) =

an t n ,

n i φ, e h¯ H t ψ = bn t ,

and they coincides, i.e. an = bn ∀n ∈ N. Proof. By Theorem 9, the functional integral Iti (φ, ψ) can be written as T −1 φ, e− h¯ A T ψ, where A is the operator defined by (41) and (42). Under the stated assumptions, the vector T ψ belongs to the domain of An ∀n ∈ N, and An T ψ belongs to S2 ⊂ D(T −1 ) ∀n ∈ N. In particular, for any N ∈ N one has t

N

−1 t t n −1 1 − T φ, e− h¯ A T ψ = T φ, An T ψ + RN , h¯ n! n=0


1049

1

t 1 t N RN = uN −1 T −1 φ, AN e− h¯ A(1−u) T ψ du. − h¯ N − 1! 0

The remainder RN can easily be estimated by means of Schwartz inequality and one has: |RN |

|t|N |h¯ |−N T −1 φ AN T ψ = O |t|N . N!

As T ψ ∈ S2 , one has An T ψ = (HT )n T ψ = i n T H n ψ , so that by Eq. (39)

T

−1

φ, e

− ht¯ A

N it n 1 − φ, H n ψ + O t N . Tψ = n! h¯

n=0

Analogously

φ, e

− ith¯ H

N it n 1

− φ, H n ψ + RN ψ = , n! h¯

n=0

RN

1

it it N 1 − = uN −1 φ, H N e− h¯ H (1−u) ψ du, N − 1! h¯ 0

= O(t N ), and one can easily verify that the asymptotic expansion in powers of t of with RN it

Iti (φ, ψ) and of φ, e− h¯ H ψ coincide.

2

Remark 6. The power series of the variable t are not convergent, but only asymptotic. This fact implies that the result Theorem 10 is not sufficient to deduce the equality between Iti (φ, ψ) and i

φ, e h¯ H t ψ. Let us consider now couples of vectors φ, ψ satisfying the assumptions of Theorem 1 (so that the result of Theorem 4 holds) and such that the functional integral (28) is well defined. In fact, it is always possible to find a dense set of vectors in L2 (Rd ) satisfying both conditions. For instance, if M = 2, it is sufficient to take ψ ∈ S1 and φ ∈ S2 , while if M 3 the fulfillment of hypothesis of Theorem 1 implies that the integral (28) is well defined. Under these conditions, it is possible to interpret the functional integral (28) as an asymptotic weak solution of the Schrödinger equation, in the sense of the following theorem. Theorem 11. Under the assumptions above, the asymptotic expansion in powers of the coupling constant λ as λ → 0 of the functional integral representation (28) coincides with the corresponding asymptotic expansion (14) of the matrix elements of the Schrödinger group. Moreover the latter is Borel summable. Proof. By expanding the functional integral (28) in powers of λ one has

1050


Iti (φ, ψ) =

N −1

a n λ n + RN ,

n

with 1 π an = e i 4 d n!

π φ¯ ei 4 x

Rd

×ψ

√

h¯ e

n i(M+1) π t 2M √ 2λ e h¯ ω(s) + x ds − h¯

Ct i π4

ω(t) + e

i π4

0

x W (dω) dx

and π λN RN = ei 4 d (N − 1)!

× e−u

e

i(M+1) π 2 h¯

1

N −1

(1 − u)

π φ¯ ei 4 x

Rd

0

λ t √ ¯ ω(s)+x|2M ds 0| h

N i(M+1) π t √ 2M 2λ e ds − h¯ ω(s) + x h¯

Ct

ψ

0

√ i π π h¯ e 4 ω(t) + ei 4 x W (dω) dx du.

It is easy to verify that RN = O(λN ). Moreover, the coefficients an can be written as: i(M+1) π n t t 2λ π e 1 iπd an = e 4 − . . . ds1 . . . dsn φ¯ ei 4 x n! h¯ ×

0

Rd

0

√ √ √ h¯ ω(s1 ) + x 2M ds . . . h¯ ω(sn ) + x 2M ds ψ h¯ ei π4 ω(t) + ei π4 x W (dω) dx.

Ct

By the result of Corollary 1, the latter coincides with the coefficient an of the asymptotic expansion (14). On the other hand, by Theorem 4, the matrix elements of the Schrödinger group can be obtained in terms of the Borel sum of the asymptotic expansion an λn . 2 Remark 7. By a direct computation it is possible to verify that in the case M = 2 (the quartic oscillator case) the functional integral Iti (φ, ψ) can be extended to an analytic function of the variable λ in the sector D0,π of the complex plane and satisfies there an estimate of the following form N −1 i n an λ AC N |λ|N N !. It (φ, ψ) − n

By Watson–Nevanlinna’s theorem, it is possible to recover Iti (φ, ψ) in terms of the coefficients i

an in the asymptotic expansion. This result, combined with the analogous result for φ, e− h¯ H t ψ, i is not sufficient to deduce the equality Iti (φ, ψ) = φ, e− h¯ H t ψ, as the two function are defined as the Borel sums of the same asymptotic expansion but on different regions of the complex plane (the left-hand side on D0,π and the right-hand side on Dπ,o ). Indeed, let us consider two functions f1 (z) and f2 (z) of the complex variable z, defined and holomorphic resp. in D0,π


1051

and Dπ,0 , admitting as z → 0 the same asymptotic expansion and estimate uniformly in their analyticity domains:

f1 (z) ∼

f2 (z) ∼

N −1 an zn A1 C1N |z|N N !, f1 (z) − n N −1 n an z A2 C2N |z|N N !. f2 (z) −

an z n ,

an z n ,

n

The function f1 , f2 can be recovered by their asymptotic expansion an zn by means of the following procedure. Let us define two functions g1 (z) and g2 (z) of the complex variable z ∈ D−π/2,π/2 defined by g1 (z) := f1 (iz),

g2 (z) := f2 (−iz),

z ∈ D−π/2,π/2 .

g1 and g2 admit the following asymptotic expansion and estimate: g1 (z) ∼

g2 (z) ∼

n

N −1 n n i an z A1 C1N |z|N N !, g1 (z) − n N −1 (−i)n an zn A2 C2N |z|N N !. g2 (z) − n

n

i an z ,

(−i)n an zn ,

By Theorem 3 they are both Borel summable, i.e. formally: 1 g1 (z) = z

∞

e−t/z

i n an n!

t n dt,

(43)

0

g2 (z) =

1 z

∞

e−t/z

(−i)n an n!

t n dt.

(44)

0

One would have f1 (z) = f2 (z) for z ∈ R+ , if g1 (iρ) = g2 (−iρ) for ρ ∈ R+ , however the Borel summability of the asymptotic expansion an zn , in particular the relations (43) and (44), are by themselves not sufficient to deduce this equality. Remark 8. In [4] the asymptotic expansion in powers of the parameter h¯ of a finite-dimensional analogue of the integral Iti (φ, ψ) has been studied and its Borel summability has been proved. Acknowledgments The hospitality of the Mathematics Institutes of Trento and of Bonn Universities is gratefully acknowledged, as well as the financial support of the F. Severi fellowship of I.N.d.A.M.

1052


References [1] S. Albeverio, R. Høegh-Krohn, S. Mazzucchi, Mathematical theory of Feynman path integrals, in: An Introduction, 2nd and enlarged edition, in: Lecture Notes in Math., vol. 523, Springer-Verlag, Berlin/Heidelberg, 2008. [2] S. Albeverio, S. Mazzucchi, Generalized infinite-dimensional Fresnel integrals, C. R. Math. Acad. Sci. Paris 338 (3) (2004) 255–259. [3] S. Albeverio, S. Mazzucchi, Some new developments in the theory of path integrals, with applications to quantum theory, J. Stat. Phys. 115 (112) (2004) 191–215. [4] S. Albeverio, S. Mazzucchi, Generalized Fresnel integrals, Bull. Sci. Math. 129 (1) (2005) 1–23. [5] S. Albeverio, S. Mazzucchi, Feynman path integrals for polynomially growing potentials, J. Funct. Anal. 221 (1) (2005) 83–121. [6] S. Albeverio, S. Mazzucchi, Feynman path integrals for the time dependent quartic oscillator, C. R. Math. Acad. Sci. Paris 341 (10) (2005) 647–650. [7] S. Albeverio, S. Mazzucchi, The time dependent quartic oscillator—A Feynman path integral approach, J. Funct. Anal. 238 (2) (2006) 471–488. [8] S. Albeverio, S. Mazzucchi, The trace formula for the heat semigroup with polynomial potential, SFB-611-Preprint No. 332, Bonn, 2007. [9] C. Bender, T. Wu, Anharmonic oscillator, Phys. Rev. (2) 184 (1969) 1231–1260. [10] R.H. Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. Phys. 39 (1960) 126–140. [11] I.M. Davies, A. Truman, On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender–Wu formula for x 2N -anharmonic oscillators, J. Math. Phys. 24 (2) (1983) 255–266. [12] H. Doss, Sur une Résolution Stochastique de l’Equation de Schrödinger à Coefficients Analytiques, Comm. Math. Phys. 73 (1980) 247–264. [13] K.-J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, in: S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, R. Schnaubelt (Eds.), Grad. Texts in Math., vol. 194, Springer-Verlag, New York, 2000. [14] G.H. Hardy, Divergent Series, Oxford University Press, London, 1963. [15] T. Hida, H.H. Kuo, J. Potthoff, L. Streit, White Noise, Kluwer, Dordrecht, 1995. [16] G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford University Press, New York, 2000. [17] G. Kallianpur, D. Kannan, R.L. Karandikar, Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron Martin formula, Ann. Inst. H. Poincaré Probab. Th. 21 (1985) 323–361. [18] T. Kato, On some Schrödinger operators with a singular complex potential, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5 (1) (1978) 105–114. [19] S. Mazzucchi, Feynman path integrals for the inverse quartic oscillator, J. Math. Phys. 49 (9) (2008) 093502 (15 pp.). [20] S. Mazzucchi, Mathematical Feynman Path Integrals and Applications, World Scientific Publishing, Singapore, 2009. [21] E. Nelson, Feynman integrals and the Schrödinger equation, J. Math. Phys. 5 (1964) 332–343. [22] F. Nevanlinna, Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. (A) 12 (3) (1919) 1–81. [23] M. Reed, B. Simon, Methods of modern mathematical physics, II, in: Fourier Analysis, Self-Adjointness, Academic Press/Harcourt Brace Jovanovich Publishers, New York/London, 1975. [24] M. Reed, B. Simon, Methods of modern mathematical physics, IV, in: Analysis of Operators, Academic Press/ Harcourt Brace Jovanovich Publishers, New York/London, 1978. [25] B. Simon, Coupling constant analyticity for the anharmonic oscillator, Ann. Physics 58 (1970) 76–136. [26] B. Simon, Functional Integration and Quantum Physics, second ed., Amer. Math. Soc. Chelsea Publishing, Providence, RI, 2005. [27] A. Sokal, An improvement of Watson’s theorem on Borel summability, J. Math. Phys. 21 (1980) 261–263. [28] K. Yajima, Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys. 181 (1996) 605–629.


Local minimizers of the Ginzburg–Landau functional with prescribed degrees Mickaël Dos Santos Université de Lyon, Université Lyon 1, INSA de Lyon, F-69621, Ecole Centrale de Lyon, CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne cedex, France Received 19 February 2009; accepted 23 February 2009 Available online 17 March 2009 Communicated by H. Brezis

Abstract We consider, in bounded multiply connected domain D ⊂ R2 , the Ginzburg–Landau en a smooth 1 1 2 ergy Eε (u) = 2 D |∇u| + 2 D (1 − |u|2 )2 subject to prescribed degree conditions on each com4ε ponent of ∂D. In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg– Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: Eε (u) has, in domains D with 2, 3, . . . holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control. © 2009 Elsevier Inc. All rights reserved. Keywords: Ginzburg–Landau functional; Prescribed degrees; Local minimizers

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.023

1054

M. Dos Santos / Journal of Functional Analysis 257 (2009) 1053–1091

1. Introduction This article deals with the existence problem of local minimizers of the Ginzburg–Landau functional with prescribed degrees in a 2D perforated domain D. The domain we consider is of the form D = Ω \ i∈NN ωi , where N ∈ N∗ , Ω and the ωi ’s are simply connected, bounded and smooth open sets of R2 . We assume that ωi ⊂ Ω and ωi ∩ ωj = ∅ for i, j ∈ NN := {1, . . . , N}, i = j . The Ginzburg–Landau functional is 2 1 1 Eε (u, D) := 1 − |u|2 dx |∇u|2 dx + 2 (1) 2 4ε D

D

with u : D → C R2 and ε is a positive parameter (the inverse of κ, the Ginzburg–Landau parameter). When there is no ambiguity we will write Eε (u) instead of Eε (u, D). Functions we will consider belong to the class J = u ∈ H 1 (D, C) s.t. |u| = 1 on ∂D . Clearly, J is closed under weak H 1 -convergence. This functional is a simplified version of the Ginzburg–Landau functional which arise in superconductivity (or superfluidity) to model the state of a superconductor submitted to a magnetic field (see, e.g., [10] or [9]). The simplified version of the Ginzburg–Landau functional considered in (1) ignores the magnetic field. The issue we consider in this paper is existence of local minimizers with prescribed degrees on ∂D. We next formulate rigorously the problem discussed in this paper. To this purpose, we start by defining properly the degrees of a map u ∈ J . For γ ∈ {∂Ω, . . . , ∂ωN } and u ∈ J we let 1 degγ u = u × ∂τ u dτ. 2π γ

Here: • each γ is directly (counterclockwise) oriented, • τ = ν ⊥ , τ is the tangential vector of γ and ν the outward normal to Ω if γ = ∂Ω or ωi if γ = ∂ωi , • ∂τ = τ · ∇, the tangential derivative and “·” stands for the scalar product in R2 , • “×” stands for the vectorial product in C, (z1 + ız2 ) × (w1 + ıw2 ) := z1 w2 − z2 w1 , z1 , z2 , w1 , w2 ∈ R, • the integral over γ should be understood using the duality between H 1/2 (γ ) and H −1/2 (γ ) (see, e.g., [4, Definition 1]). It is known that degγ u is an integer see [4] (the introduction) or [6]. We denote the (total) degree of u ∈ J in D by deg(u, D) = deg∂ω1 (u), . . . , deg∂ωN (u), deg∂Ω (u) ∈ ZN × Z.


1055

For (p, q) ∈ ZN × Z, we are interested in the minimization of Eε in Jp,q := u ∈ J s.t. deg(u, D) = (p, q) . There is a huge literature devoted to the minimization of Eε . In a simply connected domain Ω, the minimization problem of Eε with the Dirichlet boundary condition g ∈ C ∞ (∂Ω, S 1 ) is studied in details in [7]. Eε has a minimizer for each ε > 0. This minimizer need not to be unique. In this framework, when deg∂Ω (g) = 0, the authors studied the asymptotic behaviour of a sequence of minimizers (when εn ↓ 0) and point out the existence (up to subsequence) of a finite set of singularities of the limit. Other types of boundary conditions were studied, like Dirichlet condition g ∈ C ∞ (∂Ω, C \ {0}) (in a simply connected domain Ω) in [1] and later for g ∈ C ∞ (∂Ω, C) (see [2]). If the boundary data is not u|∂ D , but a given set of degrees, then the existence of local minimizers is not trivial. Indeed, one can show that Jp,q is not closed under weak H 1 -convergence (see next section), so that one cannot apply the direct method in the calculus of variations in order to derive existence of minimizer. Actually this is not just a technical difficulty, since in general the infimum of Eε in Jp,q is not attained, we need more assumptions like the value of the H 1 -capacity of D (see [3] and [4]). Minimizers u of Eε in Jp,q , if they do exist, satisfy the equation ⎧ u ⎪ −u = 2 1 − |u|2 in D, ⎪ ⎨ ε |u| = 1 on ∂D, ⎪ ⎪ u × ∂ u = 0 on ∂D, ν ⎩ deg(u, D) = (p, q)

(2)

∂ = ν · ∇. where ∂ν denotes the normal derivative, i.e., ∂ν = ∂ν Existence of local minimizers of Eε is obtained following the same lines as in [5]. It turns out that, even if the infimum of Eε in Jp,q is not attained, (2) may have solutions. This was established by Berlyand and Rybalko when D has a single hole, i.e., when N = 1. Our main result is the following generalisation of the main result in [5]:

Theorem 1. Let (p, q) ∈ ZN × Z and let M ∈ N∗ , there is ε1 (p, q, M) > 0 s.t. for ε < ε1 , there are at least M locally minimizing solutions. Actually, we will prove a more precise form of Theorem 1 (see Theorem 2), whose statement relies on the notion of approximate bulk degree introduced in [5] and generalised in the next section. The main difference with respect to [5] stems in the construction of the test functions with energy control in Section 6. In a sense that will be explained in details in Section 6, our construction is local, while the one in [5] is global. We also simplify and unify some proofs in [5]. We do not know whether the conclusion of Theorem 1 still holds when D has no holes at all. That is, we do not know whether for a simply connected domain Ω, a given d ∈ Z∗ and small ε, the problem ⎧ −u = εu2 (1 − |u|2 ) in Ω, ⎪ ⎪ ⎨ on ∂Ω, u × ∂ν u = 0 (3) ⎪ |u| = 1 on ∂Ω, ⎪ ⎩ deg∂Ω (u) = d

1056


has solutions. Existence of a solution of (3) is clear when Ω is a disc, say Ω = D(0, R) (it suffices z d z d ) with u|∂Ω = ( |z| ) ). to consider a solution of −u = εu2 (1 − |u|2 ) of the form u(z) = f (|z|)( |z| However, we do not know the answer when Ω is not radially symmetric anymore. 2. The approximate bulk degree This section is a straightforward adaptation of [5]. Existence of (local) minimizers for Eε in Jp,q is not straightforward since Jp,q is not closed under weak H 1 -convergence. A typical example (see [4]) is a sequence (Mn )n s.t. Mn : D(0, 1) → D(0, 1), x →

x − (1 − 1/n) , (1 − 1/n)x − 1

where D(0, 1) ⊂ C is the open unit disc centred at the origin. Then Mn 1 in H 1 , degS 1 (Mn ) = 1 and degS 1 (1) = 0. To obtain local minimizers, Berlyand and Rybalko (in [5]) devised a tool: the approximate bulk degree. We adapt this tool for a multiply connected domain. We consider, for i ∈ NN := {1, . . . , N}, Vi the unique solution of

−Vi = 0 Vi = 1 Vi = 0

in D, on ∂D \ ∂ωi , on ∂ωi .

For u ∈ J := {u ∈ H 1 (D, C), |u| = 1 on ∂D}, we set, noting ∂k u = 1 abdegi (u, D) = 2π

(4)

∂ ∂xk u

u × (∂1 Vi ∂2 u − ∂2 Vi ∂1 u) dx, D

abdeg(u, D) = abdeg1 (u, D), . . . , abdegN (u, D) .

(5)

Following [5], we call abdeg(u, D) the approximate bulk degree of u. abdegi : J → R, in general, is not an integer (unlike the degree). However, we have Proposition 1. (1) If u ∈ H 1 (D, S 1 ), then abdegi (u, D) = deg∂ωi (u); (2) Let Λ, ε > 0 and u, v ∈ J s.t. Eε (u), Eε (v) Λ, then abdeg (u) − abdeg (v) 2 Vi 1 Λ1/2 u − v 2 ; i i C (D ) L (D ) π

(6)

(3) Let Λ > 0 and (uε )ε>0 ⊂ J s.t. for all ε > 0, Eε (uε ) Λ, then dist abdeg(uε ), ZN → 0 when ε → 0.

(7)


1057

Proof of Proposition 1 is postponed to Appendix B. We define for d = (d1 , . . . , dN ) ∈ ZN , p = (p1 , . . . , pN ) ∈ ZN and q ∈ Z, 1 d d . Jp,q = Jp,q (D) := u ∈ Jp,q ; abdeg(u) − d ∞ := max di − abdegi (u) 3 i∈NN d in never empty for (p, q, d) ∈ ZN × Z × ZN . The following result states that Jp,q For i ∈ {0, . . . , N}, we denote ei = (δi,1 , . . . , δi,N , δi,0 ) ∈ ZN +1 where

δi,k =

1 if i = k, 0 otherwise.

It is the Kronecker symbol. d = ∅. Proposition 2. Let (p, q, d) ∈ ZN × Z × ZN . Then Jp,q

Proof. For i ∈ {0, . . . , N}, there is Mni ∈ J(pi −di )ei if i = 0 and Mn0 ∈ J(q− dj )e0 s.t. Mni 1 in H 1 and |Mni | 1 (Lemmas 6.1 and 6.2 in [4]). Let Ed := u ∈ H 1 D, S 1 ; deg(u, D) = (d, d) ,

d = (d1 , . . . , dN ), d =

N

dj .

j =1

i We note that, Ed = ∅, see, e.g., [7]. Let u ∈ Ed and un := u N i=0 Mn . Then we will prove that, d for large n, we have, up to subsequence, that un ∈ Jp,q . Indeed, up to subsequence, un u in H 1 ,

un ∈ Jp,q .

Using the fact that abdeg(u) = d and the weak H 1 -continuity of the approximate bulk degree, d . 2 we obtain for n sufficiently large, that un ∈ Jp,q d , i.e., We denote mε (p, q, d) the infimum of Eε on Jp,q

mε (p, q, d) = inf Eε (u). d u∈Jp,q

We may now state a refined version of Theorem 1. Theorem 2. Let d ∈ (N∗ )N . Then, for all (p1 , . . . , pN , q) ∈ ZN +1 s.t. q d and pi di , there is ε2 = ε2 (p, q, d) > 0 s.t. for 0 < ε < ε2 , mε (p, q, d) is attained. Moreover, we have the following estimate mε (p, q, d) = I0 (d, D) + π(d1 − p1 + · · · + dN − pN + d − q) − oε (1),

oε (1) − −−→ 0. ε→0

N ∗ )N s.t. p d and For i i further use, a configuration of degrees (p, q, d) ∈ Z × Z × (N N q di will be called a “good configuration”. Noting that, for d = d˜ ∈ Z and (p, q) ∈ ZN ×Z, d ∩ J d˜ = ∅, we are led to we have Jp,q p,q

1058


Proof of Theorem 1. Let (p, q) ∈ ZN × Z and set for k ∈ N∗ , d = max max |pi |, |q| i

and dk = (d + k, . . . , d + k).

dk We apply Theorem 2 to the class Jp,q . We obtain the existence of

ε1 (p, q, M) = min ε2 (p, q, dk ) > 0 k∈NM

s.t. for ε < ε1 , k ∈ NM , mε (p, q, dk ) is achieved by ukε . Noting the continuity of the degree and of the approximate bulk degree for the strong dk H 1 -convergence, there exists Vεk ⊂ Jp,q ⊂ J an open (for H 1 -norm) neighbourhood of ukε . It follows easily that Eε ukε = min Eε (u). u∈Vεk

Then ukε ∈ Jp,q is a local minimizer of Eε in J (for H 1 -norm) for 0 < ε < ε1 (p, q, M).

2

3. Basic facts of the Ginzburg–Landau theory It is well known (cf. [4, Lemma 4.4, p. 22]) that the local minimizers of Eε in Jp,q satisfy 1 u 1 − |u|2 in D, 2 ε and u × ∂ν u = 0 on ∂D.

−u = |u| = 1

(8) (9)

Eq. (8) and the Dirichlet condition on the modulus in (9) are classical. The Neumann condition on the phase in (9) is less standard but it is for example stated in [4]. Eq. (8) combined with the boundary condition on ∂D implies, via a maximum principle, that |u| 1 in D.

(10)

One of the questions in the Ginzburg–Landau model is the location of the vortices of stable solutions (i.e., local minimizers of Eε ). We will define ad hoc a vortex as an isolated zero x of u with nonzero degree on small circles around x. The following result shows that, under energy bound assumptions on solutions of (8), vortices are expelled to the boundary when ε → 0. Lemma 1. (See [8].) Let Λ > 0 and let u be a solution of (8) satisfying (10) and the energy bound Eε (u) Λ. Then with C, Ck and ε3 depending only on Λ, D, we have, for 0 < ε < ε3 and x ∈ D, 2 1 − u(x)

Cε 2 dist2 (x, ∂D)

(11)


1059

and k D u(x)

Ck distk (x, ∂D)

When u is smooth in D and ρ = |u| > 0, the map

u ρ

(12)

.

admits a lifting θ , i.e., we may write

u = ρeıθ , where θ is a smooth (and locally defined) real function on D and ∇θ is a globally defined smooth vector field. Using (8) and (9), we have

div ρ 2 ∇θ = 0 in B, on ∂D, ∂ν θ = 0 −ρ + |∇θ |2 ρ + ε12 ρ ρ 2 − 1 = 0 ρ=1

(13) in B, on ∂D,

(14)

here, B = {x ∈ D; u(x) = 0}. We will need later the following. Lemma 2. (See [5].) Let u be a solution of (8) and (9). Let G ⊂ D be an open Lipschitz set s.t. u does not vanish in G. Write, in G, u = ρv with ρ = |u|. Let w ∈ H 1 (G, C) be s.t. |tr∂G w| ≡ 1. Then Eε (ρw, G) = Eε (u, G) + Lε (w, G), with 1 Lε (w, G) = 2

1 ρ |∇w| dx − 2 2

2

G

1 |w| ρ |∇v| dx + 2 4ε 2 2

2

G

2 ρ 4 1 − |w|2 dx.

G

For further use, we note that we may write, locally in G, u = ρeıθ , so that v = eıθ . It turns out that ∇θ is smooth and globally defined in G. In terms of ∇θ , we may rewrite Lε (w, G) =

1 2

ρ 2 |∇w|2 dx − G

1 2

|w|2 ρ 2 |∇θ |2 dx + G

1 4ε 2

2 ρ 4 1 − |w|2 dx.

G

For u a solution of (8) and (9), we can consider (see Lemma 7 in [5]) h the unique globally defined solution of ⎧ ⎨ ∇ ⊥ h = u × ∇u in D, (15) h=1 on ∂Ω, ⎩ on ∂ωi , h = ki

1060


where ki ’s are real constants uniquely defined by the first two equations in (15). Here u × ∂1 u −∂2 h is the orthogonal gradient of h and u × ∇u = . ∇ ⊥h = ∂1 h u × ∂2 u It is easy to show that ⎧ ∇h = −ρ 2 ∇ ⊥ θ ⎪ ⎪ ⎨ 1 div 2 ∇h = 0 ⎪ ρ ⎪ ⎩ h = 2∂1 u × ∂2 u

in B, in B,

(16)

in B,

here, B = {x ∈ D; u(x) = 0}. In [7], Brezis, Bethuel and Hélein consider the minimization of E(u) = 12 D |∇u|2 dx, the Dirichlet functional, in the class Ed := u ∈ H 1 D, S 1 ; deg(u, D) = (d, d) ; here, d = dk . Theorem I.1 in [7] gives the existence of a unique solution (up to multiplication by an S 1 -constant) for the minimization of E in Ed . We denote u0 this solution. This u0 is also a solution of −v = v|∇v|2 in D, on ∂D. v × ∂ν v = 0 Moreover, we have 1 I0 (d, D) := min E(u) = u∈Ed 2 with h0 the unique solution of ⎧ h0 = 0 ⎪ ⎪ ⎪ ⎪ h =1 ⎪ ⎨ 0 h0 = Cst ⎪ ⎪ ⎪ ∂ν h0 dσ = 2πdk ⎪ ⎪ ⎩

|∇h0 |2 dx D

in D, on ∂Ω, on ∂ωk , k ∈ {1, . . . , N}, for k ∈ {1, . . . , N}.

∂ωk

One may prove that h0 is the (globally defined) harmonic conjugate of a local lifting of u0 . 4. Energy needed to change degrees We denote

æ : ZN × Z × ZN × Z → N, N |di − pi | + |d − q|. (d, d), (p, q) → i=1

(17)

(18)


1061

The next result quantifies the energy needed to change degrees in the weak limit. Lemma 3. (See [4, Lemma 1].) Let (un )n ⊂ Jp,q be a sequence weakly converging in H 1 to u. Then lim inf E(un ) E(u) + π æ deg(u, D), (p, q)

(19)

lim inf Eε (un ) Eε (u) + π æ deg(u, D), (p, q) .

(20)

n

and n

The next lemma is proved in [5]. Lemma 4. Let d = (d1 , . . . , dN ), p = (p1 , . . . , pN ) ∈ ZN , q ∈ Z. There is oε (1) − −−→ 0 (deε→0 d pending of (p, q, d)) s.t. for u ∈ Jp,q we have Eε (u) I0 (d, D) + π æ (d, d), (p, q) − oε (1). Here, d :=

(21)

di .

We present below a simpler proof than the original one in [5]. Proof. Let (p, q, d) ∈ ZN × Z × ZN . We argue by contradiction and we suppose that there are d s.t. δ > 0, εn ↓ 0 and (un ) ⊂ Jp,q Eεn (un ) I0 (d, D) + π æ (d, d), (p, q) − δ.

(22)

Since (un )n is bounded in H 1 , there is some u s.t., up to subsequence, un u in H 1 and un → u in L4 . Using the strong convergence in L4 , (22) and Proposition 1, we have d . u ∈ H 1 (D, S 1 ) ∩ Jd,d To conclude, we apply Lemma 3 I0 (d, D) + π æ (d, d), (p, q) − δ lim inf Eεn (un ) n

lim inf E(un ) n E(u) + π æ (d, d), (p, q) I0 (d, D) + π æ (d, d), (p, q) which is a contradiction.

2

One may easily proved (see Lemma 14 in Appendix C) that for η > 0, i ∈ {0, . . . , N } and u ∈ Jdeg(u,D) , there are v± ∈ Jdeg(u,D)±ei s.t. Eε (v± ) Eε (u) + π + η.

1062


The key ingredient is a sharper result which holds under two additional hypotheses. In order to unify the notations, we use the notation ω0 for Ω. We may now state the main ingredient in the proof of Theorem 2. Lemma 5. Let u ∈ Jp,q be a solution of (8), (9). Assume that 1 1 , abdegj (u) ∈ dj − , dj + 3 3

∀j ∈ NN .

(23)

Let i ∈ {0, . . . , N} and assume that there is some point x i ∈ ∂ωi s.t. u × ∂τ u(x i ) > 0. Then there is u˜ ∈ J(p,q)−ei s.t. Eε (u) ˜ < Eε (u) + π, 1 1 , abdegj (u) ˜ ∈ dj − , dj + 3 3

∀j ∈ NN .

The proof of Lemma 5 is postponed to Section 6. We also have an upper bound for mε (p, q, d). Lemma 6. Let ε > 0 and (p, q, d) ∈ ZN × Z × ZN . Then mε (p, q, d) I0 (d, D) + π æ (d, d), (p, q) .

(24)

To prove Lemma 6, we need the following Lemma 7. Let u ∈ J , ε > 0 and δ = (δ1 , . . . , δN , δ0 ) ∈ ZN +1 . For all η > 0, there is uδη ∈ Jdeg(u,D)+δ s.t.

Eε uδη Eε (u) + π

|δi | + η

(25)

i∈{0,...,N}

and u − uδ

η L2 (D )

= oη (1),

oη (1) − −−→ 0. η→0

The proof of Lemma 7 is postponed to Appendix C. Proof. We prove that for η > 0 small, we have mε (p, q, d) I0 (d, D) + π æ (d, d), (p, q) + η. We denote u0 ∈ Ed s.t. E(u0 ) = I0 (d, D). Then abdegi (u0 ) = di . Using Lemma 7 with δ = (p, q) − (d, d), there is uη s.t. uη ∈ J(p,q) and Eε (uη ) Eε (u0 ) + π æ (d, d), (p, q) + η = I0 (d, D) + π æ (d, d), (p, q) + η.

(26)


1063

d Furthermore, by (26), u0 − uη L2 (D) = oη (1). For η small, by Proposition 1, we have u0 ∈ Jp,q which proves the lemma. 2

5. A family with bounded energy converges In this section we discuss: d (ε ↓ 0) with 1. the asymptotic behaviour of a sequence of solutions of (8), (9), (uεn )n ⊂ Jp,q n bounded energy, i.e., Eεn (uεn ) Λ, d , 2. the asymptotic behaviour of a minimizing sequence of Eε in Jp,q 3. a fundamental lemma. d with u a solution of (8), (9), s.t. for Λ > 0, we have Proposition 3. Let εn ↓ 0, (uεn )n ⊂ Jp,q εn

Eεn (uεn ) Λ. Then, denoting hεn the unique solution of (15) with u = uεn , we have hεn h0

in H 1 (D),

(27)

in H 1 (D),

(28)

where h0 is the unique solution of (18). Up to subsequence, it holds uεn u0

where u0 ∈ Ed is the unique solution of (17) up to multiplication by an S 1 -constant. Proof. Using the energy bound on uεn and a Poincaré type inequality, we have, up to subsequence, hεn h

in H 1 .

