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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, LOCAL RIGIDITY OF HYPERBOLIC 3-MANIFOLDS AFTER DEHN SURGERY KEVIN P. ...

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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1,

LOCAL RIGIDITY OF HYPERBOLIC 3-MANIFOLDS AFTER DEHN SURGERY KEVIN P. SCANNELL

Abstract It is well known that some lattices in SO(n, 1) can be nontrivially deformed when included in SO(n +1, 1) (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in SO(3, 1) which are locally rigid in SO(4, 1) by considering closed hyperbolic 3-manifolds obtained by Dehn filling on hyperbolic two-bridge knots. We generalize this result to Dehn filling on a more general class of one-cusped finite volume hyperbolic 3-manifolds, allowing us to produce the first examples of closed hyperbolic 3-manifolds which contain embedded quasi-Fuchsian surfaces but are locally rigid in SO(4, 1). 1. Introduction This paper continues our study of the local deformation theory of rank-one lattices, which began with [28]. We are particularly interested in the local deformation space of representations of an SO(3, 1) lattice when viewed as a “Fuchsian” subgroup of SO(4, 1). The first examples of such deformations were given by B. Apanasov [2], [4] around the same time that W. Thurston introduced his closely related notion of bending deformations of Fuchsian groups (see [30, §8.7.3]). Examples in all dimensions and detailed discussion can be found in [13] (see also [22], [23]). Generalizations of bending have been considered by various authors (see, e.g., [3], [5], [29]) and typically involve either intersecting totally geodesic surfaces or a family of totally geodesic surfaces with a common boundary geodesic. Kapovich conjectured in [16] that a closed hyperbolic 3-orbifold admits a nontrivial deformation in O(4, 1) if and only if it contains an embedded quasi-Fuchsian suborbifold. In [28] we gave examples of infinitesimal deformations for infinitely many two-generator, closed hyperbolic 3-manifolds with zero first Betti number. One of these examples is non-Haken, and its fundamental group contains no nonelementary Fuchsian subgroups, providing an infinitesimal counterexample to one half of DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 28 April 2000. Revision received 22 May 2001. 2000 Mathematics Subject Classification. Primary 57M50; Secondary 22E40, 57N10, 57N16, 57M25.

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Kapovich’s conjecture. One of the goals of this paper is to give counterexamples to the other half, by constructing infinitely many closed hyperbolic 3-manifolds that contain quasi-Fuchsian surfaces but are locally rigid in SO(4, 1). These manifolds also serve as counterexamples to a second conjecture of Kapovich [15], [17], namely, that for a closed 3-manifold, the deformation space of flat conformal structures should have only finitely many components. That this statement is false for hyperbolic nmanifolds, n ≥ 5, was known previously (see [20]). 5.2 There exist infinitely many closed hyperbolic 3-manifolds that contain embedded quasi-Fuchsian surfaces and that are locally rigid in SO(4, 1). The deformation space of flat conformal structures for these manifolds contains an infinite set of isolated points. THEOREM

Our method for producing these examples rests on the following general rigidity theorem. THEOREM 4.4 Let M ≈ 0\H3 be a complete, orientable, hyperbolic 3-manifold of finite volume with one cusp. If P H 1 (0, so(4, 1)) = 0, then there exist infinitely many closed hyperbolic 3-manifolds obtained by Dehn filling on M which are locally rigid in SO(4, 1).

When M is the complement of a hyperbolic two-bridge knot in S3 , this result was obtained by Kapovich in [18]. In this case, the vanishing of the parabolic cohomology P H 1 (0, so(4, 1)) follows easily from the fact that 0 is generated by two parabolic elements (see §3). In general, this cohomology group can be computed from an ideal triangulation of M, and our calculations indicate that it vanishes quite often. We discuss some of these computations in §5. 2. Preliminaries ∼ = We first recall some of the notation from [28]. Let ρ0 : π → 0 ⊂ SO0 (3, 1) be the inclusion of a lattice in the identity component of O(3, 1), and consider the composition of ρ0 with the inclusion SO0 (3, 1) ,→ SO0 (4, 1). We are interested in describing a neighborhood of ρ0 in the representation space Hom(π, SO0 (4, 1)). This space is a real algebraic variety in a natural way, and the Zariski tangent space at ρ0 is identified with the vector space of group cocycles Z 1 (π, so(4, 1)). Here the coefficients lie in the Lie algebra of SO0 (4, 1), made into a Zπ-module via ρ0 and the adjoint action. There is an action of SO0 (4, 1) on Hom(π, SO0 (4, 1)) by conjugation; the subspace B 1 (π, so(4, 1)) of coboundaries consists of the Zariski tangent vectors along orbits