In order to establish (27), it suffices to prove that h = h0 . The set H := {h ∈ H 1 (D, R); ∂τ h ≡ 0 on ∂D and h|∂Ω ≡ 1} is closed convex in H 1 (D, R). Since (hεn )n ⊂ H, we find that h ∈ H. 2 (D, R2 ). Therefore Since Eεn (uεn ) is bounded, Lemma 1 implies that uεn is bounded in Cloc 1 1 there is some u ∈ Cloc (D, C) s.t., up to subsequence, uεn → u in Cloc (D, R2 ), L4 (D, R2 ) and weakly in H 1 (D, R2 ). Using the strong convergence in L4 and the energy bound on uεn , we find that u ∈ H 1 (D, S 1 ). It follows that ∂1 u × ∂2 u = 0 in D. On the other hand, 0 . hεn = 2∂1 uεn × ∂2 uεn → 0 in Cloc

Therefore, h is a harmonic function in D. In order to show that h = h0 , it suffices to check that ∂ν h dσ = 2πdi . ∂ωi

1064


To this end, we note that, since uεn × (∂1 Vi ∂2 uεn − ∂2 Vi ∂1 uεn ) = ∇Vi · ∇hεn , we have 2π abdegi (uεn ) = ∇Vi · ∇hεn dx − ∇V · ∇h dx = ∂ν h dσ. − − − → i n→∞ D

D

∂ D \∂ωi

Noting that, by Proposition 1, abdegi (uεn ) − −−−→ abdegi (u) = deg∂ωi (u), n→∞ abdegi (uεn ) − −−−→ di n→∞ and that 0 = D h dx = ∂ D ∂ν h dσ , we obtain

∂ν h dσ = ∂ D \∂ωi

∂ν h dσ = 2πdi = 2π deg∂ωi (u).

∂ωi

In the first integral, ν is the outward normal to D, in the second, ν is the outward normal to ωi . This proves (27). We next turn to (28). Let u0 be s.t., up to subsequence, uεn u0 in H 1 (D). Since |uεn | 1, we find that uεn × ∇uεn u0 × ∇u0

in L2 (D).

In view of (15) and (27), we have u0 × ∇u0 = ∇ ⊥ h0 . Therefore, E(u0 ) = E(h0 ) = I0 (d, D). Proposition 1 implies that u0 ∈ Ed . Then u0 is the unique, up to multiplication by an S 1 -constant, minimizer of E in Ed . 2 d be a minimizing Proposition 4. Let (p, q, d) ∈ ZN × Z × ZN . For ε > 0, let (uεn )n0 ⊂ Jp,q d . Then there is ε (p, q, d) > 0 s.t. for 0 < ε < ε , up to subsequence, sequence of Eε in Jp,q 4 4 d 1 un u in H with u which minimizes Eε in Jdeg(u, . D) d be minimizing sequences of E in J . Up to subsequence, Proof. For ε > 0, let (uεn )n ⊂ Jp,q ε using Proposition 1,

uεn uε

in H 1

d with uε ∈ Jdeg(u ε ,D ) .

Using Lemma 3, we see that {deg(uε , D), ε > 0} ⊂ ZN × Z is a finite set. Applying Lemma 6, it follows that Eε (uε ) is bounded. Therefore, with Proposition 1, there is ε4 > 0 s.t. |abdegi (uε ) − di | < 13 for all i ∈ NN and 0 < ε < ε4 . We argue by contradiction and we assume that there is ε < ε4 s.t. Eε (uε ) = mε (deg(uε , D), d ε d) + 2η, η > 0. Let u ∈ Jdeg(u ε ,D ) be s.t. Eε (u) mε (deg(u , D), d) + η. ε Using Lemma 7 with δ = (p, q) − deg(u , D), there is v ∈ Jp,q s.t. Eε (v) < Eε (u) + π æ (p, q), deg uε , D + η.


1065

Furthermore, by the construction, u − vL2 can be taken arbitrary small, so that we may further d . To summarise we have assume v ∈ Jp,q mε (p, q, d) = lim inf Eε uεn n Eε uε + π æ (p, q), deg uε , D = mε deg uε , D , d + 2η + π æ (p, q), deg uε , D Eε (u) + π æ (p, q), deg uε , D + η > Eε (v) mε (p, q, d). This contradiction completes the proof.

2

The main tool requires the following lemma. Lemma 8. Let (p, q, d) ∈ ZN × Z × ZN and Λ > 0. There is ε5 (p, q, d, Λ) > 0 s.t. for ε < ε5 d , a solution of (8) and (9) with E (u) Λ, if d > 0 (respectively d > 0), then there and u ∈ Jp,q ε i 0 is x ∈ ∂Ω (respectively x i ∈ ∂ωi ) s.t. u × ∂τ u(x 0 ) > 0 (respectively u × ∂τ u(x i ) > 0). Proof. We prove existence of x 0 ∈ ∂Ω under appropriate assumptions. Existence of x i is similar. d solutions of (8) and (9) We argue by contradiction. Assume that there are εn ↓ 0, (un ) ⊂ Jp,q with Eεn (un ) Λ s.t. un × ∂τ un 0 on ∂Ω. 1 Since q = 2π ∂Ω un × ∂τ un , we have q 0. Up to subsequence, by Proposition 3, we can assume that un → u0 a.e. with u0 the unique solution up to S 1 of (17). Let x0 ∈ ∂Ω and let γ : ∂Ω → [0, H1 (∂Ω)[ = I be s.t. γ −1 is the direct arc-length parametrization of ∂Ω with the origin at x0 . We denote θn : I → R the smooth functions s.t. un (x) = eıθn [γ (x)] ∀x ∈ ∂Ω, 0 θn (0) < 2π. Then, for all n, θn is nonincreasing and θn ∈ [θn (0) + 2πq, θn (0)] ⊂ [2πq, 2π]. Using Helly’s selection theorem, up to subsequence, we can assume that θn → θ everywhere on I with θ nonincreasing. Denote Ξ the set of discontinuity points of θ . Since θ is nonincreasing, Ξ is a countable set. Using the monotonicity of θ , we can consider the following decomposition θ = θc + θδ,

with θ c and θ δ are nonincreasing functions.

θ c is the continuous part of θ and θ δ is the jump function. The set of discontinuity points of θ δ is Ξ . For t ∈ / Ξ, θ (s+) − θ (s−) 1(−∞,s] (u). θ δ (t) = 0<s 0. Without loss of generality, we may assume that u(x 0 ) = 1. (See Fig. 1.) Then there is Υ ⊂ D, a compact neighbourhood of x 0 , simply connected and with nonempty interior, s.t.: • • • •

γ := ∂Ω ∩ ∂Υ is connected with nonempty interior; x 0 is an interior point of γ ; |∇h| > 0, ρ > 0, h 1 in Υ ; ∂ν h > 0 on γ (ν the outward normal of Ω).

It follows that, in Υ , θ is globally defined (we take the determination of θ which vanishes at x 0 ). Using the inverse function theorem, we may assume, by further restricting Υ , that there are some 0 < η, δ < 1 s.t. Υ = x ∈ D s.t. dist x, x 0 < η, 1 − δ h(x) 1, −2δ θ (x) 2δ . We may further assume that, by replacing δ by smaller value if necessary and denoting ◦

Dδ :=Υ , we have

1068

(i)


Θ := (θ, h)|Dδ : Dδ → (−2δ, 2δ) × (1 − δ, 1) is a C 1 -diffeomorphism, x → (θ, h),

(ii) ∂Dδ \ ({h = 1} ∪ {h = 1 − δ}) = ∂Dδ ∩ ({θ = −2δ} ∪ {θ = 2δ}), (iii) Dδ is a Lipschitz domain. We consider δ0 > 0 s.t. for δ < δ0 , |Dδ |1/2
0 smaller than δ) s.t.

1 in D \ Dδ , ψt (x) = e−ıθ −(1−tϕ(θ)) (30) on ∂Ω ∩ ∂Dδ , e−ıθ (1−tϕ(θ))−1 with 0 ϕ 1 a smooth, even and 2π -periodic function satisfying ϕ|(−δ/2,δ/2) ≡ 1 and ϕ|[−π,π[\(−δ,δ) ≡ 0. It is clear that ψt|∂ D ∈ C ∞ (∂D) and deg∂ωi (ψt ) = 0 for all i ∈ NN .

(31)

Expanding in Fourier series, we have e−ıθ − (1 − tϕ(θ )) (t) + t bk (t)e−(k+1)ıθ . = 1 − tb −1 e−ıθ (1 − tϕ(θ )) − 1

(32)

k=−1

−ıθ

−(1−tϕ(θ)) Noting that the real part of ee−ıθ (1−tϕ(θ))−1 is even and the imaginary part is odd, we obtain that bk (t) ∈ R for all k, t. The following lemma is proven in Appendix B.


1069

Lemma 9. We denote, for eıθ ∈ S 1 , e−ıθ − (1 − tϕ(θ )) Ψt eıθ = −ıθ e (1 − tϕ(θ )) − 1

e−ıθ − (1 − t) and Ft eıθ = −ıθ . e (1 − t) − 1

Then: (1) |Ψt − Ft | Cδ t on S 1 ; z−(1−t) = (1 − tc−1 ) + t k=−1 ck (t)zk+1 , with (2) Ft (z) = z(1−t)−1 ⎧ ⎨ (t − 2)(1 − t)k ck = 0 ⎩ 1

if k 0, if k −2, if k = −1;

(3) |bk (t) − ck (t)| C(n, δ)(1 + |k|)−n , ∀n > 0, with C(n, δ) independent of t sufficiently small. It is easy to see using Lemma 9 that, for t sufficiently small, degS 1 (Ψt ) = degS 1 (Ft ) = −1. Using the previous equality and the fact that ∂τ θ > 0 on γ , we find that deg∂Ω (ψt ) = −1.

(33)

It will be convenient to use h and θ as a shorthand for h(x) and θ (x). With these notations, we will look for ψt of the form ψt (x) = ψ˜t (h, θ ) ⎧ −(k+1)ıθ ⎪ ⎨ (1 − tf−1 (h)b−1 (t)) + t k=−1 bk (t)fk (h)e 2δ−θ ˜ = θ−δ δ + ψt (h, δ) δ ⎪ ⎩ θ+δ − + ψ˜t (h, −δ) 2δ+θ δ

δ

in Dδ ,

in Dδ+ , in Dδ− .

(34)

We impose fk (1 − δ) = 0 and fk (1) = 1 for k ∈ Z. Our aim is to show that for t > 0 small and appropriate fk ’s, the function ψt defined by (34) satisfies (30) and Lε ψt eıθ , Dδ < π. Here, Lε is the functional defined in Lemma 2, so that Eε ρψt eıθ , Dδ = Eε (u, Dδ ) + Lε ψt eıθ , Dδ . Then, considering ψt =

ψt ψt 2 |ψ t|

if |ψt | 2, if |ψt | > 2

(35)

1070


and set u˜ =

ρwt = ψt u u

in Dδ , in D \ Dδ .

In view of (35), it is straightforward that u˜ satisfies the conclusion of Lemma 5. 6.3. Upper bound for Lε (·, Dδ ). An auxiliary problem If we let w: ˜ [1 − δ, 1] × [−2δ, 2δ] be s.t. w(h(x), ˜ θ (x)) := w(x), then we have |∇w|2 =

2 ∂h w(h, |∂i w|2 = ˜ θ )∂i h + ∂θ w(h, ˜ θ )∂i θ i

i

2 2 = ρ 4 ∂h w(h, ˜ θ ) + ∂θ w(h, ˜ θ ) |∇θ |2 . Therefore, 1 Lε (w, Dδ ) = 2

2 2 2 4 ρ ∂h w(h, ˜ θ ) + ∂θ w(h, ˜ θ ) − w(h, ˜ θ ) ρ 2 |∇θ |2

Dδ

2 2 1 4 ˜ θ) dx + 2 ρ 1 − w(h, 2ε 2 2 2 2 1 ∂h w(h, ˜ θ ) + λeıθ − w(h, ˜ θ ) + ∂θ w(h, ˜ θ ) − w(h, ˜ θ ) ρ 2 |∇θ |2 dx 2 Dδ

=: Mλ (w, Dδ ), provided that |w| 2 in Dδ and λ

(36) 9 . 2ε 2 infDδ |∇θ|2

In order to simplify formulas, we will write, in what follows, the second integral in (36) as 1 2

2 |∂h w| ˜ 2 + |∂θ w| ˜ 2 − |w| ˜ 2 + λeıθ − w˜ ρ 2 |∇θ |2 dx.

Dδ

The same simplified notation will be implicitly used for similar integrals. w Claim. If we replace w by w := |w| min(|w|, 2), then Mλ does not increase. Furthermore replacing w by w does not affect the Dirichlet condition of (30). Therefore, by replacing w by w if necessary, we may assume |w| 2.

We next state a lemma which allows us to give a new form of Mλ .


1071

Lemma 10. Let f ∈ C 1 (R, R). Then, for k ∈ Z, we have

f (h) cos(kθ )ρ |∇θ | dx = 2

2

Dδ

2δ

1

1−δ f (s) ds 2 sin(kδ) 1 1−δ f (s) ds k

if k = 0, if k = 0,

1 f (h)ρ 2 |∇θ |2 dx = δ

Dδ±

f (s) ds.

1−δ

Proof. This result is easily obtained by noting that the jacobian of the change of variable x → (θ (x), h(x)) is exactly ρ 2 |∇θ |2 . 2 For w = wt = ψt eıθ where ψt of the form given by (34), we have

1 Mλ (w, Dδ ) = 2

2 |∂h w| ˜ 2 + |∂θ w| ˜ 2 − |w| ˜ 2 + λeıθ − w˜ ρ 2 |∇θ |2 dx.

Dδ

We next rewrite Mλ (wt , Dδ ). Recalling that for a sequence {ak } ⊂ R, we have 2 ıkθ ak e = ak2 + 2 ak al cos (k − l)θ . k∈Z

k∈Z

k,l∈Z k>l

Then we obtain Mλ w, Dδ =

Dδ

t 2 2 2 bk fk + fk2 k 2 + λ − 1 − t bk fk (k + 1) cos (k + 1)θ 2

− t2

k=−1

+ t2

k=−1

k∈Z

b−1 bk f−1 fk − f−1 fk (k − λ + 1) cos (k + 1)θ bk bl fk fl + (kl + λ − 1)fk fl cos (k − l)θ ρ 2 |∇θ |2 .

k,l=−1 k−l>0

Using Lemma 10 and (37), we have 1 2 2 Mλ w, Dδ = δt bk φk (fk ) − 2t bk sin (k + 1)δ fk k=−1

k∈Z

− 2t 2

k=−1

b−1 bk

sin[(k + 1)δ] k+1

1−δ

1 1−δ

f−1 fk − (k − λ + 1)f−1 fk

(37)

1072


+ 2t

2

k,l=−1 k−l>0

sin[(k − l)δ] bk bl k−l

= Rλ (w) − 2t

1

(38)

1−δ

bk sin (k + 1)δ

k=−1

fk fl + (kl + λ − 1)fk fl

1 (39)

fk

1−δ

with Rλ (w) = δt

2

bk2 φk (fk ) − 2t 2

k=−1

k∈Z

+ 2t 2

bk bl

k,l=−1 k−l>0

sin[(k + 1)δ] b−1 bk k+1

sin[(k − l)δ] k−l

1

1

f−1 fk − (k − λ + 1)f−1 fk

1−δ

fk fl + (kl + λ − 1)fk fl ,

1−δ

1 φk (f ) =

2 f + αk2 f 2

1−δ

and αk =

k 2 + λ − 1.

We next establish a similar identity for Mλ (wt , Dδ± ). Using (34), we have

Mλ wt , Dδ± =

=

1 2 1 2

2 2 2 ∂h w(h, ˜ θ ) + ∂θ w(h, ˜ θ ) − |w|2 + λeıθ − w ρ 2 |∇θ |2

Dδ±

Dδ±

∓ 2δ =

1 2δ 2

+t

2 2 ∂h ψ˜t (h, ±δ)2 2δ ∓ θ + δ −2 1 + λ(2δ ∓ θ )2 ψ˜t (h, ±δ) − 1 δ

−1

˜ Im ψt (h, ±δ) ρ 2 |∇θ |2

∂h ψ˜t (h, ±δ)2 (2δ ∓ θ )2 + 1 + λ(2δ ∓ θ )2 ψ˜t (h, ±δ) − 12 ρ 2 |∇θ |2

Dδ±

k=−1

bk (t) sin (k + 1)δ

1 fk ,

1−δ

where Im ψ denotes the imaginary part of ψ . To obtain (40), we used the identity ıθ 2 ∂θ ψe = |∂θ ψ|2 + |ψ|2 + 2ψ × ∂θ ψ.

(40)


1073

6.4. Choice of w = ψt eıθ We take fk (h) =

eαk (h−1) e−αk (h−1) + . 1 − e−2αk δ 1 − e2αk δ

(41)

With this choice, by direct computations we have φk (fk ) = αk 1 + 1 1−δ

2 e2αk δ − 1

,

2 1 1− α δ fk = αk e k +1

(42)

(43)

and for k, l ∈ Z s.t. k = ±l, 1 fk fl = 1−δ

1 − e−2(αk +αl )δ (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) −

1 αk αl

1

fk fl =

1−δ

1 − e−2(αk −αl )δ , (αk − αl )(1 − e−2αk δ )(e2αl δ − 1)

(44)

1 − e−2(αk +αl )δ (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) +

1 − e−2(αk −αl )δ . (αk − αl )(1 − e−2αk δ )(e2αl δ − 1)

(45)

Using (39)–(45), we may obtain the following estimate, whose proof is postpone to Appendix B. Lemma 11. We have Mλ (wt , Dδ ) δ − 2δt + 4t 2

ck cl

k>l>0

sin[(k − l)δ] kl + o(t). k−l k+l

(46)

6.5. End of the proof of Lemma 5 We denote

S(δ, t) :=

ck cl

k>l>0

Setting n = k − l and noting that

(n+l)l n+2l

=

l 2

+

sin[(k − l)δ] kl . k−l k+l

ln 2(n+2l) ,

we have

(47)

1074


2 sin(nδ) l . S(δ, t) = (1 − t)n l(1 − t)2l + (1 − t)n+2l sin(nδ) 2 n n + 2l (t − 2) n>0

l>0

n,l>0

Here, we have used the explicit formulae for the ck ’s, given by Lemma 9. Using Appendix A (see Appendix A.1) we find that for 0 < t < δ, we have 1 − t − cos δ cos δ (1 − t)2 S(δ, t) = arctan + arctan sin δ sin δ 2t 2 +

(1 − t + cos δ)(2 − t) + O(1). 8t sin δ

(48)

We note that 1 − t − cos δ 1 − cos δ t sin δ arctan = arctan − + O t2 sin δ sin δ 2(1 − cos δ) δ t sin δ = − + O t2 2 2(1 − cos δ)

(49)

and π cos δ = − δ. arctan sin δ 2

(50)

From (48)–(50) we infer S(δ, t)

1 sin δ 1 (1 − t + cos δ)(2 − t) − + O(1) (π − δ) + t 8 sin δ 4(1 − cos δ) 4t 2

(51)

with (1 − t + cos δ)(2 − t) sin δ (1 + cos δ) sin δ − < − = 0. 8 sin δ 4(1 − cos δ) 4 sin δ 4(1 − cos δ)

(52)

Mλ (w, Dδ ) δ − 2δt + 4t 2 S(t, δ) + o(t).

(53)

4t 2 S(t, δ) π − δ + o(t).

(54)

From (46),

Using (51) and (52),

Finally, we have combining (53) and (54), Mλ (w, Dδ ) π − 2δt + o(t) < π

for t small.

We conclude that for t sufficiently small, Ldε (w t , Dδ ) < π .

(55)


1075

6.6. Conclusion t |,2) , satisfies the desired properties i.e.: u˜ := ψu, with ψ = ψt min(|ψ |ψt |

˜ < Eε (u) + π (by (36) and (55)); • Eε (u) d • u˜ ∈ Jp,q−1 (by (29), (31) and (33)). 7. Proof of Theorem 2 The energy estimate is obtained from Lemmas 4 and 6. The proof is made by induction on K = æ (d, d), (p, q) = |d1 − p1 | + · · · + |dN − pN | + |d − q| 0. We call (p, q, d) a good configuration of degrees if (p, q, d) ∈ ZN × Z × (N∗ )N ,

pi di and q

di =: d.

i

We prove Theorem 2 when K = 0. Let (p, q, d) be a configuration s.t. æ(d, d) = 0 (⇔ p = d and q = d). d . For ε < ε , up to subsequence, For ε > 0, let (uεn )n be a minimizing sequence of Eε in Jd,d 4 ε ε 1 using Proposition 4, un → u weakly in H and strongly in L4 and uε is a (global) minimizer d of Eε in Jdeg(u, D) . Applying Lemmas 3 and 4, for ε < ε2 (d) ε4 (here, ε2 is s.t. the oε (1) of Lemma 4 is lower than π2 ), I0 (d, D) Eε uε + π æ deg uε , D , (d, d) π I0 (d, D) − + 2π æ deg uε , D , (d, d) . 2 d . It follows, æ(deg(uε , D), (d, d)) 14 which implies uε ∈ Jd,d Assuming Theorem 2 true for all configurations (p, q, d) s.t.

0 æ (p, q), (d, d) K. ˜ q, We prove it for all good configurations (p, ˜ d) s.t.

˜ q), ˜ (d, d) = K + 1. æ (p, ˜ q), ˜ q, ˜ (d, d)) = K + 1. Then there is some Let (p, ˜ d) be a good configuration s.t. æ((p, ˜ q) ˜ q) ˜ + ei , (d, d)) = K. i ∈ {0, . . . , N} s.t. ((p, ˜ + ei , d) be a good configuration and æ((p, Without loss of generality, we may assume that i = 0. Set p = p˜ and q˜ + 1 = q. By induction hypothesis, Theorem 2 holds for (p, q, d). d . Let ε < ε2 (p, q, d) and (uε )ε2 >ε>0 be a family of (global) minimizers of (Eε )ε2 >ε>0 in Jp,q 0 0 By Lemma 8, for ε < ε5 (p, q, d, Λ), there is xε ∈ ∂Ω s.t. (uε × ∂τ uε )(xε ) > 0.

1076


The third assertion in Proposition 1 and the energy bound give the existence of 0 < ε2 (p, q, d, Λ) < ε5 (p, q, d, Λ) s.t. for 0 < ε < ε2 , 1 1 . abdegi (uε ) ∈ di − , di + 3 3 d Using Lemma 5, for ε < ε2 , we have the existence of u˜ε ∈ Jp,q−1 s.t.

mε (p, q, d) + π = Eε (uε ) + π > Eε (u˜ε ) mε (p, q − 1, d). Using (24), we have, mε (p, q − 1, d) < I0 (d, D) + (K + 1)π. d Let (uεn )n be a minimizing sequence of Eε in Jp,q−1 . Using Proposition 4, for ε < ε4 , up to subsequence, uεn → uε weakly in H 1 and strongly in L2 to u which is a global minimizer of Eε d in Jdeg(u ε ,D ) .

d It suffices to show that, for ε sufficiently small, uε ∈ Jp,q−1 . The argument is similar to the 2 ε one used for K = 0. By strong L convergence, abdegi (u ) ∈ [di − 1/3, di + 1/3]. Using Lemmas 3–5, we have

I0 (d, D) + (K + 1)π > mε (p, q − 1, d) = lim inf Eε uεn by the definition of uεn n Eε uε + π æ (p, q − 1), deg uε , D (Lemma 3) ε > I0 (d, D) + π æ (p, q − 1), deg u , D π (Lemma 4) + æ (d, d), deg uε , D − 2 1 I0 (d, D) + π æ (p, q − 1), (d, d) − 2 (by the triangle inequality) 1 π. I0 (d, D) + K + 2 It follows that

æ (p, q − 1), deg uε , D

+ æ (d, d), deg uε , D = K + 1.

Since æ((p, q − 1), (d, d)) = K + 1, we must have pi deg∂ωi uε di

and q − 1 deg∂Ω uε d.

Let H := (p , q ) s.t. pi pi di , q − 1 q di ⊂ ZN × Z.

(56)


Using (56), 1 card(H ) < ∞. Then, we can define 0 < ε2 min ε2 , min ε (p , q , d) 5 (p ,q )∈H

1077

(ε5 defined in Lemma 8).

Consequently, for ε < ε2 , (p , q ) ∈ H and u ∈ Jpd ,q a solution of (8) and (9), by Lemma 8, on each connected component of ∂D, there is some x s.t. u × ∂τ u(x) > 0. Only two cases are possible: Case 1. For ε < ε2 (p, q − 1, d), deg∂ωi (uε ) = pi for all i ∈ NN and deg∂Ω (uε ) = q − 1. d In this case we have the result since for 0 < ε < ε2 (p, q − 1, d), uε ∈ Jp,q−1 .

Case 2. There is 0 < ε < ε2 s.t. deg(uε , D) ∈ H and deg(uε , D) = (p, q − 1). We denote αi = deg∂Ω uε − pi ,

α0 = deg∂Ω uε − (q − 1).

Then, αi ∈ N, i ∈ {0, . . . , N}. Let J = {i ∈ {0, . . . , N} s.t. αi > 0} = ∅. We enumerate the elements of J in (lm )m∈{1,...,|J |} s.t. lm < lm+1 , m < |J |. d Using æ(deg(uε , D), (p, q − 1)) times Lemma 5, we obtain the existence of v|J | ∈ Jp,q−1 s.t.

mε (p, q − 1, d) Eε (v|J | ) < Eε uε + π æ deg uε , D , (p, q − 1) .

(57)

Indeed, for 0 m |J | we construct inductively vm ∈ J s.t. ⎧ v0 = uε , ⎪ ⎪ ⎪ ⎨ abdeg (v ) − d < 1 , i i m 3 ⎪ ⎪ for m < |J |, vm+1 ∈ Jdeg(vm ,D)−αlm elm , ⎪ ⎩ for m < |J |, Eε (vm+1 ) < Eε (vm ) + αlm π. The map vm+1 is obtained from vm by applying αlm times Lemma 5 as follows. For j ∈ j {1, . . . , αlm }, we denote vm the global minimizer of Eε in Jdeg(v j −1 ,D)−e . For j = 0, we set lm

m

0 =v . vm m j From Lemma 5 and since vm is a global minimizer of Eε in J d

j −1

deg(vm ,D )−elm

,

j j −1 Eε vm < Eε vm + π.

(58)

α

From (58), for m < |J |, by noting that vm+1 = vmlm , we obtain Eε (vm+1 ) < Eε (vm ) + αlm π.

(59)

d For m = |J |, we have v|J | ∈ Jp,q−1 and

Eε (v|J | ) = mε (p, q − 1, d) < Eε uε + π æ deg uε , D , (p, q − 1) .

(60)

1078


On the other hand, Lemma 3 yields mε (p, q − 1, d) = lim inf Eε uεn Eε uε + π æ deg uε , D , (p, q − 1) . n

(61)

Estimate (61) contradicts (60). This contradiction achieves the proof. Acknowledgment The author would like to express his gratitude to Professor Petru Mironescu for suggesting him to study the problem treated in this paper and for his useful remarks. Appendix A. Results used in the proof of Lemma 5 A.1. Power series expansions For X ∈ C, |X| < 1, we have |X|k k

k1

= − ln 1 − |X| ,

(A.1)

1 , 1−X

(A.2)

X , (1 − X)2

(A.3)

Xk =

k0

kX k =

k1

X sin δ , (A.4) 1 − 2X cos δ + X 2 k>0 sin(kδ) cos δ X − cos δ X k = arctan + arctan , (A.5) k sin δ sin δ k>0 1 l X + cos δ X − cos δ n+2l X + Cst(δ). (A.6) − sin(nδ) = arctan n + 2l sin δ 4(1 − X 2 ) sin δ 4 sin2 δ sin(kδ)X k =

n,l>0

Proof. The first four identities are classical. We sketch the argument that leads to (A.5) and (A.6). Identities (A.5) follows from (A.4) by integration. We next prove (A.6). Let f (X) =

n,l>0

sin(nδ)

l X n+2l . n + 2l

On the one hand, by (A.3), (A.4), f (X) =

1 X 2 sin δ . sin(nδ)X n lX 2l = X (1 − X 2 )2 (1 − 2X cos δ + X 2 ) n>0

l>0


1079

On the other hand X + cos δ X 2 sin δ d 1 X − cos δ = − . arctan dX 4 sin δ(1 − X 2 ) 4 sin2 δ sin δ (1 − X 2 )2 (1 − 2X cos δ + X 2 )

2

A.2. Estimates for fk and αk Recall that we defined, in Section 6, fk and αk by eαk (h−1) e−αk (h−1) + , −2α δ 1−e k 1 − e2αk δ αk = k 2 + λ − 1.

fk (h) =

In this part, we prove the following inequalities. 1 , αk = |k| + O |k| + 1

(A.7)

fk (h) − e−|k|(1−h) C , with C independent of k ∈ Z∗ , h ∈ (1 − δ, 1), k2 f (h) − |k|e−|k|(1−h) C , with C independent of k ∈ Z∗ , h ∈ (1 − δ, 1). k |k|

(A.8) (A.9)

Proof. The first assertion is obtained using a Taylor expansion. Let gh (u) = eu(h−1) , we have fk (h) − e−|k|(1−h) gh (αk ) − gh |k| + C sup g (u)αk − |k| + C h 2 k k2 (|k|,αk )

C 1 1 C + 2 2. ek 2k k k

The proof of (A.9) is similar, one uses g˜ h (u) = ueu(h−1) instead of gh .

2

A.3. Further estimates on fk and αk We have 1 0

2 fk − αk2 fk2

1−δ

1

2 fk − k 2 fk2

1−δ

C , |k| + 1

with C independent of k ∈ Z, (A.10)

1 C fk fl , max(|k|, |l|) 1−δ

with C independent of k, l ∈ Z, s.t. |k| = |l|,

(A.11)

1080


1 fk fl C min |k|, |l| + 1 ,

with C independent of k, l ∈ Z, s.t. |k| = |l|. (A.12)

1−δ

Proof. Actually (A.11), (A.12) still hold when |k| = |l|, but this will not used in the proof of Lemma 5 and requires a separate argument. Since αk |k|, 1

2 fk − αk2 fk2

1−δ

1

2 fk − k 2 fk2 .

1−δ

By direct computations, 1 0

2 fk − αk2 fk2 =

1−δ

1

2 fk − k 2 fk2 =

1−δ

1 fk2 1−δ

4δαk2 −2α (1 − e k δ )(e2αk δ 1

1−δ

− 1)

C(δ, n) , kn

2 fk − αk2 fk2 + (λ − 1)

∀n ∈ N∗ ,

1 fk2 ,

1−δ

1 1 1 1 1 +O =O . = − 2αk 1 − e−2αk δ 1 − e2αk δ |k| + 1 |k| + 1

Which proves (A.10). For |k| = |l|, we have 1 1 − e−2(αk +αl )δ 1 − e−2(αk −αl )δ − fk fl = (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) (αk − αl )(1 − e−2αk δ )(e2αl δ − 1) 1−δ

C 1 − e−2(αk −αl )δ . + −2α δ 2α δ max(|k|, |l|) (αk − αl )(1 − e k )(e l − 1)

We assume that |k| > |l| and we consider the two following cases: αl < αk 2αl and αk > 2αl . −2xδ Noting that 1−ex is bounded for x ∈ R∗+ , we have 1 − e−2(αk −αl )δ C C (α − α )(1 − e−2αk δ )(e2αl δ − 1) e2αl δ max(|k|, |l|) if αl < αk 2αl , k l C 1 − e−2(αk −αl )δ C (α − α )(1 − e−2αk δ )(e2αl δ − 1) α − α max(|k|, |l|) if αk > 2αl . k l k l This proves (A.11).


1081

For |k| = |l|, 1 αk αl

1

fk fl =

1−δ

1 − e−2(αk −αl )δ 1 − e−2(αk +αl )δ + . (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) (αk − αl )(1 − e−2αk δ )(e2αl δ − 1)

It is clear that, αk αl (1 − e−2(αk +αl )δ ) αk αl C C min |k|, |l| + 1 . −2α δ −2α δ k l αk + αl (αk + αl )(1 − e )(1 − e )

(A.13)

As in the proof of (A.11), we have αk αl (1 − e−2(αk −αl )δ ) Cαk αl (α − α )(1 − e−2αk δ )(e2αl δ − 1) max(|k|, |l|) C min |k|, |l| + 1 . k l Inequalities (A.12) follows from (A.13) and (A.14).