RIGIDITY AND DEHN SURGERY

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of this action. For this reason, we call a nonzero cohomology class in H 1 (π, so(4, 1)) an infinitesimal deformation of ρ0 in SO0 (4, 1). An infinitesimal deformation is integrable if it is tangent to a nontrivial curve in Hom(π, SO0 (4, 1)). For any Zπ-module V , we compute the group cohomology H 1 (π, V ) in terms of the standard resolution. Thus, a 1-cocycle is a function c : π → V satisfying c(gh) = c(g) + gc(h) for all g, h ∈ π , and a 1-coboundary is a 1-cocycle of the form c(g) = (1 − g)w for some w ∈ V . When there is a possibility of confusion, we make the Zπ-module structure explicit; for example, we write Vρ for a given ρ : π → Aut(V ). When 0 is a nonuniform lattice, it contains parabolic elements, and we need a notion of cohomology classes that are trivial when restricted to cyclic parabolic subgroups. Let P Z 1 (0, V ) = c ∈ Z 1 (0, V ) for parabolic γ ∈ 0, c(γ ) ∈ im(1 − γ ) , and define the parabolic cohomology P H 1 (0, V ) = P Z 1 (0, V )/B 1 (0, V ). In [7] it is shown that P H 1 (0, so(3, 1)) = 0 for a nonuniform lattice 0 ⊂ SO0 (3, 1), giving the following analogue of the splitting lemma in [28]. LEMMA 2.1 Fix a representation ρ0 : π → SO0 (3, 1) ,→ SO0 (4, 1). The Lie algebra so(4, 1) splits as an SO0 (3, 1)-module so(4, 1) ∼ = so(3, 1) ⊕ R41 , inducing a splitting in the parabolic cohomology P H 1 π, so(4, 1) ∼ = P H 1 π, so(3, 1) ⊕ P H 1 (π, R41 ).

When ρ0 is an isomorphism onto a nonuniform lattice in SO0 (3, 1), we have P H 1 π, so(4, 1) ∼ = P H 1 (π, R41 ). The next results are standard consequences of duality for surfaces and 3-manifolds. LEMMA 2.2 ([8]) Let π be the fundamental group of a closed, orientable surface of genus g. Let G be a semisimple Lie group, and fix a representation ρ0 : π → G. Then

dim H 1 (π, g) = (2g − 2) dim G + 2 dim H 0 (π, g).

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Proof The Killing form on g is nondegenerate and Ad-invariant, and therefore gives a Gmodule isomorphism between g and its dual g∗ . By Poincaré duality, we see that H 2 (π, g) ∼ = H 0 (π, g∗ )∗ ∼ = H 0 (π, g)∗ and hence that dim H 2 (π, g) = dim H 0 (π, g). The Euler characteristic in group cohomology is independent of the representation ρ0 , so by considering the trivial representation, we see that it is equal to (2 − 2g) dim G. The lemma follows. Let M be a compact, oriented 3-manifold with a nonempty collection of boundary L ∗ components {6 j }, and let π = π1 M. We write H ∗ (∂, V ) = j H (π1 6 j , V ) for the group cohomology of the boundary components, and we write i ∗ : H 1 (π, V ) → H 1 (∂, V ) for the corresponding restriction map on first cohomology. The next proposition is an easy consequence of Lefschetz duality for 3-manifolds and has appeared in [11, §15], [16], and [12]. A geometric argument for the case G = SO0 (3, 1) ∼ = PSL(2, C) can be found in [30, §5.6] and [6]. PROPOSITION 2.3 Let M be a compact, oriented 3-manifold such that ∂ M consists of a nonempty union of tori. For any representation ρ0 : π → G of π = π1 M in a semisimple Lie group G, dim H 1 (π, g) = dim ker i ∗ + dim H 0 (∂, g).