(A.14)

2

A.4. Two fundamental estimates In this part, we let k > l 0 and prove the following: Xk,l :=

1 2kl (αk αl + kl + λ − 1)(1 − e−2(αk +αl )δ ) + O , = k+l l+1 (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ )

(A.15)

(αk αl + kl + λ − 1)(1 − e−2(αk −αl )δ ) Ce−δl . (αk − αl )(1 − e−2αk δ )(e2αl δ − 1)

(A.16)

Yk,l :=

The computations are direct: Xk,l −

2kl 1 2kl 2kl = + O − k + l (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) k + l l+1 1 k + l − (αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) + O = 2kl l+1 (k + l)(αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) 2 −lδ/2 1 ) O(k + k le + O = l+1 (k + l)(αk + αl )(1 − e−2αk δ )(1 − e−2αl δ ) 1 . =O l+1

We now turn to (A.16). If αk 2αl (or equivalently, if αk − αl

αk 2 ),

then

(αk αl + kl + λ − 1)(1 − e−2(αk −αl )δ ) kl C e−2αl δ Ce−δl . −2α δ 2α δ k l αk (αk − αl )(1 − e )(e − 1)

1082


If αk < 2αl , then (αk αl + kl + λ − 1)(1 − e−2(αk −αl )δ ) Cl 2 e−2αl δ Ce−δl . (αk − αl )(1 − e−2αk δ )(e2αl δ − 1) Appendix B. Proof of Proposition 1 and of Lemma 9 B.1. Proof of Proposition 1 1 1 The proof of (1) is direct by noticing that if u ∈ H (D, S ), then ∂1 u and ∂2 u are pointwise proportional and deg∂Ω (u) = i deg∂ωi (u),

1 k (−1) (u × ∂k u)∂3−k Vi abdegi (u, D) = 2π k=1,2

1 = 2π

D

Vi u × ∂τ u dτ = deg∂Ω (u) −

deg∂ωj (u) = deg∂ωi (u).

j =i

∂D

Proof of (2). Since Vi is locally constant on ∂D, integrating by parts,

v × (∂1 u∂2 Vi − ∂2 u∂1 Vi ) dx = D

u × (∂1 v∂2 Vi − ∂2 v∂1 Vi ) dx. D

Then 2π abdegi (u) − abdegi (v) = (u − v) × (∂1 Vi ∂2 u − ∂2 Vi ∂1 u) + (∂1 Vi ∂2 v − ∂2 Vi ∂1 v) dx D

√ 2u − vL2 (D) Vi C 1 (D) ∇uL2 (D) + ∇vL2 (D) 1/2 1/2 + Eε (v) 2u − vL2 (D) Vi C 1 (D) Eε (u) 4u − vL2 (D) Vi C 1 (D) Λ1/2 . We prove assertion (3) by showing that dist(abdegi (uε ), Z) = o(1). Using the first and the second assertion, we have dist abdegi (uε ), Z inf abdegi (uε ) − abdegi (v) v∈E0Λ

2 Vi C 1 (D) Λ1/2 inf uε − vL2 (D) π v∈E0Λ

(B.1)

where E0Λ := {u ∈ H 1 (D, S 1 ) s.t. 12 D |∇u|2 dx Λ} = ∅. Now, it suffices to show that infv∈E Λ uε − vL2 (D) → 0. We argue by contradiction and we 0 assume that there is an extraction (εn )n ↓ 0 and δ > 0 s.t. for all n, infv∈E Λ uεn − vL2 (D) > δ. 0


1083

We see that (uεn )n is bounded in H 1 . Then, up to subsequence, un converges to u ∈ H 1 (D, R2 ) weakly in H 1 and strongly in L4 . Since |uεn |2 − 1L2 (D) → 0, we have u ∈ H 1 (D, S 1 ) and by weakly convergence, ∇u2L2 (D) 2Λ. To conclude, we have u ∈ E0Λ et uεn − uL2 → 0, which is a contradiction.

B.2. Proof of Lemma 9 (1) We see easily that, with z = eıθ , we have Ψt (z) − Ft (z) (1 − ϕ(θ ))(1 − z2 ) A(θ, t) = ≡ . t [z(1 − t) − 1][z(1 − tϕ(θ )) − 1] B(θ, t)

(B.2)

The modulus of the RHS of (B.2) can be bounded by noting that • there is some m > 0 s.t. |B(θ, t)| m for each t and each θ s.t. |θ | > δ/2 mod 2π ; • there is some M > 0 s.t. |A(θ, t)| M for each t and each θ s.t. |θ | > δ/2 mod 2π ; • if |θ | δ/2 (modulo 2π ), then (Ψt − Ft )t −1 ≡ 0. (2) This assertion is a standard expansion. (3) With a classical result relating regularity of Ψt − Ft to the asymptotic behaviour of its Fourier coefficients, we have n

2n+1 π∂θ (Ψt − Ft )L∞ (S 1 ) bk (t) − ck (t) . t (1 + |k|)n Noting that, for ∂θ (Ψt − Ft )t −1 ≡ n

An (θ,t) Bn (θ,t)

• there is some mn > 0 s.t. |Bn (θ, t)| mn for each t and each θ s.t. |θ | > δ/2 mod 2π ; • there is some Mn > 0 s.t. |An (θ, t)| Mn for each t and each θ s.t. |θ | > δ/2 mod 2π ; • if |θ | δ/2 (modulo 2π ), then (Ψt − Ft )t −1 ≡ 0 we obtain the result. B.3. Proof of Lemma 11 The key argument to treat the energetic contribution of Dδ± is the following lemma. Lemma 12. 1. |ψ˜t (h, ±δ) − 1| = O(t); 2. |∂h ψ˜t (h, ±δ)| = O(t| ln t|). Proof. Using Lemma 9, (A.2) and (A.8), we have −1 ˜ −ı[(k+1)δ] t ψt (h, δ) − 1 −c−1 f−1 (h) + ck (t)fk (h)e k=−1

1084


−ı(k+1)δ bk − ck (t) fk (h)e + −(b−1 − c−1 )f−1 (h) + k=−1

−(1−h)−ıδ k C(δ) + 1 = O(1). (1 − t)e k0

We prove that |∂h ψ˜t (h, δ)| = O(t| ln t|). Using Lemma 9, (A.3) and (A.9), t −1 ∂h ψ˜t (h, δ) −c−1 f−1 + ck fk e−ı(k+1)δ k=−1

+ −(b−1 − c−1 )f−1 + (bk − ck )fk e−ı(k+1)δ k=−1

k 2 k (1 − t)e−ıδ−(1−h) + O | ln t| = O | ln t| .

2

k0

Using (39), (40) and Lemma 12, we have (with the notation of Section 6) that Mλ (wt , Dδ ) = Rλ (wt ) + o(t), where

Rλ (wt ) = δt

2

bk2 φk (fk ) − 2t 2

k=−1

k∈Z

+ 2t 2

k,l=−1 k−l>0

bk bl

sin[(k + 1)δ] b−1 bk k+1

sin[(k − l)δ] k−l

1

1

f−1 fk − (k − λ + 1)f−1 fk

1−δ

fk fl + (kl + λ − 1)fk fl .

1−δ

The proof of Lemma 12 is completed provided we establish the following estimate: Rλ (wt ) δ − 2δt + 4t 2

ck cl

k,l0 k−l>0

sin[(k − l)δ] kl + o(t). k−l k+l

(B.3)

The remaining part of this appendix is devoted to the proof of (B.3). We estimate the first term of Rλ : Using (42) and Lemma 9, we have (with C independent of t) 2 2 b φ (f ) − c φ (f ) k k k k k k C. k∈Z

With (42) and (A.7), we obtain

k∈Z

(B.4)


φk (fk ) = α 1 +

2 2αδ e −1

= |k| + O

1 |k| + 1

1085

when |k| → ∞.

(B.5)

From (A.1), (A.3) and (B.5), t2

ck2 φk (fk ) = t 2 φ−1 (f−1 ) + t 2 (t − 2)2

k∈Z

= t 2 (t − 2)2

(1 − t)2k φk (fk )

k0

k(1 − t)2k + o(t) = 1 − 2t + o(t).

(B.6)

k>0

We estimate the second term of Rλ : Using Lemma 9, (A.11) and (A.12), we have (with C independent of t) 1 sin[(k + 1)δ] f−1 fk − (k − λ + 1)f−1 fk C. (bk − ck ) k+1 k=−1

1−δ

Since b−1 (t) is bounded by a quantity independent of t, in the order to estimate the third term of the RHS of (38), we observe that there is C independent of t s.t. 1 (1 − t)k sin[(k + 1)δ] f−1 fk − (k − λ + 1)f−1 fk k+1 k0

C

k1

1−δ

(1 − t)k k

+ 1 = C | ln t| + 1 .

Finally, using Lemma 9, (44) and (45), we have 1 sin[(k + 1)δ] f b f − (k − λ + 1)f f k −1 k C | ln t| + 1 . −1 k k+1 k=−1

(B.7)

1−δ

We estimate the last term of Rλ : First, we consider the case k = −l > 0 (i.e., fk = fl ). Using (43), 0 fk 1 and (A.10), we have (with C independent of t) 1 2 2 2 sin 2kδ fk + −k + λ − 1 fk C = C(t). bk b−k 2k k>0

1−δ

It remains to estimate the last sum in Rλ , considered only over the indices k and l s.t. |k| = |l|. We start with

1086


sin[(k − l)δ] (bk bl − ck cl ) k−l

k,l=−1 k−l>0, k=−l

=

k,l=−1 k−l>0, k=−l

1

1


1−δ

sin[(k − l)δ] (bk − ck )(bl − cl ) + ck (bl − cl ) + cl (bk − ck ) k−l

fk fl + (kl + λ − 1)fk fl .

×

(B.8)

1−δ

By the assertion (3) of Lemma 9, the first sum of the RHS of (B.8) is easily bounded by a quantity independent of t. By (A.11), (A.12) and Lemma 9,

k,l=−1 k−l>0, k=−l

sin[(k − l)δ] ck (bl − cl ) k−l

C

k0, l=−1 k−l>0, k=−l

1


1−δ

(1 − t)k |bl − cl ||l| + C. k−l

On the other hand (putting n = k − l), k0, l=−1 k−l>0, k=−l

(1 − t)k |bl − cl ||l| k−l

(1 − t)k |bl − cl |l + k−l

k>l0

l0, n>0

k0, l−1

(1 − t)n |bl − cl |l + n

(1 − t)k |bl − cl ||l| k + |l|

k>0, l−1

(1 − t)k |bl − cl ||l| + O(1) k

= O | ln t| . Similarly, we may prove that

k,l=−1 k−l>0, k=−l

sin[(k − l)δ] cl (bk − ck ) k−l

1 1−δ

fk fl + (kl + λ − 1)fk fl = O | ln t| .


1087

We have thus proved that

k,l=−1 k−l>0, k=−l

sin[(k − l)δ] (bk bl − ck cl ) k−l

1

fk fl + (kl + λ − 1)fk fl = o t −1 .

1−δ

To finish the proof, it suffices to obtain k,l=−1 k−l>0, k=−l

=2

sin[(k − l)δ] ck cl k−l

ck cl

k,l0 k−l>0

1


1−δ

sin[(k − l)δ] kl + o t −1 . k−l k+l

Since cm = 0 for m < −1, it suffices to consider the case k > l 0. Under these hypotheses, we have by (44), (45), (A.15) and (A.16), k>l0

sin[(k − l)δ] ck cl k−l

=2

k>l0

ck cl

1


1−δ

sin[(k − l)δ] kl +O k−l k+l

ck cl | sin[(k − l)δ]| 1 . k−l l+1 k>l0

We conclude by noting that (1 − t)n (1 − t)2l sin[(k − l)δ] C 1 + C 1 + ln2 t . c c k l (k − l)(l + 1) n l n>0

k>l0

l>0

Appendix C. Proof of Lemma 7 Lemma 13. Let 0 < δ, η < 1, there is Mη,δ : D(0, 1) → C x → Mη,δ (x)

s.t.:

(C.1)

(i) degS 1 (Mη,δ ) = 1, (ii) 12 D(0,1) |∇Mη,δ |2 π + η, (iii) |Mη,δ | 2, (iv) if |θ | > δ mod 2π , then Mη,δ (eıθ ) = 1. Claim. Taking Mη,δ instead of Mη,δ , we obtain the same conclusions replacing the assertion (i) by degS 1 (Mη,δ ) = −1.

1088


Proof. As in Section 6, let ϕ ∈ C ∞ (R, R) be s.t. • 0 ϕ 1, • ϕ is even and 2π -periodic, • ϕ|(−δ/2,δ/2) ≡ 1 and ϕ|[−π,π[\(−δ,δ) ≡ 0. For 0 < t < δ, let Mt = M be the unique solution of ⎧ ⎨

eıθ − (1 − tϕ(θ )) , M eıθ = ıθ e (1 − tϕ(θ )) − 1 ⎩ M = 0 in D(0, 1). It follows easily that M satisfies (i), (ii) and (iv). We will prove that for t small (iii) holds. Using (32), we have eıθ − (1 − tϕ(θ )) = 1 − tb (t) + t bk (t)e(k+1)ıθ . −1 eıθ (1 − tϕ(θ )) − 1

(C.2)

k=−1

It is not difficult to see that M reıθ = 1 − tb−1 (t) + t bk (t)r |k+1| e(k+1)ıθ . k=−1

From (C.3), 1 2

2π

|∇M| = t 2

D(0,1)

2

1 dθ

0

= πt 2

dr 0

bk2 (k + 1)r 2|k+1|−2

k=−1

bk2 (k + 1) + πt 2

k0

= πt 2

|k + 1|bk2

k−2

ck2 (k + 1) + O t 2 (using Lemma 9)

k0

= π(2 − t)2 t 2

(1 − t)2k (k + 1) + O t 2

(using Lemma 9)

k0

= π + O t 2 (using (A.2) and (A.3)) π +η

for t small.

We finish the proof taking, for t small, Mη,δ = Mt .

2

Lemma 14. Let u ∈ J , i ∈ {0, . . . , N} and ε > 0. For all η > 0, there is u± η ∈ Jdeg(u,D )±ei

(C.3)


1089

Eε u± η Eε (u) + π + η

(C.4)

s.t.

and u − u± η

L2 (D )

= oη (1),

oη (1) − −−→ 0. η→0

(C.5)

Proof. We prove that for i = 0, there is u+ η ∈ Jdeg(u,D )+ei satisfying (C.4) and (C.5). In the other cases the proof is similar. Using the density of C 0 (D, C) ∩ J in J for the H 1 -norm, we may assume u ∈ C 0 (D, C) ∩ J . It suffices to prove the result for 0 < η < min{10−3 , ε 2 }. Let x 0 ∈ ∂Ω and Vη be an open regular set of D s.t.: • • • • •

∂Vη ∩ ∂D = ∅, |Vη | η2 , x 0 is an interior point of ∂Ω ∩ ∂Vη , Vη is simply connected, |u|2 1 + η2 in Vη , ∇uL2 (Vη ) η2 .

Using the Carathéodory’s theorem, there is Φ : Vη → D(0, 1), a homeomorphism s.t. Φ|Vη : Vη → D(0, 1) is a conformal mapping. Without loss of generality, we may assume that Φ(x 0 ) = 1. Let δ > 0 be s.t. for |θ | δ we have Φ −1 (eıθ ) ∈ ∂Vη ∩ ∂Ω. Let Nη ∈ J be defined by

1 Nη (x) = M (Φ(x)) η2 ,δ

if x ∈ D \ Vη , otherwise.

Here, Mη2 ,δ is defined by Lemma 13. Using the conformal invariance of the Dirichlet functional, we have 1 1 |∇Nη |2 = |∇Mη2 ,δ |2 π + η2 . (C.6) 2 2 Vη

D(0,1)

It is not difficult to see that u+ η := uNη ∈ Jdeg(u,D )+e0 . Since |Nη | 2 and Nη − 1L2 (D ) = oη (1), using the Dominated convergence theorem, we may prove that uNη → u in L2 (D) when η → 0. It follows that (C.5) holds. From (C.6) and using the following formula, ∇(uv)2 = |v|2 |∇u|2 + |u|2 |∇v|2 + 2 (v∂j u) · (u∂j v) j =1,2

we obtain

1090


1 2

Vη

+ 2 1 ∇u = η 2

|Nη |2 |∇u|2 + |u|2 |∇Nη |2 + 2

(Nη ∂j u) · (u∂j Nη )

j =1,2

Vη

1 + η2 π + η2 + 2∇u2L2 (V ) + 4 1 + η2 ∇uL2 (Vη ) ∇Nη L2 (Vη ) η

η π + . 2

(C.7)

Furthermore, we have 1 4ε 2

2 2 2 η η. 1 − u+ η 2 4ε 2

(C.8)

Vη

From (C.7) and (C.8), it follows + Eε u+ η , D = Eε (u, D \ Vη ) + Eε uη , Vη Eε (u, D) + π + η. The previous inequality completes the proof.

2

We may now prove Lemma 7. For the convenience of the reader, we recall the statement of the lemma. Lemma 7. Let u ∈ J , ε > 0 and δ = (δ1 , . . . , δN , δ0 ) ∈ ZN +1 . For all η > 0, there is uδη ∈ Jdeg(u,D)+δ s.t.


|δi | + η

(25)

i∈{0,...,N}

and u − uδ

η L2 (D )

= oη (1),

oη (1) − −−→ 0. η→0

(26)

Proof. As in the previous lemma, it suffices to prove the proposition for 0 < η < min{10−3 , ε 2 } and u ∈ C 0 (D, C) ∩ J . We construct uδη in 1 = i∈{0,...,N } |δi | steps. If 1 = 0 (which is equivalent at δ = 0ZN+1 ) then, taking uδη = u, (25) and (26) hold. Assume 1 = 0. Let Γ = {i ∈ NN s.t. δi = 0} = ∅, L = Card Γ and μ = η1 . We enumerate the elements of Γ in (in )n∈NL s.t. for n ∈ NL−1 we have in < in+1 . x Let σ be the sign function i.e. for x ∈ R∗ , σ (x) = |x| . For n ∈ NL and l ∈ N|δin | , we construct vnl ∈ Jdeg(v l−1 ,D)+σ (δi )ei n

s.t.

n

M. Dos Santos / Journal of Functional Analysis 257 (2009) 1053–1091 |δi

v00 = u,

1091

|

n−1 vn0 = vn−1 with for n = 1, δi0 = 0,

(vnl )+ μ if δin > 0, 0 l < |δin |. vnl+1 = l (vn )− μ if δin < 0,

± l Here, (vnl )± μ stands for uμ defined by Lemma 14 taking u = vn and η = μ. |δ |

It is clear that vnl is well defined and that for n ∈ NL , vn := vn in ∈ Jdeg(vn−1 ,D)+δin ein with v0 = u. Therefore, using (C.4), we have for n ∈ NL , vn ∈ Jdeg(u,D)+k∈N

δ e , n ik ik

Eε (vn ) Eε (u) + (π + μ)

|δik |.

k∈Nn

Taking n = L, we obtain that uδη = vL ∈ Jdeg(u,D)+δ ,


|δi | + η.

i∈{0,...,N }

Furthermore, uδη is obtained from u multiplying by 1 factors Nl , l ∈ N1 . Each Nl is bounded by 2 and converges to 1 in L2 -norm (when η → 0). Using the Dominated convergence theorem, we may prove that uδη satisfies (26). 2 References [1] N. André, I. Shafrir, Minimisation of a Ginzburg–Landau type functional with boundary condition which is not S 1 -valued, Calc. Var. Partial Differential Equations 7 (1998) 191–217. [2] N. André, I. Shafrir, On the minimizers of a Ginzburg–Landau type energy when the boundary condition has zeros, Adv. Differential Equations 9 (2004) 891–960. [3] L. Berlyand, D. Golovaty, V. Rybalko, Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain, C. R. Math. Acad. Sci. Paris 343 (2006) 63–68. [4] L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004. [5] L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg– Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/ publications.html. [6] H. Brezis, New questions related to the topological degree, in: The Unity of Mathematics, in: Progr. Math., vol. 244, Birkhäuser Boston, 2006. [7] H. Brezis, F. Bethuel, F. Hélein, Ginzburg–Landau Vortices, Birkhäuser, 1994. [8] P. Mironescu, Explicit bounds for solutions to a Ginzburg–Landau type equation, Rev. Roumaine Math. Pures Appl. 41 (1996) 263–271. [9] E. Sandier, S. Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Birkhäuser, 2007. [10] M. Tinkham, Introduction to Superconductivity, McGraw–Hill, New York, 1996.


Higher order spectral shift Ken Dykema 1 , Anna Skripka ∗ Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA Received 18 December 2008; accepted 25 February 2009 Available online 10 March 2009 Communicated by N. Kalton

Abstract We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62–71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735–743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein’s and Koplienko’s) ones. We also obtain higher order spectral averaging formulas generalizing the Birman–Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert–Schmidt class of the algebra. © 2009 Elsevier Inc. All rights reserved. Keywords: Spectral shift function; Taylor formula

1. Introduction Let H be a separable Hilbert space and B(H) the algebra of bounded linear operators on H. Let M be a semi-finite von Neumann algebra acting on H and τ a semi-finite normal faithful trace on M. We study how the value f (H0 ) of a function f on a self-adjoint operator H0 in M changes under a perturbation V = V ∗ ∈ M of the operator argument H0 . It is well known that * Corresponding author.

E-mail addresses: [email protected] (K. Dykema), [email protected] (A. Skripka). 1 Research supported in part by NSF grant DMS-0600814.


K. Dykema, A. Skripka / Journal of Functional Analysis 257 (2009) 1092–1132

1093

for certain functions f , the value f (H0 + V ) can be approximated by the Fréchet derivatives of the mapping H ∗ = H → f (H ) at point H0 . Theorem 1.1. (Cf. [24, Theorem 1.43, Corollary 1.45].) Let f : R → C be a bounded function such that the mapping H → f (H ) defined on self-adjoint elements of B(H) is p times continuously differentiable in the sense of Fréchet (and, hence, in the sense of Gâteaux). Let H0 = H0∗ , V = V ∗ ∈ B(H) and denote Rp,H0 ,V (f ) = f (H0 + V ) −

p−1 j =0

1 d j f (H0 + tV ). j ! dt j t=0

(1)

dp f (H0 + tV ) dt dt p

(2)

Then Rp,H0 ,V (f ) =

1 (p − 1)!

1 (1 − t)p−1 0

and Rp,H

0 ,V

(f ) = O V p .

(3)

Theorem 1.1 generalizes the Taylor approximation theorem for scalar functions. It was proved in [8] that for f ∈ C 2p (R), the operator function f is Fréchet differentiable p times on B(H), with the derivative written as an iterated operator integral. For f ∈ Wp (the set of functions C p (R) such that for each j = 0, . . . , p, the derivative f (j ) equals the Fourier transform f ∈ itλ R e dμf (j ) (λ) of a finite Borel measure μf (j ) ) and a (possibly) unbounded H0 , the differentiability of H → f (H ) in the sense of Fréchet of order p was established in [1]; in that case, the dp Gâteaux derivative dt p f (H0 + tV ) was represented as a Bochner-type multiple operator integral. 1 (R) ∩ B p (R), it is known that the Gâteaux derivative of f of order For f in the Besov class B∞1 ∞1 p exists [23], but the bound (3) has not been proved. In the scalar case (dim(H) = 1), we have that τ [Rp,H0 ,V (f )] is a bounded functional on the space of functions f (p) and τ Rp,H

0 ,V

τ (|V |p ) (p) f . (f ) ∞ p!

(4)

In the case of a nontrivial H (dim(H) > 1), it is generally hard to separate contribution of the perturbation V to the estimate for the remainder (3) from contribution of the scalar function f (p) . One of approaches to (4) is the estimate Rp,H0 ,V (f ) C(H0 , V )f (2p) ∞ , for f ∈ C 2p (R) [8], with C(H0 , V ) a constant depending on bounded self-adjoint operators H0 and V . Another approach is the estimate τ Rp,H

0 ,V

τ (|V |p ) (f ) μf (p) p!

(5)

(p) ), for τ the usual [10] (see [12] for an example when μf (p) can be replaced with f 1 ∗ ∗ trace, H0 = H0 an operator in H, V = V an operator in the Schatten p-class, and f ∈ Wp . If

1094


H0 = H0∗ is affiliated with a semi-finite von Neumann algebra M, V = V ∗ is in the τ -Schatten p-class of M, and f ∈ Wp , then the remainder Rp,H0 ,V (f ) belongs to the Schatten p-class of M as well and p 1/p ([τ (|V |p )]1/p + V )p μf (p) τ Rp,H0 ,V (f ) + Rp,H0 ,V (f ) p! see [1]. In the particular case of p = 1 or p = 2, the functional τ [Rp,H0 ,V (f )] is bounded on the space of functions f or f , respectively, and (4) holds. The measure representing the functional is absolutely continuous (with respect to Lebesgue’s measure), with the density equal to Krein’s spectral shift function ξH0 +V ,H0 or Koplienko’s spectral shift function ηH0 ,H0 +V , respectively. That is, we have

τ R1,H0 ,V (f ) =

τ R1,H

f (t)ξH0 +V ,H0 (t) dt,

0 ,V

(f ) τ |V | f ∞

(6)

R

and

τ R2,H0 ,V (f ) =

f (t)ηH0 ,H0 +V (t) dt,

2

τ R2,H ,V (f ) τ (|V | ) f ∞ . 0 2

(7)

R

Existence of ξH0 +V ,H0 , with τ (|V |) < ∞, satisfying (6) for f ∈ W1 , was proved in the setting M = B(H) in [16] (cf. also [17]) and extended to the setting of an arbitrary semi-finite von Neumann algebra M in [2,7]. Moreover, when M = B(H), the trace formula in (6) is known 1 (R) [21]. In the setting M = B(H), existence of η to hold for f ∈ B∞1 H0 ,H0 +V , with V in the Hilbert–Schmidt class, satisfying (7) for bounded rational functions f was proved in [15]. Later, it was proved in [22] that ηH0 ,H0 +V satisfies the trace formula in (7) for functions f in 1 (R) ∩ B p (R). When V is in the trace class, Koplienko’s spectral shift function can be B∞1 ∞1 written explicitly as t ηH0 ,H0 +V (t) = −

ξH0 +V ,H0 (λ) dλ + τ EH0 (−∞, t) V ,

(8)

−∞

where EH0 is the spectral measure of H0 [15]. In the context of a general M, Koplienko’s spectral shift function ηH0 ,H0 +V , with τ (|V |) < ∞, and the representation (8) are discussed in [26]. For p 3, M = B(H), and τ (|V |p ) < ∞, the distribution τ [Rp,H0 ,V (f )] is given by an L2 -function γp,H0 ,V satisfying

τ (V p ) (p) τ Rp,H0 ,V (f ) = f (0) + p!

f (p+1) (t)γp,H0 ,V (t) dt, R

for all f ∈ Wp+1 [10]. It was conjectured in [15] that there exists a Borel measure νp with the |p ) such that total variation bounded by τ (|V p!


τ Rp,H0 ,V (f ) =

1095

f (p) (t) dνp (t),

(9)

R

for bounded rational functions f . Unfortunately, the proof of (9) in [15] was based on the false claim that for V in the Schatten p-class, p > 2, the set function defined on rectangles of Rp+1 by

A1 × A2 × · · · × Ap+1 → τ E(A1 )V E(A2 )V . . . V E(Ap+1 ) ,

(10)

where E(·) is a spectral measure on R with values in B(H), extends to a (countably additive) measure of bounded variation (see a counterexample in Section 4). When V is in the Hilbert– Schmidt class of M = B(H), the set function in (10) does extend to a (countably-additive) measure of bounded variation [5,20] and thus ideas of [15] can be applied to prove existence of a measure νp satisfying (9) for bounded rational functions (see Section 7). In this case, the total variation of νp is bounded by νp

(τ (|V |2 ))p/2 . p!

Adjusting techniques of [10] then extends (9) to the functions f ∈ Wp . For M a von Neumann algebra acting on an infinite-dimensional Hilbert space H, the set function in (10), with E(·) the spectral measure attaining its values in M and V ∈ M satisfying τ (|V |2 ) < ∞, may fail to extend to a finite measure on Rp+1 for p > 2 even if τ is finite (see a counterexample in Section 4). Therefore, the approach of [15] is not applicable in the proof of (9). When M is a general semi-finite von Neumann algebra, we prove (9) for p = 3 by relating R3,H0 ,V to R2,H0 ,V , which allows to reduce the problem to the case of p = 2 (see Sections 6 and 8). We also study the case when M is finite and H0 , V ∈ M are free with respect to the finite trace τ (which is assumed normalized so that τ (1) = 1). Freeness was introduced by Voiculescu (see, for example, [28]) and amounts to a specific prescription for the values of the mixed moments of H0 and V in terms of the individual moments of H0 and V . Free perturbations have appeared in the study of quite general operators in finite von Neumann algebras, for example in the seminal work of Haagerup and Schultz [13]. Assuming freeness, we show that for all p the set function in (10) extends to a finite measure on Rp+1 (see Section 4), from which (9) can be derived. Under the assumptions that we impose to prove existence of νp satisfying (9) (see discussion in the two preceding paragraphs), we also construct a function ηp , called the pth-order spectral shift function, such that dνp (t) = ηp (t) dt, provided H0 is bounded (see statements in Section 5 and proofs in Sections 7 and 8). The spectral shift function of order p admits the recursive representation

t ηp (t) = − −∞

ηp−1 (λ) dλ +

splineλ1 ,...,λp−1 (t) dmp−1,H0 ,V (λ1 , . . . , λp−1 ),

(11)

Rp−1

where splineλ1 ,...,λp−1 is a piecewise polynomial of degree p − 2 with breakpoints λ1 , . . . , λp−1 and dmp−1,H0 ,V (λ1 , . . . , λp−1 ) is a measure on Rp−1 determined by p − 1 copies of the spectral

1096


measure of H0 intertwined with p − 1 copies of the perturbation V (see Section 5 for the precise formula). As it is noticed in Section 5, the function η2 given by (11) coincides with the function ηH0 ,H0 +V given by (8), provided τ (|V |) < ∞. The techniques of [15] that prove existence of νp when M = B(H) do not give absolute continuity of νp . We obtain ηp by analyzing the Cauchy transform of the measure νp satisfying the trace formula (9) (see Section 6). The approach of this paper, developed mainly for higher order spectral shift functions, contributes to the subject of Krein’s spectral shift function as well. In 1972, using Theorem 1.1, (2) and the double operator integral representation for the derivative d f (H0 + xV ) = dx

(1)

R2

λ1 ,λ2 (f ) EH0 +xV (dλ1 )V EH0 +xV (dλ2 ),

M.Sh. Birman and M.Z. Solomyak [4] showed that

τ f (H0 + V ) − f (H0 ) =

1

f (t)

τ EH0 +xV (dt) dx

0

R

(see [1, Theorem 6.3] for the analogous result in the context of von Neumann algebras), which along with Krein’s trace formula

τ f (H0 + V ) − f (H0 ) =

f (t)ξH0 +V ,H0 (t) dt

R

[2,7,16] implied the spectral averaging formula 1

τ EH0 +xV (dt) dx = ξH0 +V ,H0 (t) dt

(12)

0

(see [11,18,25,26] for generalizations and extensions). The operator f (H0 + V ) − f (H0 ) also admits a double operator integral representation f (H0 + V ) − f (H0 ) =

(1)

R2

λ1 ,λ2 (f ) EH0 +V (dλ1 )V EH0 (dλ2 ).

(13)

A natural question raised by M.Sh. Birman (see, e.g., [3]) asks if it is possible to deduce ex1 istence of ξH0 +V ,H0 , or equivalently, absolute continuity of the measure 0 τ [EH0 +xV (dt)] dx, directly from the double operator integral representation (13). For M a finite von Neumann algebra, we answer this question affirmatively and represent ξH0 +V ,H0 as an integral of a basic spline straightforwardly from (13) (see Section 9). A general property of a basic spline is that it has the minimal support among all the splines with the same degree, smoothness, and domain properties (see, e.g., [9]). When dim(H) < ∞, higher order spectral shift functions can be written as integrals of basic splines as well (see Section 9). By combining different representations for the remainder τ [Rp,H0 ,V (f )] in the setting of M = B(H), we prove absolute continuity of the measure


1 A →

1097

p

dx (1 − x)p−1 τ EH0 +xV (A)V

0

and derive higher order analogs of the spectral averaging formula (12) (see Section 10). Basic technical tools of the paper are discussed in Sections 2–4, main results are stated in Section 5 and then proved in Sections 6–8, additional representations for spectral shift functions are obtained in Section 9, and the Birman–Solomyak spectral averaging formula is generalized in Section 10. By saying “the standard setting” or “τ is the standard trace,” we implicitly assume that M = B(H) and τ is the usual trace defined on the trace class operators of B(H). Let Lp (M, τ ) denote the noncommutative Lp -space of (M, τ ) with the norm V p = τ (|V |p )1/p and Lp (M, τ ) = Lp (M, τ ) ∩ M the Schatten p-class of (M, τ ). The Schatten p-class is equipped with the norm · p,∞ = · p + · , where · is the operator norm. Throughout the paper, H0 and V denote self-adjoint operators in M or affiliated with M; V is mainly taken to be an element of Lp (M, τ ). Let R denote the set of rational functions on R with nonreal poles, Rb the subset of R of bounded functions. The symbol fz is reserved for the function 1 , where z ∈ C \ R. R λ → z−λ 2. Divided differences and splines Definition 2.1. The divided difference of order p is an operation on functions f of one (real) variable, which we will usually call λ, defined recursively as follows: (0)

λ1 (f ) := f (λ1 ), ⎧ (p−1) (p−1) ⎪ ⎨ λ1 ,...,λp−1 ,λp (f )−λ1 ,...,λp−1 ,λp+1 (f ) (p) λp −λp+1 λ1 ,...,λp+1 (f ) := ⎪ (p−1) ⎩∂| ∂t t=λp λ1 ,...,λp−1 ,t (f )

if λp = λp+1 , if λp = λp+1 .

Below we state selected facts on the divided difference (see, e.g., [9]). Proposition 2.2. (p)

(1) (See [9, Section 4.7, (a)].) λ1 ,...,λp+1 (f ) is symmetric in λ1 , λ2 , . . . , λp+1 . (2) (See [9, Section 4.7, (h)].) If all λ1 , λ2 , . . . , λp+1 are distinct, then (p)

λ1 ,...,λp+1 (f ) =

p+1 j =1

f (λj ) . k=j (λj − λk )

(3) (See [9, Section 4.7].) For f a sufficiently smooth function, (p) λ1 ,...,λp+1 (f ) =

i )−1 m(λ

i∈I

cij (λ1 , . . . , λp+1 )f (j ) (λi ).

j =0

Here I is the set of indices i for which λi are distinct, m(λi ) is the multiplicity of λi , and cij (λ1 , . . . , λp+1 ) ∈ C.

1098

K. Dykema, A. Skripka / Journal of Functional Analysis 257 (2009) 1092–1132 (p)

(4) (See [9, Section 4.7].) λ1 ,...,λp+1 (ap λp + ap−1 λp−1 + · · · + a1 λ + a0 ) = ap , where a0 , a1 , . . . , ap ∈ C. (5) (See [9, Section 5.2, (2.3) and (2.6)].) The basic spline with the break points λ1 , . . . , λp+1 , where at least two of the values are distinct, is defined by ⎧ 1 ⎨ |λ −λ χ(min{λ1 ,λ2 },max{λ1 ,λ2 }) (t) if p = 1, 2 1| t → p−1 ⎩ (p) if p > 1. λ1 ,...,λp+1 ((λ − t)+ ) Here the truncated power is defined by k x+ =

if x 0, if x < 0,

xk 0

for k ∈ N. The basic spline is non-negative, supported in

min{λ1 , . . . , λp+1 }, max{λ1 , . . . , λp+1 } and integrable with the integral equal to 1/p. (Often the basic spline is normalized so that its integral equals 1). (6) (See [9, Section 5.2, (2.2) and Section 4.7, (c)].) For f ∈ C p [min{λ1 , . . . , λp+1 }, max{λ1 , . . . , λp+1 }], (p)

λ1 ,...,λp+1 (f ) ⎧ ∞ (p) (p) p−1 ⎨ 1 (t)λ1 ,...,λp+1 ((λ − t)+ ) dt (p−1)! −∞ f = ⎩ 1 f (p) (λ1 ) p!

if ∃i1 , i2 such that λi1 = λi2 , if λ1 = λ2 = · · · = λp+1 .

(7) (See [9, Section 4.7, (l)].) Let f ∈ C p [a, b]. Then, for {λ1 , . . . , λp+1 } ⊂ [a, b], 1 max f (p) (λ). p! λ∈[a,b]

(p)

λ1 ,...,λp+1 (f )

Below we state useful properties of the divided difference to be used in the paper. Lemma 2.3. For z ∈ C, with Im(z) = 0, (p) λ1 ,...,λp+1

1 z−λ

=

p+1 j =1

1 , z − λj

where the divided difference is taken with respect to the real variable λ.


1099

Proof. We notice that by Definition 2.1, (1) λ1 ,λ2

1 z−λ

=

1 1 1 ( z−λ − z−λ ) 1 = (z−λ1 )(z−λ 1 2 λ1 −λ2 2) ∂ 1 1 1 ( ) = = ∂t t=λ z−t (z−λ1 )(z−λ2 ) (z−λ )2 1

1

if λ1 = λ2 , if λ1 = λ2 .

By repeating the same argument, we obtain (2) λ1 ,λ2 ,λ3

1 z−λ

=

1 . (z − λ1 )(z − λ2 )(z − λ3 )

The rest of the proof is accomplished by induction.

2

Lemma 2.4. Let D be a domain in C and f a function continuously differentiable sufficiently many times on D × R. Then for p ∈ N, (i)

(p) (p) λ1 ,...,λp+1 f (z, λ) dz = λ1 ,...,λp+1

f (z, λ) dz ,

with an appropriate choice of the constant of integration on the left-hand side; (ii) (p) (p) lim λ1 ,...,λp+1 f (z, λ) = λ1 ,...,λp+1 lim f (z, λ) ,

z→z0

z→z0

z0 ∈ D;

(iii)

∂ (p) ∂ (p) f (z, λ) , f (z, λ) = λ1 ,...,λp+1 ∂z λ1 ,...,λp+1 ∂z where the divided difference is taken with respect to the variable λ. Proof. Follows immediately from Proposition 2.2(3).

2

Corollary 2.5. For p, k ∈ N, (−1)k ∂ k k! ∂zk

p+1 j =1

1 z − λj

(p)

= λ1 ,...,λp+1

1 . (z − λ)k+1

Proof. Follows immediately from Lemma 2.3 and Lemma 2.4.

2

1100


3. Remainders of Taylor-type approximations In this section, we collect technical facts on derivatives of operator functions and remainders of the Taylor-type approximations. The following lemma is routine. Lemma 3.1. Let H0 = H0∗ be an operator in H and V = V ∗ ∈ B(H). Let Hx = H0 + xV , with x ∈ R. Then, dp (zI − Hx )−k = p! p dx

(zI − Hx )−k0 V (zI − Hx )−k1 V . . . V (zI − Hx )−kp .

1k0 ,k1 ,...,kp k k0 +k1 +···+kp =k+p

If, in addition, H0 is bounded, then dp k H = p! dx p x

k

Hxk0 V Hxk1 V . . . V Hx p ,

p k.

0k0 ,k1 ,...,kp k0 +k1 +···+kp =k−p

Lemma 3.2. Let H0 = H0∗ be an operator affiliated with M and V = V ∗ ∈ L2 (M, τ ). Then,

(−1)k d k τ (zI − H0 )−1 V (zI − H0 )−1 V (zI − H0 )−1 k k! dz 2 d 1 −k−1 = τ . (zI − H − xV ) 0 2 dx 2 x=0

(14)

Proof. Firstly, we compute the left-hand side of (14). By cyclicity of the trace,

τ (zI − H0 )−1 V (zI − H0 )−1 V (zI − H0 )−1 = τ (zI − H0 )−2 V (zI − H0 )−1 V . By continuity of the trace in the norm · 1,∞ ,

d d τ (zI − H0 )−2 V (zI − H0 )−1 V = τ (zI − H0 )−2 V (zI − H0 )−1 V . dz dz It is easy to see that dk (zI − H0 )−2 V (zI − H0 )−1 V k dz =

k j =0

k! (−1)j (j + 1)!(zI − H0 )−2−j V (−1)k−j (k − j )!(zI − H0 )−1−(k−j ) V j !(k − j )!

= (−1)k k!

k (j + 1)(zI − H0 )−2−j V (zI − H0 )−1−(k−j ) V . j =0

(15)


1101

Now we compute the right-hand side of (14). Let Hx = H0 + xV . It is routine to see that d (zI − Hx )−(k+1) = (zI − Hx )−i V (zI − Hx )−(k+2−i) , dx k+1 i=1

and hence, d 2 (zI − Hx )−(k+1) 2 dx x=0 k+1 d (zI − Hx )−i V (zI − H0 )−(k+2−i) =2 dx x=0 i=1

=2

k+1 i−1

(zI − H0 )−(i−j ) V (zI − H0 )−1−j V (zI − H0 )−(k+2−i) .

i=1 j =0

Multiplying by 1/2 and evaluating the trace in the latter expression provides 2 d 1 −k−1 τ (zI − H − xV ) 0 2 dx 2 x=0 =

k+1 i−1

τ (zI − H0 )−1−j V (zI − H0 )−(k+2−j ) V i=1 j =0

=

k k+1

τ (zI − H0 )−1−j V (zI − H0 )−2−(k−j ) V j =0 i=j

=

k

(k + 1 − j )τ (zI − H0 )−1−j V (zI − H0 )−2−(k−j ) V .

j =0

By changing the index of summation i = k − j in the latter expression and by cyclicity of the trace, we obtain k

(i + 1)τ (zI − H0 )−2−i V (zI − H0 )−1−(k−i) V .

(16)

i=0

Comparing (15) and (16) completes the proof of the lemma.

2

As a particular case of results of [23] we have the lemma below. Lemma 3.3. Let H0 = H0∗ be an operator in H and V = V ∗ ∈ B(H). Denote Hx = H0 + xV . For f ∈ Rb ,

1102


dp f (H0 + xV ) dx p (p) = p! . . . λ1 ,...,λp+1 (f ) EHx (dλ1 )V EHx (dλ2 )V . . . V EHx (dλp+1 ). R R

(17)

R

If, in addition, H0 is bounded, then (17) holds for f ∈ R. p

d Remark 3.4. It was proved in [8, Theorem 2.2] that for H0 a bounded operator, dt p f (H0 + tV ) 2p is defined when f ∈ C (R) and the derivative can be computed as an iterated operator intedp gral (17). It was proved later in [23] that the Gâteaux derivative dt p f (H0 + tV ) is defined for f p 1 in the intersection of the Besov classes B∞1 (R) ∩ B∞1 (R) and can be computed as a Bochnertype multiple operator integral.

The following lemma is a straightforward consequence of Lemma 3.1. Lemma 3.5. Let H0 = H0∗ be an operator in H and V = V ∗ ∈ B(H). Then for f a polynomial of degree m,

Rp,H0 ,V (f ) =

k

k

ak0 ,k1 ,...,kp H0 0 V H0k1 V . . . V H0 p ,

k0 ,k1 ,...,kp 0 k0 +k1 +···+kp =m−p

with ak0 ,k1 ,...,kp numbers. Lemma 3.6. Let H0 = H0∗ be an operator in H and V = V ∗ ∈ B(H). Then, Rp,H0 ,V (fz ) = (zI − H0 − V )−1 −

p−1

j (zI − H0 )−1 V (zI − H0 )−1

(18)

j =0

p = (zI − H0 − V )−1 V (zI − H0 )−1 . Proof. By Lemma 3.1, j d j (zI − H0 − xV )−1 = j !(zI − H0 − x0 V )−1 V (zI − H0 − x0 V )−1 , j dx x=x0 which gives (18). To derive (19) from (18), we use repeatedly the resolvent identity (zI − H0 − V )−1 − (zI − H0 )−1 = (zI − H0 − V )−1 V (zI − H0 )−1 . By combining (zI − H0 − V )−1 and the first summand of p−1

j (zI − H0 )−1 V (zI − H0 )−1 ,

j =0

(19)


1103

we obtain that (provided p > 1) (zI − H0 − V )−1 −

p−1

j (zI − H0 )−1 V (zI − H0 )−1

j =0 −1

= (zI − H0 − V )

−1

V (zI − H0 )

−

p−1

j (zI − H0 )−1 V (zI − H0 )−1 .

j =1

Repeating the reasoning above sufficiently many times completes the proof of (19).

2

From (18) we have the following relation between the remainders of different order. Lemma 3.7. Let H0 = H0∗ be an operator in H and V = V ∗ ∈ B(H). Then p Rp+1,H0 ,V (fz ) = Rp,H0 ,V (fz ) − (zI − H0 )−1 V (zI − H0 )−1 . The following lemma is a straightforward generalization of [10, Lemma 2.6]. Lemma 3.8. Let H0 = H0∗ , V = V ∗ ∈ B(H), and Γ = {λ: |λ| = 1 + H0 + V }. Then, for every function f analytic in a neighborhood of D = {λ: |λ| 1 + H0 + V }, 1 Rp,H0 ,V (f ) = 2πi

p −1 f (λ)(λI − H0 )−1 V (λI − H0 )−1 I − V (λI − H0 )−1 dλ.

Γ ∗

Let (S, ν) be a measure space and let Lso ∞ (S, ν, L1 (M, τ )) denote the ∗-algebra of · bounded so∗ -measurable functions F : S → L1 (M, τ ) [19]. ∗

Proposition 3.9. (See [1, Lemma 3.10]). Let F be a function in Lso ∞ (S, ν, L1 (M, τ )) uniformly L1 (M, τ )-bounded. Then S F (s) dν(s) ∈ L1 (M, τ ), τ (F (·)) is measurable and τ S

F (s) dν(s) = τ F (s) dν(s). S

Similarly to [1, Lemma 4.5, Theorem 5.7], we have the following differentiation formula for an operator function f (·), with f ∈ Wp . Lemma 3.10. Let H0 = H0∗ be an operator affiliated with M and V = V ∗ ∈ Lp (M, τ ). Let Hx = H0 + xV , with x ∈ R. Then, for f ∈ Wp given by f (λ) = R eitλ dμf (t), the function f (Hx ) is p times Fréchet differentiable in the norm · 1,∞ and the derivative equals the Bochner-type multiple operator integral dp f (Hx ) = p! dx p

Π (p)

ei(s0 −s1 )Hx V . . . V ei(sp−1 −sp )Hx V eisp Hx dσf (s0 , . . . , sp ). (p)

1104


Here Π (p) = (s0 , s1 , . . . , sp ) ∈ Rp+1 : |sp | · · · |s1 | |s0 |, sign(s0 ) = · · · = sign(sp ) (p)

and dσf (s0 , s1 , . . . , sp ) = ip μf (ds0 )ds1 . . . dsp . In particular, for t ∈ R, d p itHx p e = i p! ei(t−s1 )Hx V . . . V ei(sp−1 −sp )Hx V eisp Hx dsp . . . ds1 . dx p Π (p−1)

By applying Proposition 3.9 and Lemma 3.10, we obtain the following Lemma 3.11. Let H0 = H0∗ be an operator affiliated with M and V = V ∗ ∈ Lp (M, τ ). Let dp Hx = H0 + xV , with x ∈ R. Then for f ∈ Wp , we have dx p f (Hx ) ∈ L1 (M, τ ), τ

(p) dp f (H ) = p! τ ei(s0 −s1 )Hx V . . . V ei(sp−1 −sp )Hx V eisp Hx dσf (s0 , . . . , sp ) x p dx Π (p)

and p d p dx p f (Hx ) V p μf (p) p . 1

Corollary 3.12. Under the assumptions of Lemma 3.11, p p d d itHx dμf (t). τ f (H ) = τ e x dx p dx p R

Proof. The claim is proved by reducing the double integral to an iterated one and applying Lemmas 3.10 and 3.11. 2 Remark 3.13. By combining Lemma 3.11 and Theorem 1.1 (2), one obtains the estimate (5). 4. Multiple spectral measures We will need the fact that certain finitely additive “multiple spectral measures” extend to countably additive measures. Theorem 4.1. Let 2 p ∈ N and let E1 , E2 , . . . , Ep be projection-valued Borel measures from R into M. Suppose that V1 , . . . , Vp belong to L2 (M, τ ). Assume that either τ is the standard trace or p = 2. Then there is a unique (complex) Borel measure m on Rp with total variation not exceeding the product V1 2 V2 2 · · · Vp 2 , whose value on rectangles is given by

m(A1 × A2 × · · · × Ap ) = τ E1 (A1 )V1 E2 (A2 )V2 . . . Vp−1 Ep (Ap )Vp for all Borel subsets A1 , A2 , . . . , Ap of R.


1105

Proof. It is enough (see, e.g., [14, Theorem 2.12] for p = 2) to prove that the variation of the set function m on the rectangles of Rp is bounded by V1 2 V2 2 · · · Vp 2 , which can be accomplished completely analogously to the proof of [20, Theorem 1] (see also [5]). 2 Remark 4.2. For τ the standard trace, the bound for the total variation in Theorem 4.1 was proved in [20, Theorem 1]. Theorem 4.1 with τ standard was also obtained in [5]. The proof in [5] is based on the facts that a Hilbert–Schmidt operator can be approximated by finite-rank operators in the norm · 2 and that for rank-one perturbations V1 , . . . , Vp and τ the standard trace, the set function m decomposes into a product of scalar measures. It is classical that a direct product of countably additive measures always has a countably-additive extension to the σ -algebra generated by the direct product of the σ -algebras involved. The argument of [5] cannot be directly extended to the case of a general trace. For a general trace τ , the set function m is known to be of bounded variation only if p = 2. Technically, this constraint is explained by the fact that in general · p is not dominated by · 2 , as distinct from the particular case of the standard trace τ . A counterexample constructed further in this section demonstrates that p = 2 is not only a technical constraint. Corollary 4.3. Let 2 p ∈ N and let E1 , E2 , . . . , Ep be projection-valued Borel measures from R to M. Suppose that V1 , . . . , Vp belong to L2 (M, τ ). Assume that either τ is the standard trace or p = 2. Then there is a unique (complex) Borel measure m1 on Rp+1 with total variation not exceeding the product V1 2 V2 2 · · · Vp 2 , whose value on rectangles of Rp+1 is given by

m1 (A1 × A2 × · · · × Ap × Ap+1 ) = τ E1 (A1 )V1 E2 (A2 )V2 . . . Vp−1 Ep (Ap )Vp E1 (Ap+1 ) , for all Borel subsets A1 , A2 , . . . , Ap , Ap+1 of R. Proof. It is straightforward to see that

m1 (A1 × A2 × · · · × Ap × Ap+1 ) = τ E1 (A1 ∩ Ap+1 )V1 E2 (A2 )V2 . . . Vp−1 Ep (Ap )Vp . By repeating the argument of [5,20], one can see that the total variation of the set function m1 is bounded on the rectangles of Rp+1 by V1 2 V2 2 · · · Vp 2 . Thus, m1 extends to a unique complex Borel measure on Rp+1 with variation bounded by V1 2 V2 2 · · · Vp 2 . 2 Corollary 4.4. Let 2 p ∈ N, E1 , . . . , Ep projection-valued Borel measures from R into M, and τ a finite trace. Suppose that V1 , . . . , Vp−1 belong to M. Assume that either τ is the standard trace or p = 2. Then there is a unique complex Borel measure m2 on Rp with total variation not exceeding V1 2 V2 2 · · · Vp−1 2 τ (I )1/2 , whose value on rectangles is given by

m2 (A1 × A2 × · · · × Ap ) = τ E1 (A1 )V1 E2 (A2 )V2 . . . Vp−1 Ep (Ap ) for all Borel subsets A1 , A2 , . . . , Ap of R. Proof. It is an immediate consequence of Theorem 4.1 applied to Vp = I .

2

1106


In the sequel, we will work with the set functions

mp,H0 ,V (A1 × A2 × · · · × Ap ) = τ EH0 (A1 )V EH0 (A2 )V . . . V EH0 (Ap )V , (1)

mp,H0 ,V (A1 × A2 × · · · × Ap × Ap+1 )

= τ EH0 (A1 )V EH0 (A2 )V . . . V EH0 (Ap )V EH0 (Ap+1 ) , (2)

mp,H0 ,V (A1 × A2 × · · · × Ap × Ap+1 )

= τ EH0 +V (A1 )V EH0 (A2 )V . . . V EH0 (Ap )V EH0 (Ap+1 ) , and their countably-additive extensions (when they exist). Here Aj are measurable subsets of R, H0 = H0∗ is affiliated with M, and V = V ∗ ∈ L2 (M, τ ). In the next result, freeness of (zI − H0 )−1 and V means freeness of the algebra generated by the spectral projections of H0 and the unital algebra generated by V . Theorem 4.5. Let τ be a finite trace normalized by τ (I ) = 1 and let H0 = H0∗ be affiliated with M and V = V ∗ ∈ M. Assume that (zI − H0 )−1 and V are free. Then the set functions mp,H0 ,V (1) and mp,H0 ,V extend to countably additive measures of bounded variation. (1)

Proof. We prove the claim for the function mp,H0 ,V ; the case of mp,H0 ,V is completely anal ogous. Using the moment-cumulant formula (see [27, Theorem 2.17]), and that i EH0 (Ai ) = EH0 ( i Ai ) we have

mp,H0 ,V (A1 × · · · × Ap ) = τ EH0 (A1 )V . . . EH0 (Ap )V =

kK(π) [V , . . . , V ]

τ EH0 (∩i∈Bj Ai ) ,

(20)

j =1

π={B1 ,...,B }∈NC(p)

where NC(p) is the lattice of all noncrossing partitions of {1, . . . , p} and where kK(π) [V , . . . , V ] is the product of cumulants of V, associated to the block structure of the Kreweras complement K(π) of π ; thus, kK(π) [V , . . . , V ] is equal to a polynomial (that depends on π ) in p variables, evaluated at τ (V ), τ (V 2 ), . . . , τ (V p ). Given π = {B1 , . . . , B } ∈ NC(p), the measure

γp,π

: A1 × · · · × Ap → τ E H0 j =1

Ai

(21)

i∈Bj

is the push-forward of the -fold product ×1 (τ ◦ EH0 ) of the spectral distribution measure of H0 (with respect to τ ) under the mapping of R onto the product of diagonals in Rp according to the block structure B1 , . . . , B . Each such push-forward is a probability measure. Thus, we see that mp,H0 ,V is a linear combination of probability measures, and has finite total variation. 2 In some cases used in the paper, the measure m2 is known to be real-valued and the measure m non-negative.


1107

Lemma 4.6. Let τ be a finite trace. Let H0 = H0∗ be affiliated with M and V = V ∗ ∈ M. Then the measure m(2) 1,H0 ,V is real-valued. Proof. For arbitrary measurable subsets A1 and A2 of R,

τ EH0 +V (A1 )V EH0 (A2 )

= τ EH0 +V (A1 )(H0 + V )EH0 (A2 ) − τ EH0 +V (A1 )H0 EH0 (A2 )

= τ EH0 (A2 ) EH0 +V (A1 )(H0 + V ) EH0 (A2 ) − τ EH0 +V (A1 ) H0 EH0 (A2 ) EH0 +V (A1 ) , where the operators EH0 (A2 ) EH0 +V (A1 )(H0 + V ) EH0 (A2 ) and EH0 +V (A1 ) H0 EH0 (A2 ) EH0 +V (A1 ) (2)

(2)

are self-adjoint. Therefore, m1,H0 ,V (A1 × A2 ) ∈ R, and the extension of m1,H0 ,V to the Borel subsets of R2 is real-valued. 2 Lemma 4.7. Let H0 = H0∗ be affiliated with M and V = V ∗ ∈ L2 (M, τ ). Then the measure m2,H0 ,V is non-negative. Proof. For arbitrary measurable subsets A1 and A2 of R,

τ EH0 (A1 )V EH0 (A2 )V = τ EH0 (A1 )V EH0 (A2 )V EH0 (A1 ) 0, since !

" ! " EH0 (A1 )V EH0 (A2 )V EH0 (A1 )f, f = EH0 (A2 ) V EH0 (A1 )f , V EH0 (A1 )f 0,

for any f ∈ H.

2

Lemma 4.8. Let τ be a finite trace. Let H0 = H0∗ be an operator affiliated with M and V = (2) V ∗ ∈ M. Then mk,H0 ,V has no atoms on the diagonal Dk+1 = {(λ1 , λ2 , . . . , λk+1 ): λ1 = λ2 = · · · = λk+1 ∈ R} of Rk+1 . (2)

Proof. By definition of the measure mk,H0 ,V , (2)

mk,H0 ,V

k−1

(λ, λ, . . . , λ) = τ EH0 +V {λ} V EH0 {λ} V EH0 {λ} .

We will show that EH0 +V ({λ})V EH0 ({λ}) is the zero operator. Let g be an arbitrary vector in H and let h = EH0 ({λ})g. Then H0 h = λh and EH0 +V {λ} V EH0 {λ} g = EH0 +V {λ} V h = EH0 +V {λ} (H0 + V )h − EH0 +V {λ} H0 h = EH0 +V {λ} (H0 + V )h − λEH0 +V {λ} h = (H0 + V − λI )EH0 +V {λ} h = 0.

2

1108


Upon evaluating a trace, some iterated operator integrals can be written as Lebesgue integrals with respect to a “multiple spectral measure.” Lemma 4.9. Assume the hypothesis of Theorem 4.1. Assume that the spectral measures E1 , E2 , . . . , Ep correspond to self-adjoint operators H0 , H1 , . . . , Hp affiliated with M, respectively, and that V1 , V2 , . . . , Vp ∈ L2 (M, τ ). Let f1 , f2 , . . . , fp be functions in C0∞ (R) (vanishing at infinity). Then

τ f1 (H1 )V1 f2 (H2 )V2 . . . fp (Hp )Vp = f1 (λ1 )f2 (λ2 ) . . . fp (λp ) dm(λ1 , λ2 , . . . , λp ), Rp

with m as in Theorem 4.1. Proof. The result obviously holds for f1 , f2 , . . . , fp simple functions. Uniform approximation of f1 , f2 , . . . , fp ∈ C0∞ (R) by (totally bounded) simple functions completes the proof. 2 Remark 4.10. (i) The result analogous to the one of Theorem 4.1 holds for integrals with respect to the measures m1 and m2 . (ii) When the operators H0 , H1 , . . . , Hp are bounded, the functions f1 , f2 , . . . , fp can be taken in C ∞ (R). Corollary 4.11. Let H0 = H0∗ be affiliated with M and V = V ∗ ∈ L2 (M, τ ). Denote Hx := H0 + xV , x ∈ [0, 1]. Assume that either τ is standard or p = 2. Then for f ∈ Rb , p d (p) (1) τ f (H + xV ) = p! λ1 ,...,λp+1 (f ) dmp,Hx ,V (λ1 , λ2 , . . . , λp+1 ). 0 dx p Rp+1

Proof. It is enough to prove the result for f (λ) = (z − λ)−k , k ∈ N. By Lemma 3.3, dp (zI − Hx )−k p dx (p) = p! . . . λ1 ,...,λp+1 (z − λ)−k EHx (dλ1 )V EHx (dλ2 )V . . . V EHx (dλp+1 ). R R

(22)

R

By Corollary 2.5, the function λ1 ,...,λp+1 ((z − λ)−k ) is a linear combination of products p+1 ∞ j =1 fj (λj ), with fj in C0 (R) for 1 j p + 1, and hence, the trace of the expression in (22) can be written as a linear combination of integrals like in Remark 4.10 (i), where (1) m1 = mp,Hx ,V (λ1 , λ2 , . . . , λp+1 ). 2 (p)

Remark 4.12. One also has p

τ (zI − H0 )−1 V =

Rp

(p−1)

λ1 ,...,λp (fz ) dmp,H0 ,V (λ1 , λ2 , . . . , λp )


1109

and p

τ (zI − H0 − V )−1 V (zI − H0 )−1 (p) (2) = λ1 ,...,λp+1 (fz ) dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 ). Rp+1

Remark 4.13. If H0 is bounded, then Corollary 4.11 also holds for f a polynomial and for f ∈ C ∞ (R) such that f |[a,b] is a polynomial, where [a, b] ⊃ σ (H0 ) ∪ σ (H0 + V ). A counterexample. Let p 2 be an integer. Let V be a self-adjoint operator on a Hilbert space H and assume V belongs to the Schatten p-class, with respect to the usual trace Tr. Let E be a spectral measure. A crucial estimate in [15] is of the total variation of the function that is defined on product sets by A1 × · · · × Ap → Tr E(A1 )V E(A2 )V · · · E(Ap )V . Unfortunately, the estimate result in [15] is false when p 3. In this section, we provide an example, based on Hadamard matrices, having unbounded total variation. We also give a version for finite traces. Consider the self-adjoint unitary 2 × 2 matrix 1 V2 = √ 2

1 1

1 . −1

When n = 2k , consider the self-adjoint unitary n × n matrix Vn = V2 ⊗ · · · ⊗ V2 . # $% & k times

Then each such Vn is a multiple of a Hadamard matrix. Let (ej k )1j,kn be the usual system of matrix units for Mn (C). Let En be the spectral measure on the set {1, . . . , n} taking values in Mn (C), defined by En ({j }) = ejj . The following lemma can be proved directly for all n, or first in the case n = 2 and then by observing how the total variation behaves under taking tensor products of matrices and spectral measures. Lemma 4.14. For every integer p 2, and every n that is a power of 2, the set function A1 × · · · × Ap → Trn En (A1 )Vn En (A2 )Vn · · · En (Ap )Vn has total variation np/2 .

1110


Consider the von Neumann algebra M with normal trace τ given by (M, τ ) =

∞ ' k=1

Mn(k) (C), # $% & α(k)

where α(k) > 0 and this notation indicates that M is the ∞ -direct sum of the matrix algebras Mn(k) (C), where every n(k) is a power of 2 and where τ is the trace determined by τ 0 · · ⊕ 0& ⊕In(k) ⊕ 0 ⊕ 0 ⊕ · · · = α(k). # ⊕ ·$% k−1 times

We will be interested in the following two cases: (I) M embeds in B(H), for H a separable, infinite-dimensional Hilbert space, in such a way that τ is the restriction of the usual trace Tr on B(H); this is equivalent to α(k) being an integer multiple of n(k) for all k. (II) M is a finite von (Neumann algebra and τ is normalized to take value 1 on the identity; this is equivalent to ∞ k=1 α(k) = 1. Example 4.15. Consider T = t1 Vn(1) ⊕ t2 Vn(2) ⊕ · · · ∈ M, for a bounded sequence of tk 0. Then |T | = t1 In(1) ⊕ t2 In(2) ⊕ · · · , T p =

∞

p

tk α(k).

(23)

k=1

Taking the obvious diagonal spectral measure E defined on the set {(k, j ) ∈ N2 | j n(k)} by E (k, j ) = #0 ⊕ ·$% · · ⊕ 0& ⊕ejj ⊕ 0 ⊕ 0 ⊕ · · · k−1 times

and using the result of Lemma 4.14, we find that the total variation of the set function A1 × · · · × Ap → τ E(A1 )T E(A2 )T · · · E(Ap )T is ∞ k=1

p

tk

∞ α(k) p n(k)p/2 = tk α(k)n(k)(p/2)−1 . n(k) k=1

(24)


1111

Now assuming n(k) is unbounded as k → ∞ and fixing an integer p 3, it is easy to choose values of α(k) and tk such that the p-norm in (23) is finite while the total variation (24) is infinite, in both cases (I) and (II) above. Remark 4.16. The above examples also work to show that the set function

A1 × · · · × Ap+1 → τ E(A1 )T . . . T E(Ap+1 ) has infinite total variation. 5. Main results In this section we state the main results which will be proved in the next three sections. Theorem 5.1. Let 2 < p ∈ N. Let H0 be a self-adjoint operator affiliated with M = B(H) and V a self-adjoint operator in L2 (M, τ ). Then, the following assertions hold. (i) There is a unique finite real-valued measure νp on R such that the trace formula

τ Rp,H0 ,V (f ) =

f (p) (t) dνp (t)

(25)

R

holds for f ∈ Wp . The total variation of νp is bounded by var(νp ) cp :=

1 p V 2 . p!

(ii) If, in addition, H0 is bounded, then νp is absolutely continuous and supported in [a, b] ⊃ σ (H0 ) ∪ σ (H0 + V ). The density ηp of νp can be computed recursively by ηp (t) =

τ (V p−1 ) − νp−1 (−∞, t] (p − 1)! 1 (p−2) p−2 − λ1 ,...,λp−1 (λ − t)+ dmp−1,H0 ,V (λ1 , . . . , λp−1 ), (p − 1)!

(26)

Rp−1

for a.e. t ∈ R. In this case, (25) also holds for f ∈ R. Theorem 5.2. Let p ∈ {2, 3}. Let H0 = H0∗ be an operator affiliated with a von Neumann algebra M with normal faithful semi-finite trace τ and V = V ∗ an operator in L2 (M, τ ). Then, the following assertions hold. (i) There is a unique real-valued measure νp on R such that the trace formula (25) holds for f ∈ Cc∞ (R). If H0 is bounded, then νp is finite and (25) also holds for f ∈ Wp ∪ R. (ii) The measure ν2 is absolutely continuous. If, in addition, H0 is bounded, then νp is absolutely continuous for p = 3.

1112


(iii) Assume, in addition, that V ∈ L1 (M, τ ) if p = 2 or that H0 is bounded if p = 3. Then the density ηp of νp can be computed recursively by (26). Remark 5.3. When V ∈ L1 (M, τ ), Koplienko’s spectral shift function η2 = ηH0 ,H0 +V can be represented by (26), which reduces to the known formula (see [15,26]) t η2 (t) = τ (V ) −

∞ ξH0 +V ,H0 (λ) dλ −

−∞

t =−

dτ EH0 (λ)V

t

ξH0 +V ,H0 (λ) dλ + τ EH0 (−∞, t) V .

−∞

For V in the standard Hilbert–Schmidt class, no explicit formula for ηH0 ,H0 +V is known; existence of Koplienko’s spectral shift function is proved implicitly by approximation of V by finite-rank operators. Remark 5.4. Representation (8) of Koplienko’s spectral shift function via Krein’s spectral shift function was obtained by integrating by parts in the trace formula in (6) [15,26]. When V ∈ L1 (M, τ ) and f ∈ Rb (or f ∈ R if H0 is bounded), one can see from Lemma 3.1 that d |t=0 f (H0 + tV )) = τ [Vf (H0 )], and thus, τ ( dt

τ R2,H0 ,V (f ) = τ f (H0 + V ) − f (H0 ) − Vf (H0 ) . When M is finite and H0 is bounded (so that η2 is integrable and supported in a segment containing the spectra of H0 and H0 + V ), integrating by parts in Koplienko’s trace formula in (7) gives 1 τ f (H0 + V ) − f (H0 ) − Vf (H0 ) − V 2 f (H0 ) 2 t

1 2 dt. η2 (λ) dλ + τ V EH0 (−∞, t) = f (t) − 2 R

(27)

−∞

The bound for the remainder in the approximation formula (27) is O(V 22 ) since η2 1 = and η2 0 (see [15,26] for properties of η2 ).

V 22 2

Corollary 5.5. Let H0 be a self-adjoint operator in M and V a self-adjoint operator in Lp (M, τ ), where 2 < p ∈ N if M = B(H) or p = 3 if M is a general semi-finite von Neumann algebra. Then, there exists a sequence {ηp,n }n of L∞ -functions such that

τ Rp,H0 ,V (f ) = lim

f (p) (t)ηp,n (t) dt,

n→∞ R

for f ∈ Wp .