Proof (following [12]) For j = 0, 1 we have the well-known identifications H j (π, g) ∼ = H j (M; E) and H j (∂, g) ∼ = H j (∂ M; E), where E is the flat bundle associated to the representation ρ0 . Consider the following commutative diagram, suppressing the local coefficients E: i∗ δ∗ H 1 (M) −−−−→ H 1 (∂ M) −−−−→ H 2 (M, ∂ M)       y y y δ∗

i∗

H 2 (M, ∂ M)∗ −−−−→ H 1 (∂ M)∗ −−−−→

H 1 (M)∗

Here the vertical arrows are isomorphisms by duality, and the maps i ∗ and δ∗ are the duals of i ∗ and δ ∗ , respectively. We have dim im δ ∗ = dim im i ∗ = dim im i ∗ ,


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and so dim H 1 (∂ M) = dim ker δ ∗ + dim im δ ∗ = 2 dim im i ∗ . Combining this with Lemma 2.2, we obtain dim im i ∗ = dim H 0 (∂ M) and thus dim H 1 (M) = dim ker i ∗ + dim im i ∗ = dim ker i ∗ + dim H 0 (∂ M). 3. Subgroups generated by parabolics The results in this section are not required for the proof of our main result, but help to explain the significance of the parabolic cohomology hypothesis and its relationship to the results of [18]. 3.1 If 1 is a subgroup of SO0 (3, 1) generated by two parabolics, then P H 1 (1, R41 ) = 0. PROPOSITION

Proof Let α0 and β0 be the parabolic generators in SO0 (3, 1), and let c ∈ P Z 1 (1, R41 ) ⊆ P Z 1 (1, so(4, 1)), using the splitting of Lemma 2.1. Now c(α0 ) = (1 − α0 )v0 and c(β0 ) = (1 − β0 )w0 for some v0 , w0 ∈ so(4, 1), and we can define two curves of parabolics αt = exp(tv0 )α0 exp(tv0 )−1 and βt = exp(tw0 )β0 exp(tw0 )−1 in SO0 (4, 1). But any pair of unipotents in SO0 (4, 1) leaves invariant a round S2 in S3 (see [18]); hence the group generated by αt and βt is conjugate back into SO0 (3, 1). This implies that [c] ∈ P H 1 (1, so(3, 1)) and hence that [c] = 0. 3.2 If M ≈ is the complement of a hyperbolic two-bridge knot or link in S3 , then 1 P H (0, so(4, 1)) = 0. COROLLARY

0\H3

Proof It is well known that the fundamental group of a two-bridge knot or link is generated by two meridional loops, so in the hyperbolic case, 0 is generated by two parabolic elements. The corollary follows from Lemma 2.1 and the previous proposition. In fact, it is a remarkable consequence of the Smith conjecture that any finite volume, orientable, hyperbolic 3-manifold with fundamental group generated by two parabolic elements is the complement of a two-bridge link in S3 (see [1]).

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COROLLARY 3.3 Let M ≈ 0\H3 be a complete, orientable, hyperbolic 3-manifold of finite volume with at least one cusp, and let i ∗ : H 1 (0, so(4, 1)) → H 1 (∂, so(4, 1)) be the restriction homomorphism as above. Then P H 1 0, so(4, 1) = ker i ∗ .

Proof That ker i ∗ is included in P H 1 (0, so(4, 1)) is clear from the definitions. For the opposite inclusion, we may assume that the coefficients of a representative cocycle c lie in R41 by Lemma 2.1. But P H 1 (∂, R41 ) = 0 by applying Proposition 3.1 to each boundary component, so i ∗ [c] = 0. 4. Dehn surgery We proceed with the proof of the rigidity theorem in this section. The basic strategy is as follows: for a finite volume complete hyperbolic 3-manifold with one cusp, the PSL(2, C) character variety R is (real) 2-dimensional and smooth at the holonomy representation ρ0 . For the larger variety of representations into SO(4, 1), we may use the results of §2, Lemma 4.1, and the parabolic cohomology hypothesis to show that the dimension of the Zariski tangent space at ρ0 is 3. The key technical point in the proof is contained in Proposition 4.3, which shows the existence of nonintegrable tangent vectors in the tangent space at ρ0 . From this result, we argue that the set of points in R corresponding to potentially nonrigid surgered manifolds lies in a proper subvariety. But it is known that the set of points yielding closed hyperbolic manifolds is a Zariski-dense subset of R, and the theorem follows. In the proof below, we actually work with the full representation variety (before conjugation) to avoid the difficulties involved in passing to the quotient variety. The idea for the last part of the argument is taken from Kapovich’s proof for the case of two-bridge knots (see [18]). We conjecture that the “infinitely many” in the conclusion of the rigidity theorem can be replaced by “all but finitely many.” LEMMA 4.1 Fix a representation ρ0 : Z ⊕ Z → SO0 (3, 1) whose image is not generated by commuting order-two elliptics. Then dim H 0 (Z ⊕ Z, so(3, 1)) = 2. If, furthermore, the image of ρ0 is not contained in any one-parameter subgroup, we have ( 1 at a parabolic representation, 0 4 dim H (Z ⊕ Z, R1 ) = 0 otherwise.

Proof Having excluded the case of commuting order-two elliptics, it is well known that


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two nontrivial elements of SO0 (3, 1) commute if and only if their fixed point sets in S2 ≈ ∂H3 coincide. Thus, if one element is parabolic, they all are, and dim H 0 (Z ⊕ Z, so(3, 1)) = 2. Alternatively, there is an invariant geodesic line in H3 , and once again we have dim H 0 (Z ⊕ Z, so(3, 1)) = 2. In the case of a parabolic representation, the common parabolic fixed point corresponds to a fixed null vector in R41 . On the other hand, there are clearly no fixed timelike vectors, and because the image is not contained in a one-parameter subgroup of parabolics, there is no invariant H2 ⊂ H3 , hence no fixed spacelike vector. Thus, dim H 0 (Z ⊕ Z, R41 ) = 1. Otherwise, our assumption implies that ρ0 (Z ⊕ Z) contains some loxodromic elements (throughout, we use loxodromic to include purely hyperbolic elements). It follows that there can be no fixed null or timelike vectors in R41 and that any invariant H2 ⊂ H3 contains the common invariant geodesic. But this is only possible if each element is purely hyperbolic, which forces ρ0 (Z ⊕ Z) to lie within a one-parameter subgroup. We conclude that dim H 0 (Z ⊕ Z, R41 ) = 0 in this case. 4.2 Let α and β be generators of Z ⊕ Z, and let ρ ∈ Hom(Z ⊕ Z, SO0 (4, 1)) be a representation such that ρ(α) is loxodromic and −1 is not an eigenvalue of ρ(β). Then the image of ρ is conjugate into SO0 (3, 1). LEMMA

Proof View SO0 (4, 1) as acting by Möbius transformations on R3 ∪ {∞} and conjugate so that 0 and ∞ are, respectively, the repelling and attracting fixed points of ρ(α). It follows that ρ(α) is of the form x 7 → λ1 Ax for some λ1 > 1 and A ∈ SO(3). Since ρ(β) commutes with ρ(α), it leaves invariant the fixed set {0, ∞}. In fact, ρ(β) must fix 0 and ∞, for if it interchanged them, it would have an eigenvalue of −1. Thus, ρ(β) is of the form x 7 → λ2 Bx, where λ2 > 0 and B commutes with A in SO(3). Since ρ(β) does not have −1 as an eigenvalue, we can exclude the possibility that A and B are a pair of commuting half-turns, and so A and B share a common axis of rotation. We conclude that ρ(α) and ρ(β) leave invariant a common plane in R3 and therefore that ρ is conjugate into SO0 (3, 1). PROPOSITION 4.3 Let ρ0 : π → SO0 (3, 1) ,→ SO0 (4, 1) be the inclusion of a torsion-free, nonuniform