1113

Proof. Given V ∈ Lp (M, τ ), there exists a sequence of operators Vn ∈ M, which are linear combinations of τ -finite projections in M (or just finite rank operators when M = B(H)) such that limn→∞ V − Vn p = 0. Then by Lemma 3.11 and Proposition 3.9 for f ∈ Wp ,

lim τ Rp,H0 ,Vn (f ) = τ Rp,H0 ,V (f ) .

n→∞

(28)

By Theorem 5.1 in the case of M = B(H) or Theorem 5.2 in the case of a general M, respectively, applied to the τ -Hilbert–Schmidt perturbations Vn , there exists a sequence {ηp,n }n of L∞ -functions such that

τ Rp,H0 ,Vn (f ) =

f (p) (t)ηp,n (t) dt.

(29)

R

Combining (28) and (29) completes the proof.

2

Theorem 5.6. Let τ be finite and let H0 = H0∗ be affiliated with the algebra M and V = V ∗ ∈ M. Assume that (zI − H0 )−1 and V are free in the noncommutative space (M, τ ). Then for p 3 the following assertions hold. (i) There is a unique finite real-valued measure νp on R such that the trace formula (25) holds for f ∈ Wp . (ii) If, in addition, H0 is bounded, then νp is absolutely continuous and supported in [a, b] ⊃ σ (H0 ) ∪ σ (H0 + V ). The density ηp of νp can be computed recursively by (26). In this case, (25) also holds for f ∈ R. 6. Recursive formulas for the Cauchy transform Let H0 and V be self-adjoint operators in M. Assume, in addition, that V ∈ L2 (M, τ ). In this section we investigate a measure νp = νp,H0 ,V as defined in (25) for f = fz and f (t) = t p . We derive properties of the Cauchy transform of the measure νp which will be used in Section 7 to show that the measure νp+1 = νp+1,H0 ,V satisfying (25) for f ∈ R is absolutely continuous and its density can be determined explicitly via the density of νp and an integral of a spline function against a certain multiple spectral measure. In addition, for a general trace τ , the results of this section will be used in Section 8 to prove existence of an absolutely continuous measure ν3 satisfying (25) for p = 3 and find an explicit formula for the density of ν3 . Let Gν denote the Cauchy transform of a finite measure ν: Gν (z) = R

1 dν(t), z−t

Im(z) = 0.

(30)

The goal of this section is to prove the theorem below. Theorem 6.1. Let H0 = H0∗ ∈ M and V = V ∗ ∈ L2 (M, τ ). Suppose that νp is a real-valued absolutely continuous measure satisfying (25) for f = fz and f (t) = t p . Let G : C+ → C be the analytic function satisfying

1114


p (p+1) G(p+1) (z) = −G(p) τ (zI − H0 )−1 V (zI − H0 )−1 , νp (z) − (−1) lim

Im |z|→∞

G(z) = 0.

(31) (32)

Then G(z) is the Cauchy transform of the measure νp+1 satisfying (25) for f = fz , which is absolutely continuous with the density given by ηp+1 (t) =

1 (p−1) p−1 τ V p − p!νp (−∞, t] − λ1 ,...,λp (λ − t)+ dmp,H0 ,V (λ1 , . . . , λp ) , p! Rp

for a.e. t ∈ R. Lemma 6.2. Let νp be a measure satisfying (25) for f (t) = t p . Then dνp (t) =

1 p τ V . p!

R

Proof. Applying the trace formula (25) to the polynomial f (t) = t p and applying Lemma 3.1 give

) dνp (t) = τ (H0 + V ) − p

p! R

p−1

j =0

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =p−j

* p p H0 0 V H0 1 V

p . . . V H0 j

=τ Vp .

2

Lemma 6.3. Let H0 = H0∗ ∈ M and V = V ∗ ∈ Lp (M, τ ). Let νp and νp+1 be compactly supported measures. Then νp and νp+1 satisfy (25) for f = fz if and only if

p (p+1) (z) = −G(p) τ (zI − H0 )−1 V (zI − H0 )−1 . Gν(p+1) νp (z) − (−1) p+1 Proof. The result follows immediately from Lemma 3.7 upon employing the straightforward equality

(−1)p G(p) νp (z) = τ Rp,H0 ,V (fz ) .

2

Lemma 6.3 will be used to construct an absolutely continuous measure νp+1 satisfying (25) for f = fz based on the existence of an absolutely continuous measure νp satisfying (25) for f = fz . Lemma 6.4. Let H0 = H0∗ ∈ M and V = V ∗ ∈ Lp (M, τ ). Let νp be a measure satisfying (25) for f = fz and f (t) = t p . Assume, in addition, that νp is absolutely continuous with the density ηp compactly supported in [a, b]. Assume that G : C+ → C is an analytic function satisfying (31). Then G is determined by


1 G(z) = − log(z − b) τ V p − p! + (−1)(p+1)

1 p

R

1 χ[a,b] (λ) z−λ

1115

λ ηp (t) dt dλ a

p p dz , τ (zI − H0 )−1 V

···

up to a polynomial of degree p. Proof. We note that

p p

d 1 −τ (zI − H0 )−1 V (zI − H0 )−1 = τ (zI − H0 )−1 V . dz p Then by Lemma 6.3, G(z) = −

Gνp (z) dz + (−1)(p+1)

1 p

···

p p dz . τ (zI − H0 )−1 V

(33)

By the assumption of the lemma, dνp (λ) = ηp (λ) dλ, and hence, b Gνp (z) = a

1 ηp (λ) dλ. z−λ

Integrating the latter expression by parts gives Gνp (z) =

1 z−λ

λ

b b λ 1 ηp (t) dt − ηp (t) dt dλ. (z − λ)2 a

a

a

(34)

a

By Lemma 6.2, the first summand in (34) equals

1 z−b

b ηp (t) dt = a

1 1 p τ V . z − b p!

(35)

The second summand in (34) equals

d dz

b a

1 z−λ

Combining (33)–(36) completes the proof.

λ

ηp (t) dt dλ .

a

2

(36)

1116


Theorem 6.5. Let H0 = H0∗ ∈ M and V = V ∗ ∈ L2 (M, τ ). Let [a, b] be a segment containing σ (H0 ) ∪ σ (H0 + V ). Assume that either τ is standard or p = 2. Let νp be a measure compactly supported in [a, b] and satisfying (25) for f = fz and f (t) = t p . Assume, in addition, that νp is absolutely continuous with the density ηp . Then the function G(z) =

1 p!

R

−

1 τ V p − p!νp (−∞, t] z−t (p−1) p−1 λ1 ,...,λp (λ − t)+ dmp,H0 ,V (λ1 , . . . , λp ) dt

Rp

satisfies (31) and (32). Proof. Since dνp (t) = ηp (t) dt, we have λ χ[a,b] (λ)

ηp (t) dt = νp (−∞, λ] χ[a,b] (λ).

(37)

a

By Remark 4.12, we obtain the representation p

= τ (zI − H0 )−1 V

(p−1)

λ1 ,...,λp

Rp

1 dmp,H0 ,V (λ1 , . . . , λp ). z−λ

(38)

Since σ (H0 )∪σ (H0 +V ) ⊂ [a, b], the measure mp,H0 ,V is supported in [a, b]. By Lemma 2.4(i), we can interchange the order of integration in

p p dz τ (zI − H0 )−1 V

···

=

···

(p−1) λ1 ,...,λp

[a,b]p

1 dmp,H0 ,V (λ1 , . . . , λp ) dzp z−λ

and obtain

···

p p dz τ (zI − H0 )−1 V

= [a,b]p

(p−1) λ1 ,...,λp

···

1 p dz dmp,H0 ,V (λ1 , . . . , λp ), z−λ

(39)

with a suitable choice of constants of integration on the left-hand side of (39). For a reason to become clear later, we choose the antiderivatives in (39) with real constants of integration. Since

···

1 dzp = (z − λ)p−1 log(z − λ) + αp−1 zp−1 + polp−2 (z), z−λ


1117

with polp−2 (z) a polynomial of degree p − 2 and a constant αp−1 ∈ R to be fixed later, we obtain by Proposition 2.2(4) that the expression in (39) equals 1 (p − 1)!

(p−1) λ1 ,...,λp (z − λ)p−1 log(z − λ) + αp−1 dmp,H0 ,V (λ1 , . . . , λp ).

(40)

[a,b]p

By Lemma 6.4 and (37)–(40), b G(z) = − a

+

1 νp (−∞, t] dt z−t

(−1)p+1 p!

(p−1) λ1 ,...,λp ((z − λ)p−1 log(z − λ))

[a,b]p

+ (−1) log(z − b) + αp−1 dmp,H0 ,V (λ1 , . . . , λp ). p

(41)

Now we will represent the second integral in (41) as the Cauchy transform of an absolutely continuous measure. If not all λ1 , λ2 , . . . , λp coincide, then by Proposition 2.2(6) and (5), (p−1) λ1 ,...,λp (z − λ)p−1 log(z − λ) p−1 (p−1) ∂ 1 p−2 p−1 = (z − t) (λ − t) dt log(z − t) + λ ,...,λ p 1 (p − 2)! ∂t p−1 R

1 = (p − 2)! = (−1)

p−1

(p−1) p−2 (−1)p−1 (p − 1)! log(z − t) + γp−1 λ1 ,...,λp (λ − t)+ dt

R

(p − 1)

(p−1) p−2 log(z − t)λ1 ,...,λp (λ − t)+ dt +

R

1 γp−1 , (p − 1)!

(42)

with γp−1 ∈ R. By (42) and Proposition 2.2(5), we obtain (p−1) Jλ1 ,...,λp (z) = λ1 ,...,λp (z − λ)p−1 log(z − λ) + (−1)p log(z − b) + αp−1 (p−1) p−2 log(z − t) − log(z − b) λ1 ,...,λp (λ − t)+ dt = (−1)p−1 (p − 1) R

+

1 γp−1 + αp−1 . (p − 1)!

(43) (p−1)

p−2

Since in (41) we need only to consider λ1 , . . . , λp ∈ (a, b) and λ1 ,...,λp ((λ − t)+ ) is supported in [min{λ1 , . . . , λp }, max{λ1 , . . . , λp }], we obtain that in (43) it is enough to take t ∈ [a, b]. By standard computations, for t < b, the function z → log(z − t) − log(z − b) maps C+ to C−

(44)

1118


and lim iy log(iy − t) − log(iy − b) = b − t.

y→∞

(45)

1 Let αp−1 = − (p−1)! γp−1 . Then (44) and (45) along with Proposition 2.2(5) imply that Jλ1 ,...,λp in (43) maps C+ to C± (depending on the sign of (−1)p−1 ) and limy→∞ iyJλ1 ,...,λp (iy) ∈ R. By the classical theory of analytic functions, Jλ1 ,...,λp is the Cauchy transform of a finite real-valued measure. If λ1 = λ2 = · · · = λp , then

(p−1) Jλ1 ,...,λ1 (z) = λ1 ,...,λp (z − λ)p−1 log(z − λ) + (−1)p log(z − b) + αp−1 = (−1)p−1 log(z − λ1 ) − log(z − b) + αp−1 .

(46)

By (44) and (45), the function Jλ1 ,...,λ1 is also the Cauchy transform of a finite real-valued measure. Below we show that the measure generating Jλ1 ,...,λp is absolutely continuous. If all λ1 , λ2 , . . . , λp are distinct, then by Proposition 2.2(2), (z − λk )p−1 log(z − λk ) (p−1) λ1 ,...,λp (z − λ)p−1 log(z − λ) = . j =k (λk − λj ) p

k=1

(p−1)

Since λ1 ,...,λp ((z − λ)p−1 log(z − λ)) is symmetric in λ1 , λ2 , . . . , λp , we may assume without loss of generality that λ1 < λ2 < · · · < λp . Then 1 lim Im (−1)p log(t + iε − b) + αp−1 π ε→0+ (p−1) 1 lim Im λ1 ,...,λp (t + iε − λ)p−1 log(t + iε − λ) − + π ε→0 ⎧ (p p−1 ⎪ (−1)p+1 + (−1)p k=1 (λk −t) if t < λ1 , ⎪ (λk −λj ) ⎪ j = k ⎪ ⎪ ⎨ ( p−1 p p+1 + (−1)p (λk −t) k=m j =k (λk −λj ) if λm−1 t < λm , for 2 m p, = (−1) ⎪ ⎪ ⎪ ⎪ (−1)p+1 if λp t < b, ⎪ ⎩ 0 if t b.

φ(t) := −

(47)

By Proposition 2.2(2) and (4), (−1)p+1 + (−1)p

p k=1

(λk − t)p−1 (p−1) = (−1)p+1 + (−1)p λ1 ,...,λp (λ − t)p−1 = 0, j =k (λk − λj )

(48)

and hence, φ is supported in [a, b]. Combining (47) and (48) gives (p−1) p−1 φ(t) = (−1)p+1 χ(−∞,b] (t) + (−1)p λ1 ,...,λp (λ − t)+ .

(49)


1119

Similarly, with the use of Definition 2.1 and Lemma 2.4 (ii), one can see that (49) holds when some of the values λ1 , λ2 , . . . , λp repeat. Combining (41) and (49) gives b G(z) = − a

+

1 νp (−∞, t] dt z−t

1 p!

1 (p−1) p−1 χ(−∞,b] (t) − λ1 ,...,λp (λ − t)+ dt dmp,H0 ,V (λ1 , . . . , λp ). z−t

[a,b]p R

(50) Changing the order of integration in the second integral in (50) and applying Lemma 6.2 along with the fact that νp is supported in [a, b] imply the representation G(z) = R

1 τ (V p ) χ(−∞,b] (t) − νp (−∞, t] z−t p!

= R

1 − p!

(p−1) p−1 λ1 ,...,λp (λ − t)+ dmp,H0 ,V (λ1 , . . . , λp )

Rp

1 z−t

τ (V p ) − νp (−∞, t] p!

1 − p!

dt


dt.

2

Rp

Proof of Theorem 6.1. In view of Theorem 6.5, it is enough to prove that the function p 1 (p−1) p−1 τ V − p!νp (−∞, t] − λ1 ,...,λp (λ − t)+ dmp,H0 ,V (λ1 , . . . , λp ) t → p!

(51)

Rp

is real-valued. The integral


(52)

Rp

can be written as

(p−1) p−1 λ1 ,...,λp (λ − t)+ d Re mp,H0 ,V (λ1 , . . . , λp )

Rp

+i Rp

(p−1) p−1 λ1 ,...,λp (λ − t)+ d Im mp,H0 ,V (λ1 , . . . , λp ) .

(53)

1120


It is easy to see that mp,H0 ,V (dλ1 , dλ2 , . . . , dλp−1 , dλp ) = mp,H0 ,V (dλp , dλp−1 , . . . , dλ2 , dλ1 ), and hence, Im mp,H0 ,V (dλ1 , dλ2 , . . . , dλp−1 , dλp ) = − Im mp,H0 ,V (dλp , dλp−1 , . . . , dλ2 , dλ1 ) . (54) (p−1)

p−1

Along with symmetry of the divided difference λ1 ,...,λp ((λ − t)+ ) in λ1 , . . . , λp , the equality (54) implies that the second integral in (53) equals 0, and thus (52) is real-valued. We have that ν1 and η1 are real-valued. By induction, we obtain that νp and ηp are real-valued for every p ∈ N. Therefore, (51) is real-valued. 2 7. Spectral shift functions for M = B(H) Proof of Theorem 5.1(i). Let Hx = H0 + xV . The proof of the theorem will proceed in several steps. Step 1. Assume first that H0 is bounded and f ∈ R. Let [a, b] be a segment containing σ (H0 ) ∪ σ (H0 + V ). By Corollary 4.4, the finitely additive measure defined on rectangles by

(1) mp,Hx ,V (A1 × A2 × · · · × Ap × Ap+1 ) = τ EHx (A1 )V EHx (A2 )V . . . EHx (Ap )V EHx (Ap+1 ) , with A1 , . . . , Ap+1 Borel subsets of R, extends to a countably additive measure with total variap tion not exceeding V 2 . It follows from Corollary 4.11 and Remark 4.13 that

dp (p) (1) τ f (H + xV ) = p! λ1 ,...,λp+1 (f ) dmp,Hx ,V (λ1 , λ2 , . . . , λp+1 ). 0 dx p

(55)

Rp+1

By Proposition 2.2(7), (p)

1 max f (p) (λ), p! λ∈[a,b]

λ1 ,...,λp+1 (f )

which along with (55) ensures that p d p τ max f (p) (λ). dx p f (H0 + xV ) V 2 λ∈[a,b] Applying the latter estimate to the integrand in (2) guarantees that Rp,H0 ,V (f ) is a bounded p 1 functional on the space of f (p) with the norm not exceeding p! V 2 . Therefore, there exists a measure νp,H0 ,V supported in [a, b] and of variation not exceeding

τ Rp,H0 ,V (f ) =

such that

b f (p) (t) dνp,H0 ,V (t), a

for all f ∈ R.

p 1 p! V 2

(56)


1121

Step 2. We prove the claim of the theorem for H0 bounded and f ∈ Wp . Repeating the reasoning of [10, Theorem 2.8], one extends (56) from R to the set of functions R λ → eitλ , t ∈ R, as follows. By Runge’s Theorem, there exists a sequence of rational functions rn with poles off D = {λ: |λ| 1 + H0 + V } such that rn(k) (λ) → (it)k eitλ ,

λ ∈ D, k = 0, 1, 2, . . . ,

where the convergence is understood in the uniform sense. Making use of Lemma 3.8 and passing to the limit on both sides of (56) written for f ∈ R proves (56) for f (λ) = eitλ , with the same measure νp,H0 ,V as at the previous step. Finally, applying Corollary 3.12 extends (56) to the class of f ∈ Wp , with the same measure νp,H0 ,V . Step 3. Now we extend (25) to the case of an unbounded operator H0 and f ∈ Wp . This is done similarly to [10, Lemma 2.7], with replacement of iterated operator integrals by multiple operator integrals. Let H0,n = EH0 ((−n, n))H0 and Hx,n = H0,n + xV . It follows from (2) of Theorem 1.1 that 1 Rp,H0 ,V (f ) − Rp,H0,n ,V (f ) = (p − 1)!

1

dp dp f (H ) − f (H ) (1 − x)p−1 dx. x x,n dx p dx p

0

There exists a finite Borel measure μf such that f (λ) = Lemma 3.11,

Re

itλ dμ

f (t).

On the strength of

dp dp f (Hx ) − p f (Hx,n ) τ dx p dx

(p) τ ei(s0 −s1 )Hx V . . . V eisp Hx − ei(s0 −s1 )Hx,n V . . . V eisp Hx,n dσf (s0 , . . . , sp ). = p!

(57)

Π (p)

Proposition 3.9 implies that the integrand in (57) converges to 0, and hence, the whole expression in (57) converges to 0 as n → ∞. Then applying Proposition 3.9 yields

lim τ Rp,H0 ,V (f ) − Rp,H0,n ,V (f )

n→∞

1 = lim n→∞ (p − 1)!

1 p d dp τ f (H ) − f (H ) (1 − x)p−1 dx = 0. x x,n dx p dx p

(58)

0

By the result of the previous step applied to the bounded operators H0,n , there is a sequence of measures νp,H0,n ,V of variation bounded by cp , representing the functionals Rp,H0,n ,V (f ) for f ∈ Wp . Denote by Fn the distribution function of νp,H0,n ,V . By Helly’s selection theorem, there is a subsequence {Fnk }k and a function F of variation not exceeding cp such that Fnk converges to F pointwise and in L1loc (R). The trace formula (25) for bounded operators and the convergence in (58) ensure that the measure with the distribution F satisfies (25) for f ∈ Wp . 2 Proof of Theorem 5.1(ii). It is an immediate consequence of Theorem 5.1(i) and Theorem 6.1. 2

1122


8. Spectral shift functions for an arbitrary semi-finite M Proof of Theorem 5.2 for p = 2. Due to Theorem 4.1 and Corollary 4.11, the proof of existence of Koplienko’s spectral shift function η2 for a Hilbert–Schmidt perturbation V [15, Lemma 3.3] (cf. also [6]) can be extended to the case of a τ -Hilbert–Schmidt perturbation. 2 The proof of Theorem 5.2 for p = 3 will be based on the fact (see the lemma below) that if a measure (possibly complex-valued) satisfies (25) for f = fz , then it satisfies (25) for any f ∈ Rb . Lemma 8.1. Let H0 = H0∗ be an operator affiliated with M and V = V ∗ ∈ L2 (M, τ ). Let νp , with p = 3, be a Borel measure satisfying Rp,H0 ,V (fz ) = p! R

1 dνp (t). (z − t)p+1

Then, for all f ∈ Rb , Rp,H0 ,V (f ) =

f (p) (t) dνp (t).

(59)

R

If, in addition, H0 is bounded and νp is compactly supported, then (59) holds for f ∈ R. To prove Lemma 8.1, we need a simple lemma below. Lemma 8.2. Assume that the trace formula (25) holds for f = fz with a finite measure νp . Then, ) −1

p G(p) νp (z) = (−1) τ

(zI − H0 − V )

−

p−1

−1

(zI − H0 )

j V (zI − H0 )−1

* (60)

j =0

p

= (−1)p τ (zI − H0 − V )−1 V (zI − H0 )−1 .

(61)

Proof. Differentiating the integral in (30) gives p G(p) νp (z) = (−1) p! R

1 dνp (t), (z − t)p+1

Im(z) = 0.

(62)

Applying the trace formula (25) to f = fz ensures ) −1

τ (zI − H0 − V )

−

p−1

−1

(zI − H0 )

j =0

* −1 j V (zI − H0 ) = p! R

1 dνp (t). (z − t)p+1

(63)

Comparing (63) with (62) completes the proof of (60); comparing (63) with (19) of Lemma 3.6 completes the proof of (61). 2


1123

Proof of Lemma 8.1. Step 1. Assume that H0 is bounded. We prove the claim for f a polynomial. For z ∈ C \ R, with |z| large enough, Gνp (z) =

∞

z−(k+1)

k=0

t k dνp (t), R

and hence, (−1)p G(p) νp (z) =

∞

z−(k+p+1) (k + 1)(k + 2) . . . (k + p)

k=0

t k dνp (t).

(64)

R

On the other hand, (−1)p G(p) νp (z) ) −1

= τ (zI − H0 − V )

−

p−1

−1

(zI − H0 )

j V (zI − H0 )−1

*

j =0

) * p−1 1 H0 −1 H0 −1 j H0 + V −1 1 I− V I− I− =τ − . z z z z zj +1

(65)

j =0

Employing the power series expansion in (65) gives ) p

(−1)

G(p) νp (z) = τ

∞ 1 H0 + V m z z m=0

−

p−1 ∞ j =0 i=0

) =τ

∞

1 zj +1

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =i

H0 z

k0 k1 kj * H0 H0 V V ...V z z

z−(m+1) (H0 + V )m

m=0

−

p−1 ∞

z

j =0 i=0

*

−(j +1)

z

−i

k H0 0 V H0k1 V

k . . . V H0 j

.

(66)

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =i

By expanding (H0 + V )m one can see that τ

) p−1 m=0

= 0.

z−(m+1) (H0 + V )m −

p−1 p−1−j j =0

i=0

z−(j +1)

* k z−i H0 0 V H0k1 V

k . . . V H0 j

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =i

(67)

1124


Subtracting (67) from (66) yields (−1)p G(p) νp (z) ) ∞ p−1 ∞ =τ z−(m+1) (H0 + V )m − z−(i+j +1) j =0 i=p−j

m=p

) =τ

∞

z

−(m+1)

(H0 + V ) − m

m=p

*

k H0 0 V H0k1 V

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =i

p−1

j =0

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =m−j

k . . . V H0 j

*

k H0 0 V H0k1 V

k . . . V H0 j

(68)

.

By the continuity of the trace τ , (68) can be rewritten as (−1)p G(p) νp (z) =

∞

)

z−(m+1) τ (H0 + V )m −

m=p

=

∞

j =0

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =m−j

) z

−(k+p+1)

*

p−1

τ (H0 + V )

k+p

k=0

−

k H0 0 V H0k1 V

k . . . V H0 j

*

p−1

j =0

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =k+p−j

k H0 0 V H0k1 V

k . . . V H0 j

.

(69)

(p)

By comparing the representations for (−1)p Gνp (z) of (64) and (69), we obtain that for any k ∈ {0} ∪ N, ) τ (H0 + V )

k+p

−

p−1

j =0

k0 ,k1 ,...,kj 0 k0 +k1 +···+kj =k+p−j

* k H0 0 V H0k1 V

k . . . V H0 j

= (k + 1)(k + 2) . . . (k + p)

t k dνp (t), R

along with Lemma 3.1 proving the trace formula (25) for all polynomials. We note that under the assumptions of Step 1, p can be any natural number. Step 2. Assume that f ∈ Rb , with H0 not necessarily bounded. It is enough to prove the 1 statement for f (t) = (z−t) k+1 , k ∈ {0} ∪ N. Applying Lemma 3.7 gives p! R

1 dνp (t) = (p − 1)! (z − t)p+1

R

p−1 1 dνp−1 (t) − τ (zI − H0 )−1 V (zI − H0 )−1 . p (z − t) (70)

Differentiating (70) k times with respect to z gives


1125

1 dνp (t) (z − t)p+1+k

(−1) (p + k)! k

R

p−1 1 d k dν (t) − τ (zI − H0 )−1 V (zI − H0 )−1 . p−1 (z − t)p+k dzk

= (−1) (p − 1 + k)! k

R

(71) Dividing by (−1)k k! on both sides of (71) implies (p + k)! k!

R

=

1 dνp (t) (z − t)p+1+k

(p − 1 + k)! k!

p−1 1 (−1)k d k dν (t) − τ (zI − H0 )−1 V (zI − H0 )−1 . p−1 p+k k (z − t) k! dz

R

(72) Making use of the representation Rp−1,H0 ,V

1 (z − t)k+1

=

(p − 1 + k)! k!

R

1 dνp−1 (t) (z − t)p+k

(see Theorem 5.2 for Koplienko’s spectral shift function) and Lemma 3.2 converts (72) to (p + k)! k!

R

1 dνp (t) (z − t)p+1+k

= Rp−1,H0 ,V

1 (z − t)k+1

2 1 d −k−1 − τ . (zI − H − xV ) 0 2 dx 2 x=0

(73)

2 1 d −k−1 − τ (zI − H . − xV ) 0 2 dx 2 x=0

(74)

By (1), Rp,H0 ,V

1 (z − t)k+1

= Rp−1,H0 ,V

1 (z − t)k+1

Comparing (73) and (74) completes the proof of (25) for f (t) =

1 . (z−t)k+1

2

Proof of Theorem 5.2 for p = 3. When H0 is bounded, Lemma 8.1 and Theorem 6.1 prove the theorem for f ∈ R. Repeating the argument of Step 2 from the proof of Theorem 5.1(i) for τ the standard trace extends (ii) and (iii) of Theorem 5.2 to f ∈ Wp for H0 bounded. Repeating the argument of Step 3 from the proof of Theorem 5.1(i) on each segment of R extends (i) to f ∈ Cc∞ (R) for H0 unbounded. 2

1126


Proof of Theorem 5.6. (i) Due to Theorem 4.5, there exists a bounded measure νp satisfying the trace formula (25) for f ∈ Wp . The proof repeats the proof of Theorem 5.1 for the standard trace. (ii) Using the moment-cumulant formula (see [27, Theorem 2.17]), we have

p−1

= τ (zI − H0 )−1 V

kK(π) [V , . . . , V ]

τ (zI − H0 )−|Bj | ,

(75)

j =1

π={B1 ,...,B }∈NC(p−1)

where (see the proof of Theorem 4.5 for a bit of explanation, or [27, Theorem 2.17] for a thorough description) kK(π) [V , . . . , V ] is a polynomial of τ (V ), τ (V 2 ), . . . , τ (V p−1 ). Since for b 1,

τ (zI − H0 )−b =

1 τ EH0 (dλ1 ) · · · EH0 (dλ1 ) , (z − λ1 ) . . . (z − λb )

Rb

we have

τ (zI − H0 )−b = j =1

Rp−1

1 dγp−1,π (λ1 , . . . , λp ), (z − λ1 ) · · · (z − λp )

where γp−1,π is the measure described at (21). Combining (75) and (20) gives p−1

τ (zI − H0 )−1 V =

(p−2) λ1 ,...,λp−1

Rp−1

1 dmp−1,H0 ,V (λ1 , . . . , λp−1 ). z−λ

Following the lines in the proof of Theorem 6.1 completes the proof of the absolute continuity of νp and repeating the proof of Lemma 8.1, Step 1, proves (25) for f a polynomial. 2 9. Spectral shift functions via basic splines We represent the density of the measure νp provided by Theorem 5.1 as an integral of a basic spline against a certain multiple spectral measure when H0 and V are matrices. In addition, we show that existence of Krein’s spectral shift function can be derived from the representation of the Cauchy transform via basic splines when M is finite. The representation of the Cauchy transform via basic splines, in its turn, follows from the double integral representation of f (H0 + V ) − f (H0 ). Lemma 9.1. Let dim(H) < ∞ and H0 = H0∗ , V = V ∗ ∈ M = B(H). Then the Cauchy transform of the measure νp satisfying (25) equals G(p) νp (z) =

1 dp (p) p p−1 (−1) (z − λ) λ1 ,...,λp+1 log(z − λ) dzp (p − 1)! Rp+1

*

(2) dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 )

,

Im(z) = 0.


1127

Proof. Upon applying Remark 4.12 and Lemma 8.2, we obtain p G(p) νp (z) = (−1)

(p) λ1 ,...,λp+1

Rp+1

1 (2) dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 ). z−λ

By Lemma 2.4, one of the antiderivatives of order p of the function (p) z → λ1 ,...,λp+1

1 z−λ

equals (p) λ1 ,...,λp+1

1 p−1 p−1 , log(z − λ) − cp−1 (z − λ) (z − λ) (p − 1)!

where cp−1 is a constant. Applying Proposition 2.2(4) gives (p) λ1 ,...,λp+1 cp−1 (z − λ)p−1 = 0, 2

completing the proof of the lemma.

Lemma 9.2. Let Dp+1 = {(λ1 , λ2 , . . . , λp+1 ): λ1 = λ2 = · · · = λp+1 ∈ R}. Then, for any (λ1 , λ2 , . . . , λp+1 ) ∈ Rp+1 \ Dp+1 and z ∈ C \ R, 1 p−1 (z − λ) log(z − λ) (p − 1)! 1 (−1)p (p) p−1 = λ1 ,...,λp+1 (λ − t)+ dt. (p − 1)! z−t

(p) λ1 ,...,λp+1

R

Proof. By Proposition 2.2(6),

1 (z − λ)p−1 log(z − λ) (p − 1)! p 1 ∂ 1 (p) p−1 p−1 (z − t) = log(z − t) λ1 ,...,λp+1 (λ − t)+ dt p (p − 1)! ∂t (p − 1)!

(p)

λ1 ,...,λp+1

R

=

1 (p − 1)!

R

(−1)p (p) p−1 λ1 ,...,λp+1 (λ − t)+ dt. z−t

2

Theorem 9.3. Let dim(H) < ∞ and H0 = H0∗ , V = V ∗ ∈ M = B(H). Then the Cauchy transform of the measure νp satisfying (25) equals

1128


Gνp (z) = R

1 z−t

1 (p − 1)!

(p) p−1 (2) λ1 ,...,λp+1 (λ − t)+ dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 )

dt.

Rp+1 \Dp+1

Proof. By Lemmas 9.1 and 9.2, Gνp (z) = polp (z) 1 + (p − 1)! +

1 (p − 1)!

Rp+1 \Dp+1

Dp+1

R

1 (p) p−1 (2) (λ − t)+ dt dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 ) z − t λ1 ,...,λp+1

1 (2) dmp,H0 ,V (λ, λ, . . . , λ), z−λ

(76)

where polp (z) is a polynomial of degree p. As stated in Proposition 2.2(5), the basic spline (p)

p−1

λ1 ,...,λp+1 ((λ − t)+ ) is non-negative and integrable, with the L1 -norm equal to 1/p. By Corol(2)

lary 4.4, the measure mp,H0 ,V has bounded variation. On one hand, it guarantees that the first integral in (76) is O(1/ Im(z)) as Im(z) → +∞. On the other hand, it allows to change the order of integration in the first integral in (76). By Lemma 4.8, the second integral in (76) equals 0. Comparing the asymptotics of Gνp (z) and the integrals in (76) as Im(z) → +∞ implies that polp (z) = 0, completing the proof of the theorem. 2 Corollary 9.4. Let dim(H) < ∞ and H0 = H0∗ , V = V ∗ ∈ M = B(H). Then the density of the measure νp satisfying (25) equals ηp (t) =

1 (p − 1)!

(p) p−1 (2) λ1 ,...,λp+1 (λ − t)+ dmp,H0 ,V (λ1 , λ2 , . . . , λp+1 ),

Rp+1 \Dp+1

for a.e. t ∈ R. Proof. By Theorem 9.3, the Cauchy transforms of νp and ηp (t)dt coincide. This implies (see Step 1 in the proof of Lemma 8.1) that the functionals given by νp and ηp (t)dt coincide on the polynomials defined on [a, b], where [a, b] contains the spectra of H0 and H0 + V . Hence, dνp = ηp (t)dt. 2 Below, we prove absolute continuity of ν1 by techniques different from those of [16]. Theorem 9.5. Let τ be finite. Let H0 = H0∗ be an operator affiliated with M and V = V ∗ ∈ M. The trace formula (25) with p = 1 holds for every f ∈ W1 , with ν1 absolutely continuous. The density η1 of ν1 is given by the formula


1 (2) χ(min{λ,μ},max{λ,μ}) (t) dm1,H0 ,V (λ, μ), |μ − λ|

η1 (t) = R2 \D2

1129

(77)

for a.e. t ∈ R. If, in addition, H0 is bounded, then (25) holds for f ∈ R. Proof. Repeating the argument in the proof of Theorem 9.3 leads to the formula (76). By (2) Lemma 4.6, the measure m1,H0 ,V is real-valued. Then by Lemma 4.8 and the Poisson inversion, for any x ∈ R, lim Im

ε→0+

D2

1 (2) dm1,H0 ,V (λ, λ) = 0, x + iε − λ

proving Krein’s trace formula for f = fz with η1 given by (77). Adjusting the argument in the proof of Lemma 8.1, Step 2, extends (25) to f ∈ Rb . Repeating the argument in the proof of [29, Lemma 8.3.2] extends the result of the theorem from f ∈ R to f ∈ W1 with the same absolutely continuous measure dν1 (t) = η1 (t) dt. 2 10. Higher order spectral averaging formulas Theorem 10.1. Assume that H0 = H0∗ ∈ M and either τ is standard or p = 2. Let V ∈ L2 (M, τ ). Then the measure 1

p

dx (1 − x)p−1 τ EH0 +xV (dt)V

0

is absolutely continuous with the density equal to ηp (t)(p − 1)! 1

(p) p−1 (1) λ1 ,...,λp+1 (λ − t)+ dmp,H0 +xV ,V (λ1 , . . . , λp+1 ) dx.

(1 − x)p−1

−p

Rp+1 \Dp+1

0

Proof. Let [a, b] ⊃ σ (H0 ) ∪ σ (H0 + V ). Then by Theorem 1.1 (2) and Remark 4.13,

τ Rp,H0 ,V (f ) 1 = (p − 1)! 1 = (p − 1)!

1 (1 − x)

p−1

dp τ f (H0 + xV ) dx dx p

0

1 (1 − x)p−1 p! 0

Rp+1

(p)

(1)

λ1 ,...,λp+1 (f )dmp,H0 +xV ,V (λ1 , . . . , λp+1 ) dx,

1130


for f ∈ Cc∞ (R) such that f |[a,b] coincides with a polynomial. Applying Proposition 2.2(6) and then changing the order of integration yield

τ Rp,H0 ,V (f ) p = (p − 1)!

1 (1 − x)p−1 0

× Rp+1 \Dp+1

1 + (p − 1)! =

f (p) (t) R

(p) p−1 f (p) (t)λ1 ,...,λp+1 (λ − t)+ dt

1

f (p) (λ) dmp,H0 +xV ,V (λ, . . . , λ) dx.

Dp+1

0

p (p − 1)!

(p) p−1 (1) λ1 ,...,λp+1 (λ − t)+ dmp,H0 +xV ,V (λ1 , . . . , λp+1 ) dx

(1 − x)

dt

Rp+1 \Dp+1

0

f

(1)

(1 − x)

p−1

p−1

+

(1)

dmp,H0 +xV ,V (λ1 , . . . , λp+1 ) dx

R

1 ×

(p)

R

1 (t) (p − 1)!

1

p

dx. (1 − x)p−1 τ EH0 +xV (dt)V

(78)

0

Along with Theorem 5.1 in the case of M = B(H) or Theorem 5.2 in the case of a general M, respectively, (78) implies that f (p) (t) R

1 (p − 1)!

1

p

dx (1 − x)p−1 τ EH0 +xV (dt)V

0

=

f (p) (t)ηp (t) dt − R

R

×

p f (p) (t) (p − 1)!

1 (1 − x)p−1 0

(p) p−1 (1) λ1 ,...,λp+1 (λ − t)+ dmp,H0 +xV ,V (λ1 , . . . , λp+1 ) dx

dt,

(79)

Rp+1 \Dp+1

from which the statement of the theorem follows.

2

Remark 10.2. The assertion of Theorem 10.1 remains true if τ is finite, H0 and V are free in (M, τ ), and p 2.


1131

Remark 10.3. The argument in the proof of Theorem 10.1 can be repeated for p = 1, provided (1) V ∈ L1 (M, τ ). Since m1,H0 +xV ,V (A1 × A2 ) = τ [EH0 +xV (A1 ∩ A2 )V ], for A1 , A2 ∈ R, one has (1)

that m1,H0 +xV ,V (Rp+1 \ Dp+1 ) = 0. Therefore, (78) converts to

τ f (H0 + V ) − f (H0 ) =

1

f (t)τ EH0 +xV (dt)V dx

0 R

=

1

f (t) R

τ EH0 +xV (dt)V dx.

0

Along with Krein’s trace formula the latter implies that

1 0

τ [EH0 +xV (dt)V ] dx = η1 (t) dt.

Acknowledgments The authors thank Bojan Popov for helpful conversations about splines. References [1] N.A. Azamov, A.L. Carey, P.G. Dodds, F.A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math., in press. [2] N.A. Azamov, P.G. Dodds, F.A. Sukochev, The Krein spectral shift function in semifinite von Neumann algebras, Integral Equations Operator Theory 55 (2006) 347–362. [3] M.Sh. Birman, Spectral shift function and double operator integrals, in: Linear and Complex Analysis Problem, Book 3, in: V.P. Havin, et al. (Eds.), Lecture Notes in Math., vol. 1573, Springer-Verlag, Berlin, 1994, pp. 272–273. [4] M.Sh. Birman, M.Z. Solomyak, Remarks on the spectral shift function, Zap. Nauchn. Sem. LOMI 27 (1972) 33–46 (in Russian); translation in: J. Soviet Math. 3 (1975) 408–419. [5] M.Sh. Birman, M.Z. Solomyak, Tensor product of a finite number of spectral measures is always a spectral measure, Integral Equations Operator Theory 24 (1996) 179–187. [6] J.M. Bouclet, Trace formulae for relatively Hilbert–Schmidt perturbations, Asymptot. Anal. 32 (2002) 257–291. [7] R.W. Carey, J.D. Pincus, Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977) 153–218. [8] Yu.L. Daletskii, S.G. Krein, Integration and differentiation of functions of Hermitian operators and application to the theory perturbations, in: Trudy Sem. Funktsion. Anal., vol. 1, Voronezh Gos. Univ., 1956, pp. 81–105 (in Russian). [9] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss., vol. 303, Springer-Verlag, Berlin, 1993. [10] M. Dostani`c, Trace formula for nonnuclear perturbations of selfadjoint operators, Publ. Inst. Math. 54 (68) (1993) 71–79. [11] F. Gesztesy, K.A. Makarov, S.N. Naboko, The spectral shift operator, in: J. Dittrich, P. Exner, M. Tater (Eds.), Mathematical Results in Quantum Mechanics, in: Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 59–90. [12] F. Gesztesy, A. Pushnitski, B. Simon, On the Koplienko spectral shift function, I. Basics, Zh. Mat. Fiz. Anal. Geom. 4 (1) (2008) 63–107. [13] U. Haagerup, H. Schultz, Invariant subspaces for operators in a general II 1 -factor, preprint arXiv:math/0611256. [14] S. Karni, E. Mezbach, On the extension of bimeasures, J. Anal. Math. 55 (1990) 1–16. [15] L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62–71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735–743. [16] M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian). [17] I.M. Lifshits, On a problem of the theory of perturbations connected with quantum statistics, Uspekhi Mat. Nauk 7 (1952) 171–180 (in Russian).

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[18] K.A. Makarov, A. Skripka, Some applications of the perturbation determinant in finite von Neumann algebras, Canad. J. Math., in press. [19] B. de Pagter, F.A. Sukochev, Differentiation of operator functions in non-commutative Lp -spaces, J. Funct. Anal. 212 (1) (2004) 28–75. [20] B.S. Pavlov, On multidimensional operator integrals, in: Problems of Mathematial Analysis, vol. 2, Leningrad State Univ., 1969, pp. 99–121 (in Russian). [21] V.V. Peller, Hankel operators in the perturbation theory of unbounded self-adjoint operators, in: Analysis and Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529–544. [22] V.V. Peller, An extension of the Koplienko–Neidhardt trace formulae, J. Funct. Anal. 221 (2005) 456–481. [23] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 223 (2006) 515–544. [24] J.T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach Science Publishers, New York, London, Paris, 1969. [25] B. Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc. 126 (5) (1998) 1409–1413. [26] A. Skripka, Trace inequalities and spectral shift, Oper. Matrices, in press. [27] R. Speicher, Free calculus, in: Quantum Probability Communications, vol. XII, Grenoble, 1998, World Sci. Publ., River Edge, NJ, 2003, pp. 209–235. [28] D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monogr. Ser., vol. 1, Amer. Math. Soc., 1992. [29] D.R. Yafaev, Mathematical Scattering Theory: General Theory, Amer. Math. Soc., Providence, RI, 1992.


A duality principle for groups Dorin Dutkay a , Deguang Han a,∗ , David Larson b a Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States b Department of Mathematics, Texas A&M University, College Station, TX, United States

Received 16 February 2009; accepted 9 March 2009 Available online 24 March 2009 Communicated by D. Voiculescu

Abstract The duality principle for Gabor frames states that a Gabor sequence obtained by a time–frequency lattice is a frame for L2 (Rd ) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II1 factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras. © 2009 Elsevier Inc. All rights reserved. Keywords: Group representations; Frame vectors; Bessel vectors; Duality principle; Von Neumann algebras; II1 factors

1. Introduction Motivated by the duality principle for Gabor representations in time–frequency analysis we establish a general duality theory for frame representations of infinite countable groups, and build its connection with the classification problem [2] of II1 factors. We start by recalling some notations and definitions about frames. * Corresponding author.

E-mail addresses: [email protected] (D. Dutkay), [email protected] (D. Han), [email protected] (D. Larson). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.007

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D. Dutkay et al. / Journal of Functional Analysis 257 (2009) 1133–1143

A frame [6] for a Hilbert space H is a sequence {xn } in H with the property that there exist positive constants A, B > 0 such that Ax2

x, xn 2 Bx2

(1.1)

g∈G

holds for every x ∈ H . A tight frame refers to the case when A = B, and a Parseval frame refers to the case when A = B = 1. In the case that (1.1) hold only for all x ∈ span{xn }, then we say that {xn } is a frame sequence, i.e., it is a frame for its closed linear span. If we only require the right-hand side of the inequality (1.1), then {xn } is called a Bessel sequence. One of the well studied classes of frames is the class of Gabor (or Weyl-Heisenberg) frames: Let K = AZd and L = BZd be two full-rank lattices in Rd , and let g ∈ L2 (Rd ) and Λ = L × K. Then the Gabor (or Weyl–Heisenberg) family is the following collection of functions in L2 (Rd ): G(g, Λ) = G(g, L, K) := e2πi,x g(x − κ) ∈ L, κ ∈ K . For convenience, we write gλ = gκ, = e2πi,x g(x − κ), where λ = (κ, ). If E and Tκ are the modulation and translation unitary operators defined by E f (x) = e2πi,x f (x) and Tκ f (x) = f (x − κ) for all f ∈ L2 (Rd ). Then we have gκ, = E Tκ g. The well-known Ron–Shen duality principle states that a Gabor sequence G(g, Λ) is a frame (respectively, Parseval frame) for L2 (Rd ) if and only if the adjoint Gabor sequence G(g, Λo ) is a Riesz sequence (respectively, orthonormal sequence), where Λo = (B t )−1 Zd × (At )−1 Zd is the adjoint lattice of Λ. Gabor frames can be viewed as frames obtained by projective unitary representations of the abelian group Zd × Zd . Let Λ = AZd × BZd with A and B being d × d invertible real matrices. The Gabor representation πΛ defined by (m, n) → EAm TBn is not necessarily a unitary representation of the group Zd × Zd . But it is a projective unitary representation of Zd × Zd . Recall (cf. [25]) that a projective unitary representation π for a countable group G is a mapping g → π(g) from G into the group U (H ) of all the unitary operators on a separable Hilbert space H such that π(g)π(h) = μ(g, h)π(gh) for all g, h ∈ G, where μ(g, h) is a scalar-valued function on G × G taking values in the circle group T. This function μ(g, h) is then called a multiplier or 2-cocycle of π . In this case we also say that π is a μ-projective unitary representation. It is clear from the definition that we have (i) μ(g1 , g2 g3 )μ(g2 , g3 ) = μ(g1 g2 , g3 )μ(g1 , g2 ) for all g1 , g2 , g3 ∈ G, (ii) μ(g, e) = μ(e, g) = 1 for all g ∈ G, where e denotes the group unit of G. Any function μ : G × G → T satisfying (i)–(ii) above will be called a multiplier for G. It follows from (i) and (ii) that we also have (iii) μ(g, g −1 ) = μ(g −1 , g) holds for all g ∈ G.


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Examples of projective unitary representations include unitary group representations (i.e., μ ≡ 1) and the Gabor representations in time–frequency analysis. Similar to the group unitary representation case, the left and right regular projective representations with a prescribed multiplier μ for G can be defined by λg χh = μ(g, h)χgh , and

ρg χh = μ h, g −1 χhg −1 ,

h ∈ G,

h ∈ G,

where {χg : g ∈ G} is the standard orthonormal basis for 2 (G). Clearly, λg and rg are unitary operators on 2 (G). Moreover, λ is a μ-projective unitary representation of G with multiplier μ(g, h) and ρ is a projective unitary representation of G with multiplier μ(g, h). The representations λ and ρ are called the left regular μ-projective representation and the right regular μ-projective representation, respectively, of G. Let L and R be the von Neumann algebras generated by λ and ρ, respectively. It is known (cf. [9]), similarly to the case for regular group representations, that both R and L are finite von Neumann algebras, and that R is the commutant of L. Moreover, if for each e = u ∈ G, either {vuv −1 : v ∈ G} or {μ(vuv −1 , v)μ(v, u): v ∈ G} is an infinite set, then both L and R are factor von Neumann algebras. Notations. In this paper for a subset M of a Hilbert space H and a subset A of B(H ) of all the bounded linear operators on H , we will use [M] to denote the closed linear span of M, and A to denote the commutant {T ∈ B(H ): T A = AY, ∀A ∈ A} of A. So we have L = λ(G)

= ρ(G) and R = ρ(G)

= λ(G) . We also use M N to denote two ∗-isomorphic von Neumann algebras M and N . For any projection P ∈ λ(G) (where λ is the left regular projective representation) we use λ|P to denote the restriction of λ to P H . We refer to [17] for any other notations and terminologies about von Neumann algebras used in this paper. Given a projective unitary representation π of a countable group G on a Hilbert space H , a vector ξ ∈ H is called a complete frame vector (resp. complete tight frame vector, complete Parseval frame vector) for π if {π(g)ξ }g∈G (here we view this as a sequence indexed by G) is a frame (resp. tight frame, Parseval frame) for the whole Hilbert space H , and is just called a frame vector (resp. tight frame vector, Parseval frame vector) for π if {π(g)ξ }g∈G is a frame sequence (resp. tight frame sequence, Parseval frame sequence). A Bessel vector for π is a vector ξ ∈ H such that {π(g)ξ }g∈G is Bessel. We will use Bπ to denote the set of all the Bessel vectors of π . For x ∈ H , let Θx be the analysis operator for {π(g)x}g∈G (see Section 2). It is useful to note that if ξ and η are Bessel vectors for π , then Θη∗ Θξ commutes with π(G). Thus, if ξ is a −1/2

complete frame vector for π , then η := Sξ ξ is a complete Parseval frame vector for π , where Sξ = Θξ∗ Θξ and is called the frame operator for ξ (or Bessel operator if ξ is a Bessel vector). Hence, a projective unitary representation has a complete frame vector if and only if it has a complete Parseval frame vector. In this paper the terminology frame representation refers to a projective unitary representation that admits a complete frame vector. Proposition 1.1. (See [9,23].) Let π be a projective unitary representation π of a countable group G on a Hilbert space H . Then π is frame representation if and only if π is unitarily equivalent to a subrepresentation of the left regular projective unitary representation of G. Consequently, if π is frame representation, then both π(G) and π(G)

are finite von Neumann algebras.

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The duality principle for Gabor frames was independent and essentially simultaneous discovered by Daubechies, H. Landau, and Z. Landau [3], Janssen [16], and Ron and Shen [24], and the techniques used in these three articles to prove the duality principle are completely different. We refer to [15] for more details about this principle and its important applications. For Gabor representations, let Λo be the adjoint lattice of a lattice Λ. The well-known density theorem (cf. [14]) implies that one of two projective unitary representations πΛ and πΛo for the group G = Zd × Zd must be a frame representation and the other admits a Riesz vector. So we can always assume that πΛ is a frame representation of Zd × Zd and hence πΛ(o) admits a Riesz vector. Moreover, we also have πΛ (G) = πΛ(o) (G)

, and both representations share the same Bessel vectors. Rephrasing the duality principle in terms of Gabor representations, it states that {πΛ (m, n)g}m,n∈Zd is a frame for L2 (Rd ) if and only if {πΛ(o) (m, n)g}m,n∈Zd is a Riesz sequence. Our first main result reveals that this duality principle is not accidental and in fact it is a general principle for any commutant pairs of projective unitary representations. Definition 1.1. Two projective unitary representations π and σ of a countable group G on the same Hilbert space H are called a commutant pair if π(G) = σ (G)

. Theorem 1.2. Let π be a frame representation and (π, σ ) be a commutant pair of projective unitary representations of G on H such that π has a complete frame vector which is also a Bessel vector for σ . Then (i) {π(g)ξ }g∈G is a frame sequence if and only if {σ (g)ξ }g∈G is a frame sequence, (ii) if, in addition, assuming that σ admits a Riesz sequence, then {π(g)ξ }g∈G is a frame (respectively, a Parseval frame) for H if and only if {σ (g)ξ }g∈G is a Riesz sequence (respectively, an orthonormal sequence). For a frame representation π , we will call (π, σ ) a dual pair if (π, σ ) is a commutant pair such that π has a complete frame vector which is also a Bessel vector for σ , and σ admits a Riesz sequence. We remark that this duality property is not symmetric for π and σ since π is assumed to be a frame representation and σ in general is not. Theorem 1.2 naturally leads to the following existence problem: Problem 1. Let G be a infinite countable group and μ be a multiplier for G. Does every μprojective frame representation π of G admit a dual pair (π, σ )? While we maybe able to answer this problem for some special classes of groups, this is in general open due to its connections (See Theorem 1.4) with the classification problem of II1 factors which is one of the big problems in von Neumann algebra theory. It has been a longstanding unsolved problem to decide whether the factors obtained from the free groups with n and m generators respectively are isomorphic if n is not equal to m with both n, m > 1. This problem was one of the inspirations for Voiculescu’s theory of free probability. Recall that the fundamental group F (M) of a type II1 factor M is an invariant that was considered by Murray and von Neumann in connection with their notion of continuous dimension in [18], where they proved that F (M) = R∗+ when M is isomorphic to a hyperfinite type II1 factor, and more generally when it splits off such a factor. For free groups Fn of n-generators, by using Voiculescu’s free probability theory [26], Radulescu [21,22] showed that the fundamental group F (M) = R∗+ for


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M = λ(F∞ ) . But the problem of calculating F (M) for M = λ(Fn ) with 2 n < ∞ remains a central open problem in the classification of II1 factors, and it can be rephrased as: Problem 2. Let Fn (n > 1) be the free group of n-generators and P ∈ λ(Fn ) is a nontrivial projection. Is λ(Fn ) ∗-isomorphic to P λ(Fn ) P ? It was proved independently by Dykema [7] and Radulescu [22] that either all the von Neumann algebras P λ(Fn ) P (0 = P ∈ λ(Fn ) ) are ∗-isomorphic, or no two of them are ∗isomorphic. Our second main result established the equivalence of these two problems for free groups. Theorem 1.3. Let π = λ|P be a subrepresentation of the left regular representation of an ICC (infinite conjugate class) group G and P ∈ λ(G) be a projection. Then the following are equivalent: (i) λ(G) and P λ(G) P are isomorphic von Neumann algebras; (ii) there exists a group representation σ such that (π, σ ) form a dual pair. The above theorem implies that the answer to Problem 1 is negative in general, but is affirmative for amenable ICC groups. Theorem 1.4. Let G be a countable group and λ be its left regular unitary representation (i.e. μ ≡ 1). Then we have (i) If G is either an abelian group or an amenable ICC group, then for every projection 0 = P ∈ λ(G) , there exists a unitary representation σ of G such that (λ|P , σ ) is a dual pair. (ii) There exist ICC groups (e.g., G = Z2 SL(2, Z)), such that none of the nontrivial subrepresentations λ|P admits a dual pair. We will give the proof of Theorem 1.2 in Section 2 and the proof requires some resent work by the present authors including the results on parameterizations and dilations of frame vectors [10–12], and some result results on the “duality properties” for π -orthogonal and π -weakly equivalent vectors [13]. The proofs for Theorems 1.3 and 1.4 will provided in Section 3, and additionally we will also discuss some concrete examples including the subspace duality principle for Gabor representations. 2. The duality principle We need a series of preparations in order to prove Theorem 1.2. For any projective representation π of a countable group G on a Hilbert space H and x ∈ H , the analysis operator Θx for x from D(Θx )(⊆ H ) to 2 (G) is defined by Θx (y) =

y, π(g)x χg ,

g∈G

where D(Θx ) = {y ∈ H : g∈G |y, π(g)x|2 < ∞} is the domain space of Θx . Clearly, Bπ ⊆ D(Θx ) holds for every x ∈ H . In the case that Bπ is dense in H , we have that Θx is a densely de-

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fined and closable linear operator from Bπ to 2 (G) (cf. [8]). Moreover, x ∈ Bπ if and only if Θx is a bounded linear operator on H , which in turn is equivalent to the condition that D(Θx ) = H . Lemma 2.1. (See [8].) Let π be a projective representation of a countable group G on a Hilbert space H such that Bπ is dense in H . Then for any x ∈ H , there exists ξ ∈ Bπ such that (i) {π(g)ξ : g ∈ G} is a Parseval frame for [π(G)x]; (ii) Θξ (H ) = [Θx (Bπ )]. Lemma 2.2. Assume that π is a projective representation of a countable group G on a Hilbert space H such that π admits a Riesz sequence and Bπ is dense in H . If [Θξ (H )] = 2 (G), then there exists 0 = x ∈ H such that [Θx (H )] ⊥ [Θξ (H )]. Proof. Assume that {π(g)η}g∈G is a Riesz sequence. Then we have that Θη (H ) = 2 (G) and Θη is invertible when restricted to [π(G)η]. Let P be the orthogonal projection from 2 (G) onto [Θξ (H )]. Then P ∈ λ(G) and P = I . Let x = θη−1 P ⊥ χe . Then x = 0 and [Θx (H )] ⊥ [Θξ (H )]. 2 Lemma 2.3. (See [8,11].) Assume that π is a projective representation of a countable group G on a Hilbert space H such that π admits a complete frame vector ξ . If {π(g)η}g∈G is a frame sequence, then there exists a vector h ∈ H such that η and h are π -orthogonal and {π(g)(η + h)}g∈G is a frame for H . Two other concepts are needed. Definition 2.1. Suppose π is a projective unitary representation of a countable group G on a separable Hilbert space H such that the set Bπ of Bessel vectors for π is dense in H . We will say that two vectors x and y in H are π -orthogonal if the ranges of Θx and Θy are orthogonal, and that they are π -weakly equivalent if the closures of the ranges of Θx and Θy are the same. The following result obtained in [13] characterizes the π -orthogonality and π -weakly equivalence in terms of the commutant of π(G). Lemma 2.4. Let π be a projective representation of a countable group G on a Hilbert space H such that Bπ is dense in H , and let x, y ∈ H . Then (i) x and y are π -orthogonal if and only if [π(G) x] ⊥ [π(G) y] (or equivalently, x ⊥ π(G) y); (ii) x and y are π -weakly equivalent if and only if [π(G) x] = [π(G) y]. We also need the following parameterization result [10–12]. Lemma 2.5. Let π be a projective representation of a countable group G on a Hilbert space H and {π(g)ξ }g∈G is a Parseval frame for H . Then (i) {π(g)η}g∈G is a Parseval frame for H if and only if there is a unitary operator U ∈ π(G)

such that η = U ξ ;


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(ii) {π(g)η}g∈G is a frame for H if and only if there is an invertible operator U ∈ π(G)

such that η = U ξ ; (iii) {π(g)η}g∈G is a Bessel sequence if and only if there is an operator U ∈ π(G)

such that η = U ξ , i.e., Bπ = π(G)

ξ . There are several other interesting parametrization results for frame vectors. In particular, all the frame vectors can be parameterized by a special class of k-tuple of operators from π(G) [4], where k is the cyclic multiplicity of π(G) . As a consequence of Lemma 2.5 we have Corollary 2.6. Let π be a frame representation of a countable group G on a Hilbert space H . Then (i) Bπ is dense in H ; (ii) π has a complete frame vector which is also a Bessel vector for σ if and only if Bπ ⊆ Bσ . Proof. (i) follows immediately from Lemma 2.5(iii). For (ii), assume that {π(g)ξ }g∈G is a frame for H and {σ (g)ξ }g∈G is also Bessel. Then for every η ∈ Bπ , we have by Lemma 2.5(iii) there is A ∈ π(G)

such that η = Aξ . Thus {σ (g)η}g∈G = A{σ (g)ξ }g∈G is Bessel, and so η ∈ Bσ . Therefore we get Bπ ⊆ Bσ . The other direction is trivial. 2 Now we are ready to prove Theorem 1.2. We divide the proof into two propositions. Proposition 2.7. Let π be a frame representation and (π, σ ) be a commutant pair of projective unitary representations of G on H such that π has a complete frame vector which is also a Bessel vector for σ . Then {π(g)ξ }g∈G is a frame sequence (respectively, a Parseval frame sequence) if and only if {σ (g)ξ }g∈G is a frame sequence (respectively, a Parseval frame sequence). Proof. “⇒”: Assume that {π(g)ξ }g∈G is a frame sequence. Since π is a frame representation, by the dilation result (Lemma 2.3), there exists h ∈ H such that (ξ, h) are π -orthogonal and {π(g)(ξ + h)}g∈G is a frame for H . If we prove that {σ (g)(ξ + h)}g∈G is a frame sequence, then {σ (g)ξ }g∈G is a frame sequence. In fact, using the π -orthogonality of ξ and h and Lemma 2.4, we get that [π(G) ξ ] ⊥ [π(G) h], and hence [σ (G)ξ ] ⊥ [σ (G)h] since σ (G)

= π(G) . Therefore, projecting {σ (g)(ξ + h)}g∈G onto [σ (G)ξ ] we get that {σ (g)ξ }g∈G is a frame sequence as claimed. Thus, without losing the generality, we can assume that {π(g)ξ }g∈G is a frame for H . By Corollary 2.6, we have ξ ∈ Bπ ⊆ Bσ . From Lemma 2.1 we can choose η ∈ [σ (G)ξ ] =: M such that ξ and η are σ -weakly equivalent and {σ (g)η}g∈G is a Parseval frame for [σ (G)ξ ]. By the parameterization theorem (Lemma 2.5) there exists an operator A ∈ σ (G)

|M such that ξ = Aη. Assume that C is the lower frame bound for {π(g)ξ }g∈G . Then for every x ∈ M we have x2

2 1 1 x, π(g)Aη 2 x, π(g)ξ = C C g∈G

g∈G

2 2 1 1 ∗ A x, π(g)η = A∗ x . = C C g∈G

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Thus A∗ is bounded from below and therefore it is invertible since σ (G)

|M is a finite von Neumann algebra (Proposition 1.1). This implies that A is invertible (on M) and so {σ (g)ξ }g∈G (= {Aπ(g)η}g∈G ) is a frame for M. “⇐”: Assume that {σ (g)ξ }g∈G is a frame sequence. Applying Lemma 2.1 again there exists η ∈ [π(G)ξ ] such that η and ξ are π -weakly equivalent, and {π(g)η}g∈G is a Parseval frame for [π(G)ξ ]. Using the converse statement proved above, we get that {σ (g)η}g∈G is a frame sequence for M := [σ (G)η]. Since ξ are π -weakly equivalent, we have by Lemma 2.4 that [π(G) ξ ] = [π(G) η] and so M = [σ (G)η] = [σ (G)ξ ]. Thus {σ (g)η}g∈G is a frame for [σ (G)ξ ]. By the parameterization theorem (Lemma 2.5), there exists an invertible operator A ∈ σ (G)

|M such that ξ = Aη. Extending A to an invertible operator B in σ (G)

, we have Aη = Bη, and so π(g)ξ = π(g)Aη = π(g)Bη = Bπ(g)η. Thus {π(g)ξ }g∈G is a frame sequence since {π(g)η}g∈G is a frame sequence and B is bounded invertible. For the Parseval frame sequence case, all the operators A and B involved in the parameterization are unitary operators and the rest of the argument is identical to the frame sequence case. 2 Proposition 2.8. Let π be a frame representation of G on H . Assume that (π, σ ) is a commutant pair of projective unitary representations of G on H such that such that π has a complete frame vector which is also a Bessel vector for σ . If σ admits a Riesz sequence, then (i) {π(g)ξ }g∈G is a frame for H if and only if {σ (g)ξ }g∈G is a Riesz sequence. (ii) {π(g)ξ }g∈G is a Parseval frame for H if and only if {σ (g)ξ }g∈G is an orthonormal sequence. Proof. (i) “⇒”: Assume that {π(g)ξ }g∈G is a frame for H . Then from Proposition 2.7 we have that {σ (g)ξ }g∈G is a frame sequence. Thus, in order to show that {σ (g)ξ }g∈G is a Riesz sequence, it suffices to show that [Θσ,ξ (H )] = 2 (G), where Θσ,ξ is the analysis operator of {σ (g)ξ }g∈G . We prove this by contradiction. Assume that [Θσ,ξ (H )] = 2 (G). Then, by Lemma 2.2, there is a vector 0 = x ∈ H such that Θσ,x (H ) ⊥ Θσ,ξ (H ). Since Bσ is dense in H (recall that Bπ is dense in H since π is a frame representation), we get by Lemma 2.4 that [σ (G) x] ⊥ [σ (G) ξ ] and so [π(G)x] ⊥ [π(G)ξ ] since σ (G) = π(G)

. On the other hand, since {π(g)ξ }g∈G is a frame for H , we have [π(G)ξ ] = H and so we have x = 0, a contradiction. “⇐”: Assume that {σ (g)ξ }g∈G is a Riesz sequence. Then, again by Proposition 2.7 we {π(g)ξ }g∈G is a frame sequence. So we only need to show that [π(G)ξ ] = H . Let η ⊥ [π(G)ξ ]. So we have [π(G)η] ⊥ [π(G)ξ ]. By Lemma 2.4, we have that Θσ,η (H ) ⊥ Θσ,ξ (H ). But Θσ,ξ (H ) = 2 (G) since {σ (g)ξ }g∈G is a Riesz sequence. This implies that η = 0, and so [π(G)ξ ] = H , as claimed. (ii) Replace “frame” by “Parseval frame”, and “Riesz” by “orthonormal”, the rest is exactly the same as in (i). 2


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3. The existence problem We will divide the proof of Theorem 1.4 into two cases: The abelian group case and the ICC group case. We deal the abelian group first, and start with an simple example when G = Z. Example 3.1. Consider the unitary representation of Z defined by π(n) = Me2πint on the Hilbert space L2 [0, 1/2]. Then σ (n) = Me2πi2nt is another unitary representation of Z on L2 [0, 1/2]. Note that {σ (n)1[0,1/2] }n∈Z is an orthogonal basis for L2 [0, 1/2]. We have that σ (Z)

is maximal abelian and hence σ (Z)

= M∞ = π(Z) . Moreover a function f ∈ L2 [0, 1/2] is a Bessel vector for π (respectively, σ ) if and only if f ∈ L∞ [0, 1/2]. So π and σ share the same Bessel vectors. Therefore (π, σ ) is a commutant pair with the property that Bπ = Bσ , and σ admits a Riesz sequence. It turns out the this example is generic for abelian countable discrete group. Proposition 3.1. Let π be a unitary frame representation of an abelian infinite countable discrete group G on H . Then there exists a group representation σ such that (π, σ ) is a dual pair. ˆ be the dual group of G. Then G ˆ is a compact space. Let μ be the unique Haar Proof. Let G ˆ measure of G. Any frame representation π of G is unitarily equivalent to a representation of the ˆ with positive measure, and eg is defined by form: g → eg |E , where E is a measurable subset of G ˆ So without losing the generality, we can assume that π(g) = eg |E . eg (χ) = g, χ for all χ ∈ G. 1 Let ν(F ) := μ(E) μ(F ) for any measurable subset F of E. Then both μ and ν are Borel probability measures without any atoms. Hence (see [5]) there is a measure preserving bijection ˆ Define a unitary representation σ of G on L2 (E) by ψ from E onto G. σ (g)f (χ) = eg ψ(χ) f (χ),

f ∈ L2 (E).

Then by the same arguments as in Example 3.1 we have that {σ (g)1E }g∈G is an orthogonal basis for L2 (E), and (π, σ ) satisfies all the requirements of this theorem. 2 Proof of Theorem 1.3. “(i) ⇒ (ii)”: Let Φ : λ(G) → P λ(G) P be an isomorphism between the two von Neumann algebras. Note that tr(A) = Aχe , χe is a normalized normal trace for λ(G) . Define τ on λ(G) by τ (A) =

1 tr Φ(A) , tr(P )

∀A ∈ λ(G) .