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lattice. Then the representation variety Hom(π, SO0 (4, 1)) is singular at ρ0 . Indeed, given any v ∈ Z 1 (π, R41 ) such that i ∗ [v] 6 = 0 ∈ H 1 (∂, R41 ), there is a nonintegrable cocycle in Z 1 (π, so(3, 1)) ⊕ hvi. Proof First, note that the existence of a cocycle v satisfying the properties in the statement of the proposition follows from Lemma 4.1 and Proposition 2.3. Since i ∗ [v] 6= 0, we may fix one cusp end T such that i ∗ [v] 6 = 0 in H 1 (π1 T, R41 ). Let α and β be generators for π1 T . A generic deformation ωt ∈ Hom(π, SO0 (3, 1)) of ω0 = ρ0 has the property that ωt (α) is loxodromic for small t > 0 (see [27]); fix one such deformation, and let ω˙ ∈ Z 1 (π, so(3, 1)) be the tangent vector at t = 0. Since the loxodromic elements are open in SO0 (4, 1), there exists > 0 such that if ρt integrates ω˙ + v, then ρt (α) is loxodromic for small t > 0. Furthermore, since ρ0 (β) is unipotent, ρt (β) does not have −1 as an eigenvalue for small values of t. Thus, Lemma 4.2 implies that ρt |π1 T is conjugate into SO0 (3, 1) for small t. We conclude that i ∗ (ω˙ + v), and therefore i ∗ (v), are cohomologous to cocycles in Z 1 (π1 T, so(3, 1)), a contradiction. We are now ready to prove the main theorem. THEOREM 4.4 Let M ≈ 0\H3 be a complete, orientable, hyperbolic 3-manifold of finite volume with one cusp. If P H 1 (0, so(4, 1)) = 0, then there exist infinitely many closed hyperbolic 3-manifolds obtained by Dehn filling on M which are locally rigid in SO(4, 1).

Proof By abuse of notation we also write M for the compact manifold with boundary whose interior is homeomorphic to 0\H3 . As usual, we let π = π1 M, and we fix a holonomy ∼ =

representation ρ0 : π → 0 ⊂ SO0 (3, 1) corresponding to the complete hyperbolic structure on M. For brevity, we write Xn = Hom(π, SO0 (n, 1)) for n = 3, 4. We begin with some properties of X3 . First, recall that X3 is smooth at ρ0 (see [11, §15], [19, §8.8]); its dimension can be computed using Proposition 2.3: dim Z 1 π, so(3, 1)ρ0 = dim H 1 π, so(3, 1)ρ0 + dim B 1 π, so(3, 1)ρ0 = dim ker i ∗ + dim H 0 ∂, so(3, 1)ρ0 + 6 = 0 + 2 + 6 = 8, where ker i ∗ vanishes by [7], and we have used Lemma 4.1 for the Z ⊕ Z centralizer in SO0 (3, 1). (Of course, since SO0 (3, 1) ∼ = PSL(2, C), it is more common to realize


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X3 as a smooth 4-dimensional complex variety, but as we are in the context of an inclusion into the noncomplex group SO0 (4, 1), it is convenient to work only with the real algebraic structure.) A neighborhood of ρ0 in X3 consists of representations with Zariski dense image in SO0 (3, 1); in particular, they all have trivial centralizer in SO0 (4, 1). It follows that the image e X3 of the conjugation map X3 × SO0 (4, 1) → X4 (i.e., the set of representations with image conjugate into SO0 (3, 1)) is smooth and 12-dimensional near ρ0 . On the other hand, the dimension of the Zariski tangent space to X4 can be computed in a similar fashion: dim Z 1 π, so(4, 1)ρ0 = dim H 1 π, so(4, 1)ρ0 + dim B 1 π, so(4, 1)ρ0 = dim ker i ∗ + dim H 0 ∂, so(4, 1)ρ0 + 10 = 0 + 3 + 10 = 13, where we have used the parabolic cohomology hypothesis for the vanishing of ker i ∗ and Lemma 4.1 for the Z ⊕ Z centralizer in SO0 (4, 1). Because X4 is singular at ρ0 (see Proposition 4.3), the dimension of X4 as a real algebraic variety is strictly less than 13. Since e X3 is smooth and 12-dimensional, we may conclude that the dimension of X4 is precisely 12. Thus, at a representation ρ ∈ X4 near ρ0 , we have dim Z 1 (π, so(4, 1)ρ ) = 12 + ε(ρ), where ε(ρ) = 0, 1. A calculation like the one above shows that dim H 1 (π, so(4, 1)ρ ) = 2 + ε(ρ) and therefore that dim H 1 (π, (R41 )ρ ) = ε(ρ) for ρ ∈ X3 . To dispose of the possibility that there is an open neighborhood of ρ0 where ε(ρ) = 1, we replace X4 with the reduced variety Y4 defined by the ideal of polynomials vanishing on X4 . These two varieties coincide as point sets, but the Zariski tangent spaces of Y4 are a priori smaller, and we are able to conclude that its singular subvariety B has positive codimension (see [32]). We next write R for the 2-dimensional character variety of representations in SO0 (3, 1) up to conjugation. Fix a basis for the homology of the boundary torus ∂ M, and write κ(ρ) = ( p, q) for the generalized Dehn surgery invariant associated to a representation ρ as in [27]. The set of conjugacy classes of ρ ∈ R such that κ(ρ) is a pair of relatively prime integers clusters at ρ0 and is Zariski-dense (see [18], [27]); the same statement holds in e X3 . Since B has positive codimension, we conclude that there are infinitely many closed hyperbolic 3-manifolds obtained by ( p, q) filling on M such that the corresponding representation ρ is not contained in B. Fix one such representation ρ ∈ / B with κ = κ(ρ) = ( p, q), and let Mκ be the closed manifold obtained by ( p, q) filling on M. The representation ρ factors through π1 Mκ to give the holonomy ρ of the complete hyperbolic structure on Mκ , so we have Z 1 (π1 Mκ , (R41 )ρ ) ⊆ Z 1 (π, (R41 )ρ ). The image ρ(π) = ρ(π1 Mκ ) is a lattice in SO0 (3, 1) and therefore has trivial centralizer in SO0 (4, 1). This means that

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B 1 (π1 Mκ , (R41 )ρ ) = B 1 (π, (R41 )ρ ) ∼ = R41 , and so H 1 π1 Mκ , (R41 )ρ ⊆ H 1 π, (R41 )ρ . Since H 1 (π1 Mκ , so(4, 1)ρ ) = H 1 (π1 Mκ , (R41 )ρ ) by the splitting lemma in [28], we see by [31] that ρ is locally rigid whenever ε(ρ) = 0. On the other hand, if there exists a nontrivial integrable deformation of ρ with tangent vector v ∈ H 1 (π1 Mκ , (R41 )ρ ), it would also define a nontrivial curve in Y4 starting at ρ. We conclude from this that the Zariski tangent space to Y4 at ρ is at least 13-dimensional and therefore that ρ ∈ B, a contradiction. 5. Examples and remarks The hypothesis of vanishing parabolic cohomology in the main theorem is closely related to the Menasco-Reid conjecture (see [25]), which states that no hyperbolic knot complement in S3 contains a closed, embedded, totally geodesic surface. Of course, the existence of such a surface implies the existence of a nontrivial class in P H 1 (0, so(4, 1)) by bending. The converse is false, however, as we have shown that the family of Turk’s head links (starting with 818 ; see [28]) all have nonvanishing parabolic cohomology, while it is observed in [25] that no closed 3-braid can contain a closed embedded totally geodesic surface. The Fibonacci manifolds discussed in [28] are the two-fold branched covers of the Turk’s head links, and the respective cohomology calculations are closely related. We should also remark that computer-assisted calculations are possible using the Fox calculus and group representations computed from ideal triangulations with SnapPea. As an example, among knots in S3 with fewer than eleven crossings, we have found only three that have nontrivial parabolic cohomology. (Two of these are the Turk’s head links 818 and 10123 , and the third is 1099 in the standard tables.) Thus, vanishing results of this kind appear to be a promising approach to the Menasco-Reid conjecture; in addition, they can be used to produce many interesting closed examples in light of our main theorem. Indeed, one of our main goals was to find counterexamples to the conjectures of Kapovich mentioned in the introduction. We may now do so by considering closed manifolds obtained by Dehn filling on certain hyperbolic knots in S3 . For instance, the 3-braids e2−1 e12 e2−3 e1 e2−1 e12 and e2−1 e1 e2−3 e1 e2−2 e12 close up to the knots 1091 and 1094 , respectively, each of which is hyperbolic and satisfies P H 1 (π, so(4, 1)) = 0 using SnapPea. 5.1 All but finitely many Dehn fillings on 1091 and 1094 yield closed hyperbolic 3manifolds that contain at least one closed, embedded, quasi-Fuchsian surface. PROPOSITION