Then τ is also an normalized normal trace for λ(G) . Thus τ (·) = tr(·) since λ(G) is a factor von Neumann algebra. In particular we have that 1 tr Φ(ρg ) = τ (ρg ) = tr(ρg ) = δg,e . tr(P ) Therefore, if we define σ (g) = Φ(ρg ), then σ is a unitary representation of G such that σ (G)

= P λ(G) P = (λ(G)P ) = π(G) and σ admits an orthogonal sequence {σ (g)ξ }g∈G , where ξ = P χe . Moreover, for any A ∈ π(G)

we have that σ (g)Aξ = Aσ (g)ξ and so Aξ is a Bessel vector

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for σ . By Lemma 2.5(iii), we know that Bπ = π(G)

ξ . Thus we get Bπ ⊆ Bσ and therefore (π, σ ) is a dual pair. “(ii) ⇒ (i)”: Assume that (π, σ ) is a dual pair. Let {σ (g)ψ}g∈G be a Riesz sequence, σ1 (g) := σ (g)|M and σ2 (g) := σ (g)|M ⊥ , where M = [σ (g)ψ]. Then σ is unitarily equivalent to the group representation ζ := σ1 ⊕ σ2 acting on the Hilbert space K := M ⊕ M ⊥ . Since σ1 is unitarily equivalent to the right regular representation of G (because of the Riesz sequence), we have that σ1 (G)

λ(G) . Let q be the orthogonal projection from K onto M ⊕ 0. Then q ∈ ζ (G) . Clearly, ζ (G)

q σ1 (G)

. Since ζ (G)

is a factor, we also have that ζ (G)

ζ (G)

q, and hence σ (G)

λ(G) , i.e., λ(G) P λ(G) P since σ (G)

= P λ(G) P . 2 Remark 3.1. Although we stated the result in Theorem 1.3 for group representations, the proof works for general projective unitary representations when the von Neumann algebra generated by the left regular projective unitary representation of G is a factor. Proof of Theorem 1.4. (i) The abelian group case is proved in Proposition 3.1. If G is an amenable ICC group G, then the statement follows immediately from Theorem 1.3 and the famous result of A. Connes [2] that when G is an amenable ICC group, then λ(G) is the hyperfinite II1 factor, and we have that λ(G) and P λ(G) P are isomorphic for any non-zero projection P ∈ λ(G) . (ii) Recall that the fundamental group of a type II1 factor M is the set of numbers t > 0 for which the “amplification” of M by t is isomorphic to M, F (M) = {t > 0: M Mt }. Let G = Z2 SL(2, Z). Then, by [19,20], the fundamental group of λ(G) is {1}, which implies that von Neumann algebras P λ(G) P is not ∗-isomorphic to λ(G) for any nontrivial projection P ∈ λ(G) . Thus, by Theorem 1.3, none of the nontrivial subrepresentations λ|P admits a dual pair. 2 Example 3.2. Let G = F∞ . Using Voiculescu’s free probability theory Radulescu [21,22] proved that fundamental group F (M) = R∗+ for M = λ(F∞ ) . Therefore for λ(F∞ ) P λ(F∞ ) P for any nonzero projection P ∈ λ(F∞ ) , and thus λ|P admits a dual pair for free group F∞ . Example 3.3. Let G = Zd × Zd , and πΛ (m, n) = EAm TBn be the Gabor representation of G on L2 (Rd ) associated with the time–frequency lattice Λ = AZd × BZd . Since G is abelian, we have that the von Neumann algebra πΛ (G) is amenable (cf. [1]). Thus, if πΛ (G) is a factor, then for every πΛ invariant subspace M of L2 (Rd ), we have by the remark after the proof of Theorem 1.3 that πΛ |M admits a dual pair. Therefore the duality principle in Gabor analysis holds also for subspaces at least for the factor case (e.g., d = 1, A = a and B = b with ab irrational). In the case that A = B = I , then the Gabor representation πΛ is a unitary representation of the abelian group Zd × Zd , and so, from Proposition 3.1, the duality principle holds for subspaces for this case as well, In fact in this case a concrete representation σ can be constructed by using the Zak transform. Acknowledgments The authors thank their colleagues Ionut Chifan, Ken Dykema, Junsheng Fang, Palle Jorgensen and Roger Smith for many helpful conversations and comments on this paper. They also thank the anonymous referee for some suggestions in the final version of this paper.


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References [1] E. Bédos, R. Conti, On twisted Fourier analysis and convergence of Fourier series on discrete groups, preprint, 2008. [2] A. Connes, Classification of injective factors. Cases II 1 , II ∞ , III λ , λ = 1, Ann. of Math. 104 (1976) 73–115. [3] I. Daubechies, H. Landau, Z. Landau, Gabor time–frequency lattices and the Wexler–Raz identity, J. Fourier Anal. Appl. 1 (1995) 437–478. [4] D. Dutkay, D. Han, G. Picioroaga, Frames for ICC groups, J. Funct. Anal., in press. [5] J. Dixmier, Von Neumann Algebras, with a Preface by E.C. Lance, Translated from the second French edition by F. Jellett North-Holland Math. Library, vol. 27, North-Holland Publishing Co., Amsterdam–New York, 1981, xxxviii+437 pp. [6] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. [7] K. Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994) 123–135. [8] J.-P. Gabardo, D. Han, Subspace Weyl–Heisenberg frames, J. Fourier Anal. Appl. 7 (2001) 419–433. [9] J.-P. Gabardo, D. Han, Frame representations for group-like unitary operator systems, J. Operator Theory 49 (2003) 223–244. [10] J.-P. Gabardo, D. Han, The uniqueness of the dual of Weyl–Heisenberg subspace frames, Appl. Comput. Harmon. Anal. 17 (2004) 226–240. [11] D. Han, Frame representations and parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc. 360 (2008) 3307–3326. [12] D. Han, D. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 697 (2000). [13] D. Han, D. Larson, Frame duality properties for projective unitary representations, Bull. London Math. Soc. 40 (2008) 685–695. [14] D. Han, Y. Wang, Lattice tiling and Weyl–Heisenberg frames, Geom. Funct. Anal. 11 (2001) 742–758. [15] C. Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2007) 113–166. [16] A. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 1 (1995) 403–436. [17] R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, vols. I and II, Academic Press, Inc., 1983–1985. [18] F.J. Murray, J. von Neumann, On rings of operators. IV, Ann. of Math. 44 (1943) 716–808. [19] S. Popa, On the fundamental group of type II1 factors, Proc. Natl. Acad. Sci. 101 (2004) 723–726. [20] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. 163 (2006) 809–899. [21] F. Radulescu, The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R+ \ {0}, J. Amer. Math. Soc. 5 (1992) 517–532. [22] F. Radulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994) 347–389. [23] M.A. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981) 403– 413. [24] A. Ron, Z. Shen, Weyl–Heisenberg frames and Riesz bases in L2 (Rd ), Duke Math. J. 89 (1997) 237–282. [25] V.S. Varadarajan, Geometry of Quantum Theory, second ed., Springer-Verlag, New York–Berlin, 1985. [26] D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, CRM Monogr. Ser., Amer. Math. Soc., Providence, RI, 1992, vi+70.


Energy image density property and the lent particle method for Poisson measures Nicolas Bouleau a,∗ , Laurent Denis b a Ecole des Ponts, ParisTech, Paris-Est, 6 Avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France b Equipe Analyse et Probabilités, Université d’Evry-Val-d’Essonne, Boulevard François Mitterrand,

91025 Evry Cedex, France Received 7 February 2009; accepted 9 March 2009 Available online 2 April 2009 Communicated by Paul Malliavin

Abstract We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the energy image density property for Dirichlet forms. The associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle and taking it back after some calculation. © 2009 Elsevier Inc. All rights reserved. Keywords: Poisson functionals; Dirichlet forms; Energy image density; Lévy processes; Gradient

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Energy Image Density property (EID) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. A sufficient condition on (Rr , B(Rr )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The case of a product structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The case of structures obtained by injective images . . . . . . . . . . . . . . . . . . . . . . Dirichlet structure on the Poisson space related to a Dirichlet structure on the states space . 3.1. Density lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Construction using the Friedrichs’ argument . . . . . . . . . . . . . . . . . . . . . . . . . . .


E-mail addresses: [email protected] (N. Bouleau), [email protected] (L. Denis). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.004

. . . . . . . .

. . . . . . . .

1144 1146 1146 1150 1151 1151 1152 1153

N. Bouleau, L. Denis / Journal of Functional Analysis 257 (2009) 1144–1174

3.2.1. Basic formulas and pre-generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Particle-wise product of a Poisson measure and a probability . . . . . . . . . . . 3.2.3. Gradient and welldefinedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Upper structure and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Link with the Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Extension of the representation of the gradient and the lent particle method . . . . . . . . 3.3.1. Extension of the representation of the gradient . . . . . . . . . . . . . . . . . . . . . 3.3.2. The lent particle method: first application . . . . . . . . . . . . . . . . . . . . . . . . . 4. (EID) property on the upper space from (EID) property on the bottom space and the domain Dloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Upper bound of a process on [0, t] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Regularizing properties of Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. A regular case violating Hörmander conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1153 1155 1156 1159 1159 1162 1162 1164 1166 1168 1168 1169 1171 1173

1. Introduction The aim of this article is to improve some tools provided by Dirichlet forms for studying the regularity of Poisson functionals. First, the energy image density property (EID) which guarantees the existence of a density for Rd -valued random variables whose carré du champ matrix is almost surely regular. Second, the Lipschitz functional calculus for a local gradient satisfying the chain rule, which yields regularity results for functionals of Lévy processes. For a local Dirichlet structure with carré du champ, the energy image density property is always true for real-valued functions in the domain of the form (Bouleau [5], Bouleau and Hirsch [10, Chap. I, §7]). It has been conjectured in 1986 (Bouleau and Hirsch [9, p. 251]) that (EID) was true for any Rd -valued function whose components are in the domain of the form for any local Dirichlet structure with carré du champ. This has been shown for the Wiener space equipped with the Ornstein–Uhlenbeck form and for some other structures by Bouleau and Hirsch (cf. [10, Chap. II, §5 and Chap. V, Example 2.2.4]) and also for the Poisson space by A. Coquio [12] when the intensity measure is the Lebesgue measure on an open set, but this conjecture being at present neither refuted nor proved in full generality, it has to be established in every particular setting. We will proceed in two steps: first (Section 2) we prove sufficient conditions for (EID) based mainly on a study of Shiqi Song [31] using a characterization of Albeverio and Röckner [2], then (Section 4) we show that the Dirichlet structure on the Poisson space obtained from a Dirichlet structure on the states space inherits from that one the (EID) property. If we think a local Dirichlet structure with carré du champ (X, X , ν, d, γ ) as a description of the Markovian movement of a particle on the space (X, X ) whose transition semi-group pt is symmetric with respect to the measure ν and strongly continuous on L2 (ν), the construction of the Poisson measure allows to associate to this structure a structure on the Poisson space (Ω, A, P, D, Γ ) which describes similarly the movement of a family of independent identical particles whose initial law is the Poisson measure with intensity ν. This construction is ancient and may be performed in several ways. The simplest one, from the point of view of Dirichlet forms, is based on products and follows faithfully the probabilistic construction (Bouleau [6], Denis [14], Bouleau [7, Chap. VI, §3]).

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The cuts that this method introduces are harmless for the functional calculus with the carré du champ Γ , but it does not clearly show what happens for the generator and its domain. Another way consists in using the transition semi-groups (Martin-Löf [20], Wu [33], partially Bichteler, Gravereaux and Jacod [4], Surgailis [32]). It is supposed that there exists a Markov process xt with values in X whose transition semi-group πt is a version of pt (cf. Ma and Röckner [21, Chap. IV, §3]), the process starting at the point z is denoted by xt (z) and a probability space (W, W, Π) is considered where a family (xt (z))z∈X of independent processes is realized. For a symmetric function F , the new semi-group Pt is directly defined by (Pt F )(z1 , . . . , zn , . . .) =

F xt (z1 ), . . . , xt (zn ), . . . dΠ.

Choosing as initial law the Poisson measure with intensity ν on (X, X ), it is possible to show the symmetry and the strong continuity of Pt . This method, based on a deep physical intuition, often used in the study of infinite systems of particles, needs a careful formalization in order to prevent any drawback from the fact that the mapping X z → xt (z) is not measurable in general due to the independence. For extensions of this method see [19]. In any case, the formulas involving the carré du champ and the gradient require computations and key results on the configuration space from which the construction may be performed as starting point. From this point of view the works are based either on the chaos decomposition (Nualart and Vives [25]) and provide tools in analogy with the Malliavin calculus on Wiener space, but non-local (Picard [26], Ishikawa and Kunita [17], Picard [27]) or on the expression of the generator on a sufficiently rich class and Friedrichs’ argument (cf. what may be called the German school in spite of its cosmopolitanism, especially [1] and [22]). We will follow a way close to this last one. Several representations of the gradient are possible (Privault [28]) and we will propose here a new one with the advantages of both locality (chain rule) and simplicity on usual functionals. It provides a new method of computing the carré du champ Γ —the lent particle method—whose efficiency is displayed on some examples. With respect to the announcement [8] we have introduced a clearer new notation, the operator ε − being shared from the integration by N . Applications to stochastic differential equations driven by Lévy processes will be gathered in an other article. 2. The Energy Image Density property (EID) In this section we give sufficient conditions for a Dirichlet structure to fulfill (EID) property. These conditions concern finite dimensional cases and will be extended to the infinite dimensional setting of Poisson measures in Section 4. For each positive integer d, we denote by B(Rd ) the Borel σ -field on Rd and by λd the Lebesgue measure on (Rd , B(Rd )) and as usually when no confusion is possible, we shall denote it by dx. For f measurable f∗ ν denotes the image of the measure ν by f . For a σ -finite measure Λ on some measurable space, a Dirichlet form on L2 (Λ) with carré du champ γ is said to satisfy (EID) if for any d and for any Rd -valued function U whose components are in the domain of the form U∗ det γ U, U t · Λ λd where det denotes the determinant.


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2.1. A sufficient condition on (Rr , B(Rr )) Given r ∈ N∗ , for any B(Rr )-measurable function u : Rr → R, all i ∈ {1, . . . , r} and all x = (x1 , . . . , xi−1 , xi+1 , . . . , xr ) ∈ Rr−1 , we consider u(i) x : R → R the function defined by (i) ux (s) = u (x, s)i ,

∀s ∈ R,

where (x, s)i = (x1 , . . . , xi−1 , s, xi+1 , . . . , xr ). Conversely if x = (x1 , . . . , xr ) belongs to Rr we set x i = (x1 , . . . , xi−1 , xi+1 , . . . , xr ). Then following standard notation, for any B(R) measurable function ρ : R → R+ , we denote by R(ρ) the largest open set on which ρ −1 is locally integrable. Finally, we are given k : Rr → R+ a Borel function and ξ = (ξij )1i,j r an Rr×r -valued and symmetric Borel function. We make the following assumptions which generalize Hamza’s condition (cf. Fukushima, Oshima and Takeda [16, Chap. 3, §3.1 (3◦ ), p. 105]): Hypotheses (HG): (i) (i) 1. For any i ∈ {1, . . . , r} and λr−1 -almost all x ∈ {y ∈ Rr−1 : R ky (s) ds > 0}, kx = 0, (i) λ1 -a.e. on R \ R(kx ). 2. There exists an open set O ⊂ Rr such that λr (Rr \ O) = 0 and ξ is locally elliptic on O in the sense that for any compact subset K, in O, there exists a positive constant cK such that

∀x ∈ K, ∀c ∈ Rr ,

r

ξij (x)ci cj cK |c|2 .

i,j =1

Following Albeverio and Röckner, Theorems 3.2 and 5.3 in [2] and also Röckner and Wielens, Section 4 in [29], we consider d the set of B(Rr )-measurable functions u in L2 (k dx), such that (i) (i) for any i ∈ {1, . . . , r}, and λr−1 -almost all x ∈ Rr−1 , ux has an absolute continuous version u˜ x (i) ∂u ∂u on R(kx ) (defined λ1 -a.e.) and such that i,j ξij ∂x ∈ L1 (k dx), where i ∂xj (i)

d u˜ ∂u = x . ∂xi ds Sometimes, we will simply denote ∂x∂ i by ∂i . And we consider the following bilinear form on d: ∀u, v ∈ d,

e[u, v] =

1 2

Rr

ξij (x)∂i u(x)∂j v(x)k(x) dx.

i,j

As usual we shall simply denote e[u, u] by e[u]. We have

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Proposition 1. (d, e) is a local Dirichlet form on L2 (k dx) which admits a carré du champ operator γ given by ξij ∂i u∂j v. ∀u, v ∈ d, γ [u, v] = i,j

Proof. All is clear excepted the fact that e is a closed form on d. To prove it, let us consider a sequence (un )n∈N∗ of elements in d which converges to u in L2 (k dx) and such that limn,m→+∞ e[un − um ] = 0. Let W ⊂ O, an open subset which satisfies W ⊂ O and such that W is compact. Let dW be the set of B(Rr )-measurable functions u in L2 (1W × k dx), such that for any (i) (i) i ∈ {1, . . . , r}, and λr−1 -almost all x ∈ Rr−1 , ux has an absolute continuous version u˜ x on (i) ∂u ∂u R((1W × k)x ) and such that i,j ξij ∂x ∈ L1 (1W × k dx), equipped with the bilinear form i ∂xj ∀u, v ∈ dW ,

1 eW [u, v] = 2

W

i

1 ∂i u(x)∂i v(x)k(x) dx = 2

∇u(x) · ∇v(x)k(x) dx. W

One can easily verify, since W is an open set, that for all x ∈ Rr−1 (i) (i) Sxi (W ) ∩ R kx ⊂ R (1W × k)x ,

(1)

where Sxi (W ) is the open set {s ∈ R: (x, s)i ∈ W }. Then it is clear that the function 1W × k satisfies property 1 of (HG) and as a consequence of Theorems 3.2 and 5.3 in [2], (dW , eW ) is a Dirichlet form on L2 (1W × k dx). We have for all n, m ∈ N

2 1

1 ∇un (x) − ∇um (x) k(x) dx e(un − um ), eW (un − um ) = 2 cW W

as (d, eW ) is a closed form, we conclude that u belongs to dW . Consider now an exhaustive sequence (Wm ), of relatively compact open sets in O such that for all m ∈ N, W m ⊂ Wm+1 ⊂ O. We have that for all m, u belongs to dWm hence by Theorems 3.2 (i) and 5.3 in [2], for all i ∈ {1, . . . , r}, and λr−1 -almost all x ∈ Rr−1 , ux has an absolute continuous +∞ (i) version on m=1 R((1Wm × k)x ). Using relation (1), we have (i) +∞ (i) +∞ (i) Sxi (Wm ) ∩ R kx ⊂ R (1Wm × k)x . Sxi (O) ∩ R kx = m=1

m=1

(i) (i) 1 As λr (Rr \ O) = 0, we get that for almost all x ∈ Rr−1 , +∞ m=1 R((1Wm × k)x ) = R(kx ) λ -a.e. Moreover, by a diagonal extraction, we have that a subsequence of (∇un ) converges k dx-a.e. to ∇u, so by Fatou’s lemma, we conclude that u ∈ d and then limn→+∞ e[un − u] = 0, which is the desired result. 2 For any d ∈ N∗ , if u = (u1 , . . . , ud ) belongs to dd , we shall denote by γ [u] the matrix (γ [ui , uj ])1i,j d .


1149

Theorem 2. (EID) property: the structure (Rr , B(Rr ), k dx, d, γ ) satisfies u∗ det γ [u] · k dx λd .

∀d ∈ N∗ , ∀u ∈ dd ,

Proof. Let us mention that a proof was given by S. Song in [31, Theorem 16], in the more general case of classical Dirichlet forms. Following his ideas, we present here a shorter proof. The proof is based on the co-area formula stated by H. Federer in [15, Theorems 3.2.5 and 3.2.12]. We first introduce the subset A ⊂ Rr :

(i) A = x ∈ Rr : xi ∈ R kx i , i = 1, . . . , r . As a consequence of property 1 of (HG), Ac k(x) dx = 0. Let u = (u1 , . . . , ud ) ∈ dd . We follow the notation and definitions introduced by Bouleau and Hirsch in [10, Chap. II, Section 5.1]. Thanks to Theorem 3.2 in [2] and Stepanoff’s theorem (see Theorem 3.1.9 in [15] or Remark 5.1.2, Chap. II in [10]), it is clear that for almost derivatives rall a ∈ A, the approximate ∂u 1/2 , this is equal ap ∂x exist for i = 1, . . . , r and if we set: J u = [det(( ∂ u ∂ u ) )] k i k j 1i,j d k=1 i k dx a.e. to the determinant of the approximate Jacobian matrix of u. Then, by Theorem 3.1.4 in [15], u is approximately differentiable at almost all points a in A. We denote by Hr−d the (r − d)-dimensional Hausdorff measure on Rr . As a consequence of Theorems 3.1.8, 3.1.16 and Lemma 3.1.7 in [15], for all n ∈ N∗ , there exists a map un : Rr → Rd of class C 1 such that

1 λr A \ x: u(x) = un (x) n and

∀a ∈ x: u(x) = un (x) ,

ap

∂u ∂un (a) = ap (a), ∂xi ∂xi

i = 1, . . . , r.

Assume first that d r. Let B be a Borelian set in Rd such that λr (B) = 0 . Thanks to the co-area formula we have

1B u(x) J u(x)k(x) dx

Rr

=

1B u(x) J u(x)k(x) dx

A

= lim

n→+∞ A∩{u=un }

= lim

n→+∞ A∩{u=un }

1B u(x) J u(x)k(x) dx 1B un (x) J un (x)k(x) dx

1150


1A∩{u=un } (x)1B un (x) k(x) dHr−d (x) dλr (y)

= lim

n→+∞ Rr

= lim

n→+∞ Rr

(un )−1 (y)

1B (y)

1A∩{u=un } (x)k(x) dHr−d (x) dλr (y)

(un )−1 (y)

= 0. So that, u∗ (J u · k dx) λd . We have 1/2 J u = det Du · (Du)t

and γ (u) = Du · ξ · Dut ,

∂ui where Du is the d × r matrix: ( ∂x )1id; 1kr . k As ξ(x) is symmetric and positive definite on O and λr (Rr \ O) = 0, we have

x ∈ A; J u(x) > 0 = x ∈ A; det γ (u)(x) > 0 a.e., and this ends the proof in this case. Now, if d > r, det(γ (u)) = 0 and the result is trivial.

2

2.2. The case of a product structure We consider a sequence of functions ξ i and ki , i ∈ N∗ , ki being non-negative Borel functions such that Rr ki (x) dx = 1. We assume that for all i ∈ N∗ , ξ i and ki satisfy hypotheses (HG) so that, we can construct, as for k in the previous subsection, the Dirichlet form (di , ei ) on L2 (Rr , ki dx) associated to the carré du champ operator γi given by: ∀u, v ∈ di ,

γi [u, v] =

ξkli ∂k u∂l v.

k,l

˜ e) We now consider the product Dirichlet form (d, ˜ = +∞ on the product space i=1 (di , ei ) defined ∗ ∗ ((Rr )N , (B(R r ))N ) equipped with the product probability Λ = +∞ k i=1 i dx. We denote by ∗ r N ∗ (Xn )n∈N the coordinates maps on (R ) . Let us recall that U = F (X1 , X2 , . . . , Xn , . . .) belongs to d˜ if and only if: ∗

∗

1. U belongs to L2 ((Rr )N , (B(Rr ))N , Λ). ∗ 2. For all k ∈ N∗ and Λ-almost all (x1 , . . . , xk−1 , xk+1 , . . .) in (Rr )N , F (x1 , . . . , xk−1 , ·, xk+1 , . . .) to dk . belongs 3. e(U ˜ ) = k (Rr )N∗ ek (F (X1 (x), . . . , Xk−1 (x), ·, Xk+1 (x), . . .))Λ(dx) < +∞. Where as usual, the form ek acts only on the kth coordinate. ˜ e) It is also well known that (d, ˜ admits a carré du champ γ˜ given by γ˜ [U ] =

k

γk F (X1 , . . . , Xk−1 , ·, Xk+1 , . . .) (Xk ).


1151

To prove that (EID) is satisfied by this structure, wefirst prove that it is satisfied for a finite product. So, for all n ∈ N∗ , we consider (d˜ n , eñ ) = ni=1 (di , ei )defined on the product space ((Rr )n , (B(R r ))n ) equipped with the product probability Λn = ni=1 ki dx. By restriction, we keep the same notation as the one introduced for the infinite product. We know that this structure admits a carré du champ operator γñ given by γñ = ni=1 γi . Lemma 3. For all n ∈ N∗ , the Dirichlet structure (d˜ n , eñ ) satisfies (EID): ∀d ∈ N∗ , ∀U ∈ (d˜ n )d ,

U∗ det γñ [U ] · Λn λd .

Proof. The proof consists in remarking that this is nothing but a particular case of Theorem 2 on Rnd , ξ being replaced by Ξ , the diagonal matrix of the ξ i , and the density being the product density. 2 As a consequence of Chapter V, Proposition 2.2.3 in Bouleau and Hirsch [10], we have ˜ e) Theorem 4. The Dirichlet structure (d, ˜ satisfies (EID): ∀d ∈ N∗ , ∀U ∈ d˜ d ,

U∗ det γ˜ [U ] · Λ λd .

2.3. The case of structures obtained by injective images The following result could be extended to more general images (see Bouleau and Hirsch [10, Chapter V, §1.3, p. 196 et seq.]). We give the statement in the most useful form for Poisson measures and processes with independent increments. Let (Rp \ {0}, B(Rp \ {0}), ν, d, γ ) be a Dirichlet structure on Rp \ {0} satisfying(EID). Thus ν is σ -finite, γ is the carré du champ operator and the Dirichlet form is e[u] = 1/2 γ [u] dν. Let U : Rp \ {0} → Rq \ {0} be an injective map such that U ∈ dq . Then U∗ ν is σ -finite. If we put

dU = ϕ ∈ L2 (U∗ ν): ϕ ◦ U ∈ d , eU [ϕ] = e[ϕ ◦ U ], d U∗ (γ [ϕ ◦ U ].ν) , γU [ϕ] = d U∗ ν we have Proposition 5. The term (Rq \ {0}, B(Rq \ {0}), U∗ ν, dU , γU ) is a Dirichlet structure satisfying (EID). Proof. (a) That (Rq \ {0}, B(Rq \ {0}), U∗ ν, dU , γU ) be a Dirichlet structure is general and does not use the injectivity of U (cf. the case ν finite in Bouleau and Hirsch [10, Chap. V, §1, p. 186 et seq.]). (b) By the injectivity of U , we see that for ϕ ∈ dU γU [ϕ] ◦ U = γ [ϕ ◦ U ] ν-a.s.

1152


so that if f ∈ (dU )r f∗ det γU [f ] · U∗ ν = (f ◦ U )∗ det γ [f ◦ U ] · ν which proves (EID) for the image structure.

2

Remark 1. Applying this result yields examples of Dirichlet structures on Rn satisfying (EID) whose measures are carried by a (Lipschitzian) curve in Rn or, under some hypotheses, a countable union of such curves, and therefore without density. 3. Dirichlet structure on the Poisson space related to a Dirichlet structure on the states space Let (X, X , ν, d, γ ) be a local symmetric Dirichlet structure which admits a carré du champ operator i.e. (X, X , ν) is a measured space called the bottom space, ν is σ -finite and the bilinear form 1 e[f, g] = γ [f, g] dν 2 is a local Dirichlet form with domain d ⊂ L2 (ν) and carré du champ operator γ (see Bouleau and Hirsch [10, Chap. I]). We assume that for all x ∈ X, {x} belongs to X and that ν is diffuse (ν({x}) = 0 ∀x). The generator associated to this Dirichlet structure is denoted by a, its domain is D(a) ⊂ d and it generates the Markovian strongly continuous semi-group (pt )t0 on L2 (ν). Our aim is to study, thanks to Dirichlet forms methods, functionals of a Poisson measure N , associated to (X, X , ν). It is defined on the probability space (Ω, A, P) where Ω is the configuration space, the set of measures which are countable sum of Dirac measures on X, A is the sigma-field generated by N and P is the law of N (see Neveu [24]). The probability space (Ω, A, P) is called the upper space. 3.1. Density lemmas Let (F, F , μ) be a probability space such that for all x ∈ F , {x} belongs to F and μ is x2 , . . . , xn the coordinates maps on (F n , F ⊗n , μ×n ) and diffuse. Let n ∈ N∗ . We denote by x1 , we consider the random measure m = ni=1 εxi . Lemma 6. Let S be the symmetric sub-sigma-field in F ⊗n and p ∈ [1, +∞[. Sets {m(g1 ) · · · m(gn ): gi ∈ L∞ (μ) ∀i = 1, . . . , n} and {em(g) : g ∈ L∞ (μ)} are both total in Lp (F n , S, μ×n ) and the set {eim(g) : g ∈ L∞ (μ)} is total in Lp (F n , S, μ×n ; C). Proof. Because μ is diffuse, the set {g1 (x1 ) · · · gn (xn ): gi ∈ L∞ (μ), gi with disjoint supports ∀i = 1, . . . , n} is total in Lp (μ×n ). Let G(x1 , . . . , xn ) be a linear combination of such functions. If F (x1 , . . . , xn ) is symmetric and belongs to Lp (μ×n ) then the distance in Lp (μ×n ) between F (x1 , . . . , xn ) and G(xσ (1) , . . . , xσ (n) ) for σ ∈ S the set of permutations on {1, . . . , n}, does not depend on σ and as a consequence is larger than the distance between F (x1 , . . . , xn ) and the 1 1 ∞ barycenter n! σ ∈S G(xσ (1) , . . . , xσ (n) ). So, the set { n! σ ∈S G(xσ (1) , . . . , xσ (n) : gi ∈ L (μ), p n ×n gi with disjoint supports ∀i = 1, . . . , n} is total in L (F , S, μ ). We conclude by using the


1153

following property: if fi , i = 1, . . . , n, are F -measurable functions with disjoint supports then: m(f1 ) · · · m(fn ) = σ ∈S f1 (xσ (1) ) · · · fn (xσ (n) ). 2 Lemma 7. Let N1 be a random Poisson measure on (F, F , μ1 ) where μ1 , the intensity of N1 , is a finite and diffuse measure, defined on some probability space (Ω1 , A1 , P1 ) where A1 = σ (N1 ). Then, for any p ∈ [1, +∞[, the set {e−N1 (f ) : f 0, f ∈ L∞ (μ1 )} is total in Lp (Ω1 , A1 , P1 ) and {eiN1 (f ) : f ∈ L∞ (μ1 )} is total in Lp (Ω1 , A1 , P1 ; C). Proof. Let us put P = N1 (F ), it is an integer-valued random variable. As {eiλP : λ ∈ R} is total in Lp (N, P(N), PP ) where PP is the law of P , for any n ∈ N∗ and any g ∈ L∞ (μ1 ), one can approximate in Lp (Ω1 , A1 , P1 ; C) the random variable 1{P =n} eiN1 (g) by a sequence of iλk P eiN1 (g) with a , λ ∈ R, k = 1 · · · K. But, as a consequence variables of the form K k k k=1 ak e of the previous lemma, we know that {1{P =n} eiN1 (f ) : f ∈ L∞ (μ1 )} is total in Lp ({P = n}, A1 |{P =n} , P1 |{P =n} ; C), which provides the result. 2 We now give the main lemma, with the notation introduced at the beginning of this section. Lemma 8. For p ∈ [1, ∞[, the set {e−N (f ) : f 0, f ∈ L1 (ν) ∩ L∞ (ν)} is total in Lp (Ω, A, P) and {eiN (f ) : f ∈ L1 (ν) ∩ L∞ (ν)} is total in Lp (Ω, A, P; C). Proof. Assume that ν is non-finite. Let (Fk )k∈N be a partition of Ω such that for all k, ν(Fk ) be finite. By restriction of N to each set Fk , we construct a sequence of independent Poisson measures (Nk ) such that N = k Nk . As any variable in Lp is the limit of variables which depend only on a finite number of Nk , we conclude thanks to the previous lemma. 2 3.2. Construction using the Friedrichs’ argument 3.2.1. Basic formulas and pre-generator We set N˜ = N − ν. Then the identity E[(N˜ (f ))2 ] = f 2 dν, for f ∈ L1 (ν) ∩ L2 (ν) can be extended uniquely to f ∈ L2 (ν) and this permits to define N˜ (f ) for f ∈ L2 (ν). The Laplace characteristic functional ˜ if E ei N (f ) = e− 1−e +if dν ,

f ∈ L2 (ν),

(2)

i ˜ = 0. N a[h] + N γ [f, h] 2

(3)

yields: Proposition 9. For all f ∈ d and all h ∈ D(a), E e

i N˜ (f )

˜

Proof. Deriving in 0 the map t → E[ei N (f +ta[h]) ], we have thanks to (2), ˜ if E ei N(f )+ (1−e +if ) dν N˜ a[h] =

if e − 1 a[h] dν,

(4)

1154


then using the fact that function x → eix − 1 is Lipschitz and vanishes in 0 and the functional calculus related to a local Dirichlet form (see Bouleau and Hirsch [10, Section I.6]) we get that the member on the right-hand side in (4) is equal to i 1 − γ eif − 1, h dν = − eif γ [f, h] dν. 2 2 We conclude by applying once more (4) with γ [f, h] instead of a[h].