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Proof Let K ⊂ S3 be one of these knots, and let M = S3 \ K . First, Thurston’s Dehn surgery theorem shows that all but finitely many Dehn fillings on K are hyperbolic. Using [24, Cor. 3.7], M contains a closed, orientable, incompressible surface 6 that remains incompressible after any nontrivial Dehn filling. We claim that when the resulting closed manifold M 0 is hyperbolic, the resulting incompressible surface 6 0 ⊂ M 0 is quasi-Fuchsian. If it were not, results of F. Bonahon and Thurston imply that 6 0 would lift to a fiber in a fibration over S1 in some finite cover of M 0 . But then 6 0 is either itself a fiber in a fibration of M 0 or separates M 0 into two twisted I -bundles over a nonorientable surface (see [26]). The first possibility can be excluded because 6 and 6 0 are separating. To exclude the second possibility, observe that 6 separates S3 into two connected components, M0 (containing the knot) and M1 (not containing the knot). The manifold M1 is not a twisted I -bundle over a nonorientable surface since H 2 (M1 , Z) ∼ = H˜ 0 (M0 , Z) = 0; thus, the same is true in any manifold obtained by Dehn filling on K . 5.2 There exist infinitely many closed hyperbolic 3-manifolds that contain embedded quasi-Fuchsian surfaces and that are locally rigid in SO(4, 1). The deformation space of flat conformal structures for these manifolds contains an infinite set of isolated points. THEOREM

Proof By the previous proposition, infinitely many Dehn fillings on 1091 or 1094 are closed hyperbolic manifolds containing quasi-Fuchsian surfaces, and these are locally rigid in SO(4, 1) by our main theorem. For the second claim, we must use Thurston’s holonomy theorem (see [10]), which states that the holonomy map hol : S (M) → Hom 0, SO(4, 1) / SO(4, 1) from the deformation space of flat conformal structures on M to the representation variety is an open map and lifts to a local homeomorphism from the space of Möbius developing maps to Hom(0, SO(4, 1)). Fix a flat conformal structure σ ∈ S (M) with ρ = hol(σ ) Fuchsian (the inclusion of an SO(3, 1) lattice). Since ρ is a stable representation (see [13, §1]), there exist neighborhoods U of σ and V of ρ, and open sets U˜ and V˜ such that U (resp., V ) is the quotient of U˜ (resp., V˜ ) by the (finite) isotropy of σ (resp., ρ), and hol lifts to a homeomorphism from U˜ to V˜ . In particular, if ρ is isolated, it follows that σ is isolated as well. In our setup, hol is actually twoto-one: the isotropy of ρ in SO(4, 1) has order two, generated by the inclusion of

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−I ∈ SO(3, 1) into SO(4, 1), while the isotropy of σ is trivial (using the main result of [14] to see that σ is not fixed by the inclusion of −I ). In [9], W. Goldman gave a construction that allows one to perform “2πngrafting” on a quasi-Fuchsian surface in a hyperbolic 3-manifold, yielding an infinite family of distinct flat conformal structures with the same (Fuchsian) holonomy representation. When the Fuchsian representation is locally rigid in SO(4, 1), as in the examples constructed above, the flat conformal structures produced by Goldman’s construction are isolated in S (M). Acknowledgments. We owe a great debt to the ideas contained in the papers [13], [16], [18], and [21], and we would like to extend a special thanks to Misha Kapovich and John Millson for their interest and encouragement. Thanks also to Bill Goldman, Tom Graber, John Hempel, Geoff Mess, and Alan Reid for useful discussions on various related topics, and to the referee for several helpful comments concerning the proof of Theorem 5.2. References [1]