2

˜

The linear combinations of variables of the form ei N (f ) with f ∈ D(a) ∩ L1 (ν) are dense in L2 (Ω, A, P; C) thanks to Lemma 8. This is a natural choice for test functions, but, for technical reason, we need in addition that γ [f ] belongs to L2 (ν). So we suppose: Bottom core hypothesis (BC). The bottom structure is such that there exists a subspace H of D(a) ∩ L1 (ν) such that ∀f ∈ H , γ [f ] ∈ L2 (ν), and the space D0 of linear combinations of ˜ ei N (f ) , f ∈ H , is dense in L2 (Ω, A, P; C). This hypothesis will be fulfilled in all cases on Rr where D(a) contains the C ∞ functions with compact support and γ operates on them. ˜ If U = p λp ei N (fp ) belongs to D0 , we put A0 [U ] =

λp e

˜ p) i N(f

p

1 ˜ i N a[fp ] − N γ [fp ] . 2

(5)

This is a natural choice as candidate for the pre-generator of the upper structure, since, as easily seen using (5), it induces the relation Γ [N (f )] = N (γ [f ]) between the carré du champ operators of the upper and the bottom structures, which is satisfied in the case ν(X) < ∞. One has to note that for the moment, A0 is not uniquely determined since a priori A0 [U ] depends on the expression of U which is possibly non-unique. Proposition 10. Let U, V ∈ D0 , U =

p λp e

i N˜ (fp )

and V =

q

˜

μq ei N(gq ) . One has

1 i N˜ (fp −gq ) λp μq e N γ [fp , gq ] −E A0 [U ]V = E 2 p,q

(6)

which is also equal to 1 E Fp Gq N γ [fp , gq ] , 2 p,q

(7)

where F and G are such that U = F (N˜ (f1 ), . . . , N˜ (fn )) and V = G(N˜ (g1 ), . . . , N˜ (gm )) and ∂F ˜ ∂G ˜ (N (f1 ), . . . , N˜ (fn ), Gq = ∂x (N (g1 ), . . . , N˜ (gm )). Fp = ∂x p q Proof. We have 1 i N˜ (fp −gq ) ˜ i N a[fp ] − N γ [fp ] . λp μq e −E A0 [U ]V = −E 2 p,q


1155

Thanks to Proposition 9, 1 ˜ p −gq ) i N˜ (fp −gq ) ˜ i N(f −E λp μq e i N a[fp ] = − E λp μq e N γ [fp , fp − gq ] 2 p,q p,q 1 ˜ λp μq ei N (fp −gq ) N γ [fp , gq ] = E 2 p,q 1 i N˜ (fp −gq ) − E λp μq e N γ [fp ] 2 p,q which gives the statement.

2

It remains to prove that A0 is uniquely determined and so is an operator acting on D0 . To thanks to the previous proposition, we just have to prove that the quantity this end, N (γ [f , g ]) does not depend on the choice of representations for U and V . In F G p q p,q p q the same spirit as Ma and Röckner (see [22]), the introduction of a gradient will yield this nondependence. Let us mention that the gradient we introduce is different from the one considered by these authors and is based on a notion that we present now. 3.2.2. Particle-wise product of a Poisson measure and a probability We are still considering N the random Poisson measure on (X, X , ν) and we are given an auxiliary probability space (R, R, ρ). We construct a random Poisson measure N ρ on (X × R, X ⊗ R, ν × ρ) such that if N = i εxi then N ρ = i ε(xi ,ri ) where (ri ) is a sequence of i.i.d. random variables independent of N whose common law is ρ. Such a random Poisson measure N ρ is sometimes called a marked Poisson measure. The construction of N ρ follows line by line the one of N . Let us recall it. We first study the case where ν is finite and we consider the probability space

N, P(N), Pν(X) × X, X ,

ν ν(X)

N∗ ,

where Pν(X) denotes the Poisson law with intensity ν(X) and we put N=

Y

εxi

with the convention

0

=0

1

i=1

where Y, x1 , . . . , xn , . . . denote the coordinates maps. We introduce the probability space ˆ = (R, R, ρ)N∗ , ˆ P) ˆ A, (Ω, and the coordinates are denoted by r1 , . . . , rn , . . . . On the probability space (N, P(N), Pν(X) ) × ν N∗ ˆ we define the random measure N ρ = Y ε(x ,r ) . It is a Poisson ˆ P), ˆ A, (X, X , ν(X) ) × (Ω, i i i=1 random measure on X × R with intensity measure ν × ρ. For f ∈ L1 (ν × ρ) f dN ρ = f (x, r) dρ(r) N (dx) P-a.e. (8) Eˆ X×R

X

R

1156


and if f ∈ L2 (ν × ρ) 2 2 2 Ê = f dN ρ f dρ dN − f dρ dN + f 2 dρ dN, X×R

X R

X

R

(9)

X R

ˆ where Eˆ stands for the expectation under the probability P. If ν is σ -finite, we extend this construction by a standard product argument. Eventually in all ˆ it is a random ˆ P), ˆ A, cases, we have constructed N on (Ω, A, P) and N ρ on (Ω, A, P) × (Ω, Poisson measure on X × R with intensity measure ν × ρ. We now are able to generalize identities (8) and (9): Proposition 11. Let F be an A ⊗ X ⊗ R measurable function such that E E R ( X |F | dν)2 dρ are both finite. Then the following relation holds Eˆ

2

=

F dN ρ X×R

2 F dρ dN

−

X R

2 F dρ

X

X×R F

2 dν dρ

and

dN +

R

F 2 dρ dN.

(10)

X R

Proof. Approximating first F by a sequence of elementary functions and then introducing a partition (Bk ) of subsets of X of finite ν-measure, this identity is seen to be a consequence of (9). 2 We denote by PN the measure PN = P(dw)Nw (dx) on (Ω × X, A ⊗ X ). Let us remark that PN and P × ν are singular because ν is diffuse. We will use the following consequence of the previous proposition: Corollary 12.Let F be an A ⊗ X ⊗ R measurable function. If F belongs to L2 (Ω × X × R, PN × ρ) and F (w, x, r)ρ(dr) = 0 for PN -almost all (w, x), then F dN ρ is well defined ˆ moreover and belongs to L2 (P × P), Eˆ

2

F dN ρ

=

F 2 dN dρ

P-a.e.

(11)

X×R

Proof. If F satisfies hypotheses of Proposition 11 then the result is clear. The general case is obtained by approximation. 2 3.2.3. Gradient and welldefinedness From now on, we assume that the Hilbert space d is separable so that (see Bouleau and Hirsch [10, Ex. 5.9, p. 242]) the bottom Dirichlet structure admits a gradient operator in the sense that there exist a separable Hilbert space H and a continuous linear map D from d into L2 (X, ν; H ) such that • ∀u ∈ d, D[u]2H = γ [u]. • If F : R → R is Lipschitz then ∀u ∈ d,

D[F ◦ u] = (F ◦ u)Du.


1157

• If F is C 1 (continuously differentiable) and Lipschitz from Rd into R (with d ∈ N) then ∀u = (u1 , . . . , ud ) ∈ dd ,

D[F ◦ u] =

d Fi ◦ u D[ui ]. i=1

As only the Hilbertian structure plays a role, we can choose for H the space L2 (R, R, ρ) where (R, R, ρ) is a probability space such that the dimension of the vector space L2 (R, R, ρ) is infinite. As usual, we identify L2 (ν; H ) and L2 (X ×R, X ⊗R, ν ×ρ) and we denote the gradient D by : ∀u ∈ d,

Du = u ∈ L2 (X × R, X ⊗ R, ν × ρ).

Without loss of generality, we assume moreover that operator takes its values in the orthogonal space of 1 in L2 (R, R, ρ), in other words we take for H the orthogonal of 1. So that we have (12) ∀u ∈ d, u dρ = 0 ν-a.e. Let us emphasize that hypothesis (12) although restriction-free, is a key property here (as in many applications to error calculus cf. [7, Chap. V, p. 225 et seq.]). Thanks to Corollary 12, it is the feature which will avoid non-local finite difference calculation on the upper space. Finally, although not necessary, we assume for simplicity that constants belong to dloc (see Bouleau and Hirsch [10, Chap. I, Definition 7.1.3]) which implies γ [1] = 0 and 1 = 0.

1 ∈ dloc

(13)

We now introduce the creation and annihilation operators ε + and ε − well known in quantum mechanics (see Meyer [23], Nualart and Vives [25], Picard [26], etc.) in the following way: ∀x, w ∈ Ω,

εx+ (w) = w1{x∈supp w} + (w + εx )1{x ∈supp / w} ,

∀x, w ∈ Ω,

εx− (w) = w1{x ∈supp / w} + (w − εx )1{x∈supp w} .

One can verify that for all w ∈ Ω, εx+ (w) = w

and εx− (w) = w − εx

for Nw -almost all x

(14)

for ν-almost all x.

(15)

and εx+ (w) = w + εx

and εx− (w) = w

We extend this operator to the functionals by setting: ε + H (w, x) = H εx+ w, x and ε − H (w, x) = H εx− w, x . The next lemma shows that the image of P × ν by ε + is nothing but PN whose image by ε − is P × ν:

1158


Lemma 13. Let H be A ⊗ X -measurable and non-negative, then E

+

ε H dν = E

H dN

and E

−

ε H dN = E

H dν.

Proof. Let us assume first that H = e−N (f ) g where f and g are non-negative and belong to L1 (ν) ∩ L2 (ν). We have: E

+

ε H dν = E

e−N (f ) e−f (x) g(x) dν(x),

and by standard calculations based on the properties of the Laplace functional we obtain that E

e−N (f ) e−f (x) g(x) dν(x) = E e−N (f ) N (g) = E

H dN.

We conclude using a monotone class argument and similarly for the second equation. Let us also remark that if F ∈ L2 (PN × ρ) satisfies + ε F (w, x, r) = F (εx+ (w), x, r) we have

ε + F dN ρ =

2

F dρ = 0 PN -a.e. then if we put

F dN ρ

P-a.e.

(16)

Indeed (ε + F − F )2 dN dρ = 0 P-a.e. because εx+ (w) = w for Nw -almost all x. Definition 14. For all F ∈ D0 , we put F =

ε − ε + F dN ρ.

Thanks to hypothesis (13) we have the following representation of F : F (w, w) ˆ =

ε − F ε·+ (w) − F (w) (x, r)N ρ(dx dr).

X×R

Let us also remark that Definition 14 makes sense because for all F ∈ D0 and P-almost all ˜ w ∈ Ω, the map y → F (εy+ (w)) − F (w) belongs to d. To see this, take F = ei N(f ) with f ∈ D(a) ∩ L1 (ν), then ˜ F εy+ (w) − F (w) = ei N(f ) eif (y) − 1 , and we know that eif − 1 ∈ d. We now proceed and obtain i N˜ (f ) e =

˜ ε − ei N(f ) eif − 1 dN ρ =

˜ ε − ei N(f )+if (if ) dN ρ


1159

and eventually

˜

ei N(f )

=

˜

ei N (f ) (if ) dN ρ.

˜ ˜ So, if F, G ∈ D0 , F = p λp ei N (fp ) , G = q μq ei N (gq ) , as fp dρ = gq dρ = 0 and thanks to Corollary 12, we have ˜ λp μq ei N(fp −gq ) Eˆ F G =

(ifp ) (igq ) dN dρ,

p,q

and so ˜ λp μq ei N(fp −gq ) N γ (fp , gq ) . Eˆ F G =

(17)

p,q

But, by Definition 14, it is clear that F does not depend on the representation of F in D0 so as ˜ a consequence of the previous identity p,q λp μq ei N (fp −gq ) N (γ (fp , gq )) depends only on F and G and thanks to (6), we conclude that A0 is well defined and is a linear operator from D0 into L2 (P). 3.2.4. Upper structure and first properties As a consequence of Proposition 10, it is clear that A0 is symmetric, non-positive on D0 therefore (see Bouleau and Hirsch [10, p. 4]) it is closable and we can consider its Friedrichs extension (A, D(A)) which generates a closed Hermitian form E with domain D ⊃ D(A) such that ∀U ∈ D(A), ∀V ∈ D,

E(U, V ) = −E A[U ]V .

Moreover, thanks to Proposition 10, it is clear that contractions operate, so (see Bouleau and Hirsch [10, Ex. 3.6, p. 16]) (D, E) is a local Dirichlet form which admits a carré du champ operator Γ . The upper structure that we have obtained (Ω, A, P, D, Γ ) satisfies the following properties: ˜ ) ∈ D and • ∀f ∈ d, N(f Γ N˜ (f ) = N γ [f ] ,

(18)

moreover the map f → N˜ (f ) is an isometry from d into D. ˜ • ∀f ∈ D(a), ei N (f ) ∈ D(A), and 1 i N(f ˜ ) i N˜ (f ) ˜ i N a[f ] − N γ [f ] . Ae =e 2

(19)

1160


• The operator (defined on D0 ) admits an extension on D, still denoted , it is a gradient associated to Γ and for all f ∈ d: N˜ (f ) =

f dN ρ.

(20)

X×R

As a gradient for the Dirichlet structure (Ω, A, P, D, Γ ), is a closed operator from L2 (P) into ˆ It satisfies the chain rule and operates on the functionals of the form Φ(N˜ (f )), Φ LipL2 (P× P). schitz f ∈ d, or more generally Ψ (N˜ (f1 ), . . . , N˜ (Fn )) with Ψ Lipschitz and C 1 and f1 , . . . , fn in d. Let us also remark that if F belongs to D0 , A[F ] = N ε − a ε + F .

(21)

3.2.5. Link with the Fock space The aim of this subsection is to make the link with other existing works and to present another approach based on the Fock space. It is independent of the rest of this article. Let g ∈ D(a) ∩ L1 (ν) such that − 12 g 0 and a[g] ∈ L1 (ν). Clearly, f = − log(1 + g) is non-negative and belongs to d. We have for all v ∈ d ∩ L1 (ν) 1 E e−N (f ) , e−N (v) = E e−N (f ) e−N (v) Γ N (f ), N (v) 2 1 = E e−N (f ) e−N (v) N γ [f, v] 2 1 −f −v ) dν γ [f, v]e−f −v dν. = e X (1−e 2 X

As a consequence of the functional calculus

γ [f, v]e−f −v dν =

X

X

γ g, e−v dν = −2

a[g]e−v dν,

X

this yields −N (f ) −N (v) a[g] −N (f ) −N (v) . E e = −E e ,e e N 1+g

(22)

Thus by Lemma 8, we obtain Proposition 15. Let g ∈ D(a) ∩ L1 (ν) such that − 12 g 0 and a[g] ∈ L1 (ν) then e

N (log(1+g))

∈ D(A)

and A eN (log(1+g)) = eN (log(1+g)) N

a[g] . 1+g

(23)


1161

Let us recall that (pt ) is the semi-group associated to the bottom structure. If g satisfies the hypotheses of the previous proposition, pt g also satisfies them. The map Ψ : t → eN (log(1+pt g)) N (log(1+g)) hence Ψ (t) = P [eN (log(1+g)) ] where is differentiable and dΨ t dt = AΨ with Ψ (0) = e (Pt ) is the strongly continuous semi-group generated by A. So, we have proved Proposition 16. Let g be a measurable function with − 12 g 0, then ∀t 0,

Pt eN (log(1+g)) = eN (log(1+pt g)) .

For any m ∈ N∗ , we denote by L2sym (X m , X ⊗m , ν ×m ) the set of symmetric functions in and we recall that ν is diffuse. For all F ∈ L2sym (X m , X ⊗m , ν ×m ), we put

L2 (X m , X ⊗m , ν ×m )

F (x1 , . . . , xm )1{∀i=j, xi =xj } N˜ (dx1 ) · · · N˜ (dxm ).

Im (F ) = Xm

One can easily verify that for all F, G ∈ L2sym (X m , X ⊗m , ν ×m ) and all n, m ∈ N∗ , E[Im (F )In (G)] = 0 if n = m and E In (F )In (G) = n!F, GL2sym (Xn ,X ⊗n ,ν ×n ) , where ·,·L2sym (Xn ,X ⊗n ,ν ×n ) denotes the scalar product in L2sym (X n , X ⊗n , ν ×n ). For all n ∈ N∗ , we consider Cn , the Poisson chaos of order n, i.e. the sub-vector space of L2 (Ω, A, P) generated by the variables In (F ), F ∈ L2sym (X n , X ⊗n , ν ×n ). The fact that

L (Ω, A, P) = R 2

+∞

Cn

n=1

has been proved by K. Ito (see [18]) in 1956. This proof is based on the fact that the set {N(E1 ) · · · N(Ek ), (Ei ) disjoint sets in X } is total in L2 (Ω, A, P). Another approach, quite natural, consists in studying carefully, for g ∈ L1 ∩ L∞ (ν), what has to be subtracted from the integral with respect to the product measure

g(x1 ) · · · g(xn ) N˜ (dx1 ) · · · N˜ (dxn )

Xn

to obtain the Poisson stochastic integral In g ⊗n =

Xn

g(x1 ) · · · g(xn )1{∀i=j, xi =xj } N˜ (dx1 ) · · · N˜ (dxn ).

1162


This can be done in an elegant way by the use of lattices of partitions and the Möbius inversion formula (see Rota and Wallstrom [30]). This leads to the following formula (observe the tilde on the first N only): n In g ⊗n = Bn,k N˜ (g), −1!N g 2 , 2!N g 3 , . . . , (−1)n−k (n − k)!N g n−k+1 , k=1

where the Bn,k are the exponential Bell polynomials given by Bn,k =

n! x c1 x c2 · · · c1 !c2 ! · · · (1!)c1 (2!)c2 · · · 1 2

the sum being taken over all the non-negative integers c1 , c2 , . . . such that c1 + 2c2 + 3c3 + · · · = n, c1 + c2 + · · · = k. In (g ⊗n ) is a homogeneous function of order n with respect to g. If we express the Taylor expansion of eN (log(1+tg)) and compute the nth derivate with respect to t thanks to the formula of the composed functions (see Comtet [11]) we obtain eN (log(1+tg))−tν(g) = 1 +

+∞ n n t n=1

n!

Bn,k N˜ (g), −1!N g 2 , . . . , (−1)n−k (n − k)!N g n−k+1 ,

k=1

this yields eN (log(1+g))−ν(g) = 1 +

+∞ 1 ⊗n In g . n!

(24)

n=1

The density of the chaos is now a consequence of Lemma 8. Conversely, one can prove formula (24) thanks to the density of the chaos, see for instance Surgailis [32]. By transportation of structure, the density of the chaos has a short proof using stochastic calculus for the Poisson process on R+ , cf. Dellacherie, Maisonneuve and Meyer [13, p. 207], see also Applebaum [3, Theorems 4.1 and 4.3]. 3.3. Extension of the representation of the gradient and the lent particle method 3.3.1. Extension of the representation of the gradient The goal of this subsection is to extend the formula of Definition 14 to any F ∈ D. To this aim, we introduce an auxiliary vector space D which is the completion of the algebraic tensor product D0 ⊗ d with respect to the norm D which is defined as follows. Considering η, a fixed strictly positive function on X such that N (η) belongs to L2 (P), we set for all H ∈ D0 ⊗ d:


1163

1 2 − − H D = E ε γ [H ] (w, x)N (dx) + E ε |H | (w, x)η(x)N (dx) X

1 2 γ [H ] (w, x)ν(dx) + E |H |(w, x)η(x)ν(dx). = E X

One has to note that if F ∈ D0 then ε + F − F ∈ D0 ⊗ d and if F =

p λp e

i N˜ (fp ) ,

we have

˜ γ ε+ F − F = λp λq ei N (fp −fq ) ei(fp −fq ) γ [fp , fq ], p,q

so that

ε γ ε + F − F dN = −

˜

λp λq ei N (fp −fq ) γ [fp , fq ] dN,

p,q

X

by the construction of Proposition 10, this last term is nothing but Γ [F ]. Thus, if F ∈ D0 then ε + F − F ∈ D and + 1 ε F − F = EΓ [F ] 2 + E D

+

ε F − F η dN

1 2E[F ] 2 + 2F L2 (P) N (η)L2 (P) . As a consequence, ε + − I admits a unique extension on D. It is a continuous linear map from D into D. Since by (13) γ [ε + F − F ] = γ [ε + F ] and (ε + F − F ) = (ε + F ) , this leads to the following theorem: Theorem 17. The formula ∀F ∈ D,

F =

ε − ε + F dN ρ,

(25)

X×R

is justified by the following decomposition: d(N ρ) ε − ((.) ) ε + −I ˆ F ∈ D −→ ε + F − F ∈ D −→ ε − ε + F ∈ L20 (PN × ρ) − → F ∈ L2 (P × P) where each operator is continuous on the range of the preceding one and where L20 (PN × ρ) is the closed set of elements G in L2 (PN × ρ) such that R G dρ = 0 PN -a.e. Moreover, we have for all F ∈ D 2 Γ [F ] = Eˆ F =

X

ε − γ ε + F dN.

1164


Proof. Let H ∈ D. There exists a sequence (Hn ) of elements in D0 ⊗ d which converges to H in D and we have for all n ∈ N

2 ε Hn dPN dρ = E −

ε − γ [Hn ] dN Hn 2D ,

therefore (Hn ) is a Cauchy sequence in L20 (PN ×ρ) hence converges to an element in L20 (PN ×ρ) that we denote by ε − (H ). Moreover, if K ∈ L20 (PN × ρ), we have EEˆ

2 K(w, x, r)N ρ(dx dr) = E K 2 dN dρ = K2L2 (P

X×R

N ×ρ)

.

X×R

This provides the assertion of the statement.

2

The functional calculus for and Γ involves mutually singular measures and may be followed step by step: Let us first recall that by Lemma 13 the map (w, x) → (εx+ (w), x) applied to classes of functions PN -a.e. yields classes of functions P×ν-a.e. and also the map (w, x) → (εx− (w), x) applied to classes of functions P × ν-a.e. yields classes of functions PN -a.e. But product functionals of the form F (w, x) = G(w)g(x) where G is a class P-a.e. and g a class ν-a.e. belong necessarily to a single class PN -a.e. Hence, if we applied ε + to such a ˜ functional, this yields a unique class P × ν-a.e. In particular with F = ei Nf g: ˜ ˜ ε + ei N f g = ei Nf eif g

P × ν-a.e.

from this class the operator ε − yields a class PN -a.e. ˜ ˜ ε − ei N f eif g = ei Nf g

PN -a.e.

and this result is the same as F PN -a.e. This applies to the case where F depends only on w and is defined P-a.e. then ε− ε+ F = F

PN -a.e.

Thus the functional calculus decomposes as follows: Proposition 18. Let us consider the subset of D of functionals of the form H = Φ(F1 , . . . , Fn ) with Φ ∈ C 1 ∩ Lip(Rn ) and Fi ∈ D. Putting F = (F1 , . . . , Fn ) we have the following: (a)

ε+ H

=

i

Φi ε + F ε + Fi

P × ν × ρ-a.e.,

+ + + γ ε+ H = P × ν-a.e., Φi ε F Φj ε F γ ε F i , ε + F j ij


Φi (F )ε − ε + Fi ε− ε+ H =

(b)

1165

PN × ρ-a.e.,

i

Φi (F )Φj (F )ε − γ ε + Fi , ε + Fj ε− γ ε+ H =

PN -a.e.,

ij

(c)

H =

ε − ε + H dN ρ = Φi (F )

Γ [H ] =

ε − ε + Fi dN ρ

i

ε − γ ε + H dN = Φi (F )Φj (F )

ˆ P × P-a.e.,

ε − γ ε + Fi , ε + Fj dN

P-a.e.

ij

Remark 2. The projection of the measure PN on Ω is a (possibly non σ -finite) measure equivalent to P only if ν(X) = +∞, i.e. if P{N (1) > 0} = 1. condition for exisIf ν(X) = ν < +∞, then P{N (1) = 0} = e−ν > 0, and the sufficient tence of density Γ [F ] > 0 P-a.s. is never fulfilled because Γ [F ] = ε − (γ [ε + F ]) dN vanishes on {N(1) = 0}. Conditioning arguments with respect to the set {N (1) > 0} have to be used. 3.3.2. The lent particle method: first application The preceding theorem provides a new method to study the regularity of Poisson functionals, that we present on an example. Let us consider, for instance, a real process Yt with independent increments and Lévy measure σ integrating x 2 , Yt being supposed centered without Gaussian part. We assume that σ has a density satisfying Hamza’s condition (Fukushima, Oshima and Takeda [16, p. 105]) so that a local Dirichlet structure may be constructed on R \ {0} with carré du champ γ [f ] = x 2 f 2 (x). We suppose also hypothesis (BC) (cf. Section 3.2.1). If N is the random Poisson measure with t intensity dt × σ we have 0 h(s) dYs = 1[0,t] (s)h(s)x N˜ (ds dx) and the choice done for γ t t gives Γ [ 0 h(s) dYs ] = 0 h2 (s) d[Y, Y ]s for h ∈ L2loc (dt). In order to study the regularity of the t random variable V = 0 ϕ(Ys− ) dYs where ϕ is Lipschitz and C 1 , we have two ways: (a) We may represent the gradient as Yt = B[Y,Y ]t where B is a standard auxiliary independent Brownian motion. Then by the chain rule t V =

t

ϕ (Ys− )(Ys− ) dYs +

0

ϕ(Ys− ) dB[Y ]s 0

now using (Ys− ) = (Ys )− , a classical but rather tedious stochastic calculus yields

Γ [V ] = Eˆ V

2

=

αt

t Yα2

2 ϕ (Ys− ) dYs + ϕ(Yα− )

,

(26)

]α

where Yα = Yα − Yα− . Since V has real values the energy image density property holds for V , and V has a density as soon as Γ [V ] is strictly positive a.s. what may be discussed using the relation (26). (b) An other more direct way consists in applying the theorem. For this we define by choos1 1 ing ξ such that 0 ξ(r) dr = 0 and 0 ξ 2 (r) dr = 1 and putting f = xf (x)ξ(r).

1166


1o . First step. We add a particle (α, x) i.e. a jump to Y at time α with size x what gives t

+

ε V − V = ϕ(Yα− )x +

ϕ(Ys− + x) − ϕ(Ys− ) dYs .

]α

2o . V = 0 since V does not depend on x, and t + ε V = ϕ(Yα− )x + ϕ (Ys− + x)x dYs ξ(r)

because x = xξ(r).

]α

t 3o . We compute γ [ε + V ] = (ε + V )2 dr = (ϕ(Yα− )x + ]α ϕ (Ys− + x)x dYs )2 . 4o . We take back the particle we gave, in order to compute ε − γ [ε + V ] dN . That gives

ε − γ ε + V dN =

2 t ϕ(Yα− ) + ϕ (Ys− ) dYs x 2 N (dα dx) ]α

and (26). remark that both operators F → ε + F , F → (ε + F ) are non-local, but instead F → We − + ε (ε F ) d(N ρ) and F → ε − γ [ε + F ] dN are local: taking back the lent particle gives the locality. We will deepen this example in dimension p in Section 5. 4. (EID) property on the upper space from (EID) property on the bottom space and the domain D loc From now on, we make additional hypotheses on the bottom structure (X, X , ν, d, γ ) which are stronger but satisfied in most of the examples. Hypothesis (H1): X admits a partition of the form: X = B ∪ ( +∞ k=1 Ak ) where for all k, Ak ∈ X with ν(Ak ) < +∞ and ν(B) = 0, in such a way that for any k ∈ N∗ may be defined a local Dirichlet structure with carré du champ: Sk = (Ak , X|Ak , ν|Ak , dk , γk ), with ∀f ∈ d,

f|Ak ∈ dk

and γ [f ]|Ak = γk [f|Ak ].

Hypothesis (H2): Any finite product of structures Sk satisfies (EID). Remark 3. In many examples where X is a topological space, (H1) is satisfied by choosing for (Ak ), k ∈ N∗ a regular open set. Let us remark that (H2) is satisfied for the structures studied in Section 2.


1167

The main result of this section is the following: Proposition 19. If the bottom structure (X, X , ν, d, γ ) satisfies (H1) and (H2) then the upper structure (Ω, A, P, D, Γ ) satisfies (EID). Proof. For all k ∈ N∗ , since ν(Ak ) < +∞, we consider an upper structure Sk = (Ωk , Ak , Pk , Dk , Γk ) associated to Sk as a direct application of the construction by product (see Section 3.3.2 above or Bouleau [7, Chap. VI.3]). Let k ∈ N∗ , we denote by Nk the corresponding random Poisson measure on Ak with intensity ν|Ak and we consider N ∗ the random Poisson measure on X with intensity ν, defined on the product probability space ∗ ∗ ∗ +∞ (Ωk , Ak , Pk ), Ω ,A ,P = k=1

by N∗ =

+∞

Nk .

k=1

In a natural way, we consider the product Dirichlet structure +∞ Sk . S ∗ = Ω ∗ , A∗ , P∗ , D∗ , Γ ∗ = k=1

In the third section, we have built using the Friedrichs argument, the Dirichlet structure S = (Ω, A, P, D, Γ ), let us now make the link between those structures. First of all, thanks to Theorem 2.2.1 and Proposition 2.2.2. of Chapter V in Bouleau and Hirsch [10], we know that a function ϕ in L2 (P∗ ) belongs to D∗ if and only if: 1. For all k ∈ N∗ and

n=k Pn -almost

all ξ1 , . . . , ξk1 , ξk+1 , . . . , the map

ξ → ϕ(ξ1 , . . . , ξk1 , ξ, ξk+1 , . . .)

2.

belongs to Dk . Γ [ϕ] ∈ L1 (P∗ ) and we have Γ ∗ [ϕ] = k γk [ϕ]. k k

Consider f ∈ d ∩ L1 (γ ) then clearly N (f ) =

k

Nk (f|Ak )

1168


belongs to D∗ and in the same way ˜

ei N(f ) =

˜

ei Nk (f|Ak ) ∈ D∗ .

k

Moreover, by hypothesis (H1):

2 ˜

i N˜ (f ) i N(f ˜ ) i Nl (f|Al )

k |Ak

= = Nk γ [f ]|Ak Γ e

e

Γk e ∗

k

l=k

k

˜ = N γ [f ] = Γ ei N (f ) . Thus as D0 is dense in D, we conclude that D ⊂ D∗ and Γ = Γ ∗ on D. As for all k, Sk is a product structure, thanks to hypothesis (H2) and Proposition 2.2.3 in Bouleau and Hirsch [10, Chapter V], we conclude that S ∗ satisfies (EID) hence S too. 2 Main case. Let N be a random Poisson measure on Rd with intensity measure ν satisfying one of the following conditions: (i) ν = k dx and a function ξ (the carré du champ coefficient matrix) may be chosen such that hypotheses (HG) hold (cf. Section 2.1), (ii) ν is the image by a Lipschitz injective map of a measure satisfying (HG) on Rq , q d, (iii) ν is a product of measures like (ii), then the associated Dirichlet structure (Ω, A, P, D, Γ ) constructed (cf. Section 3.2.4) with ν and the carré du champ obtained by the ξ of (i) or induced by operations (ii) or (iii) satisfies (EID). We end this section by a few remarks on the localization of this structure which permits to extend the functional calculus related to Γ or to bigger spaces than D, which is often convenient from a practical point of view. Following Bouleau and Hirsch (see [10, pp. 44–45]) we recall that Dloc denotes the set of functions F : Ω → R such that there exists a sequence (En )n∈N∗ in A such that Ω=

En

and ∀n ∈ N∗ , ∃Fn ∈ D,

Fn = F

on En .

n

Moreover if F ∈ Dloc , Γ [F ] is well defined and satisfies (EID) in the sense that F∗ Γ [F ] · P λ1 . More generally, if (Ω, A, P, D, Γ ) satisfies (EID), ∀F ∈ (Dloc )n ,

F∗ det Γ [F ] · P λn .

We can consider another space bigger than Dloc by considering a partition of Ω consisting in a sequence of sets with negligible boundary. More precisely, we denote by DLOC the set of


1169

∗ functions F : Ω → R such that there exists a sequence of disjoint sets (An )n∈N in A such that P(Ω \ n An ) = 0 and

∀n ∈ N∗ , ∃Fn ∈ D,

Fn = F

on An .

One can easily verify that it contains the localized domain of any structure S ∗ as considered in the proof of Proposition 19, that Γ is well defined on DLOC , that the functional calculus related to Γ or remains valid and that it satisfies (EID) i.e. if (Ω, A, P, D, Γ ) satisfies (EID), ∀F ∈ (DLOC )n ,

F∗ det Γ [F ] · P λn .

5. Examples 5.1. Upper bound of a process on [0, t] Let Y be a real process with stationary independent increments satisfying the hypotheses of example 3.3.2. We consider a real càdlàg process K independent of Y and put Hs = Ys + Ks . Proposition 20. If σ (R \ {0}) = +∞ and if P[supst Hs = H0 ] = 0, the random variable supst Hs possesses a density. Proof. (a) We may suppose that K satisfies supst |Ks | ∈ L2 . Indeed, if random variables Xn have densities and P[Xn = X] → 0, then X has a density. Hence the assertion is obtained by considering (Ks ∧ k) ∨ (−k). (b) Let us put M = supst Hs . Applying the lent particle method gives + ε M (α, x) = sup (Ys + Ks )1{s

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