C. C. ADAMS, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom. 4

[2]

B. N. APANASOV, “Nontriviality of Teichmüller space for Kleinian group in space” in

(1996), 181 – 206. MR 97f:57016 5

[3] [4]

[5]

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, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297 – 326. MR 88i:57006 12 , “Geometric structures on manifolds and varieties of representations” in Geometry of Group Representations (Boulder, Colo., 1987), ed. W. M. Goldman and A. R. Magid, Contemp. Math. 74, Amer. Math. Soc., Providence, 1988, 169 – 198. MR 90i:57024 11 C. D. HODGSON, Degeneration and regeneration of geometric structures on three-manifolds, Ph.D. dissertation, Princeton University, Princeton, 1986. 4, 8 C. D. HODGSON and S. P. KERCKHOFF, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998), 1 – 59. MR 99b:57030 4 D. L. JOHNSON and J. J. MILLSON, “Deformation spaces associated to compact hyperbolic manifolds” in Discrete Groups in Geometry and Analysis: Papers in Honor of G. Mostow on His Sixtieth Birthday (New Haven, Conn., 1984), ed. R. Howe, Progr. Math. 67, Birkhäuser, Boston, 1987, 48 – 106. MR 88j:22010 1, 11, 12 ` KAPOVICH, Conformal structures with Fuchsian holonomy, Soviet Math. Dokl. M. E. 38 (1989), 14 – 17. MR 90c:57007 12 , “Deformation spaces of flat conformal structures” in Proceedings of the Second Soviet-Japan Joint Symposium of Topology (Khabarovsk, Russia, 1989), Questions Answers Gen. Topology 8, Sympos. Gen. Topology, Osaka, Japan, 1990, 253 – 264. MR 91a:57019 2 , Deformations of representations of fundamental groups of three-dimensional manifolds, Siberian Math. J. 32 (1991), 33 – 38. MR 93m:57013 1, 4, 12 , “Flat conformal structures on 3-manifolds (survey)” in Proceedings of the International Conference on Algebra (Novosibirsk, Russia, 1989), Part 1, ed. L. A. Bokut, Yu. L. Ershov, and A. I. Kostrikin, Contemp. Math. 131, Amer. Math. Soc., Providence, 1992, 551 – 570. MR 93f:57015 2 , Deformations of representations of discrete subgroups of SO(3, 1), Math. Ann. 299 (1994), 341 – 354. MR 95d:57010 2, 5, 6, 9, 12 , Hyperbolic Manifolds and Discrete Groups, Progr. Math. 183, Birkhäuser, Boston, 2001. CMP 1 792 613 8 , Topological aspects of Kleinian groups in several dimensions, preprint, 1992, http://math.utah.edu/˜kapovich/eprints.html 2 ` KAPOVICH and J. J. MILLSON, On the deformation theory of representations of M. E. fundamental groups of compact hyperbolic 3-manifolds, Topology 35 (1996), 1085 – 1106. MR 97h:57029 12 C. KOUROUNIOTIS, Deformations of hyperbolic structures, Math. Proc. Cambridge Philos. Soc. 98 (1985), 247 – 261. MR 87g:32022 1 J. LAFONTAINE, Modules de structures conformes et cohomologie de groupes discrets, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 655 – 658. MR 85k:53042 1 M. T. LOZANO and J. H. PRZYTYCKI, Incompressible surfaces in the exterior of a closed 3-braid, I: Surfaces with horizontal boundary components, Math. Proc. Cambridge Philos. Soc. 98 (1985), 275 – 299. MR 87a:57013 11

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complements” in Topology ’90 (Columbus, Ohio, 1990), ed. B. N. Apanasov, W. D. Neumann, A. W. Reid, and L. C. Siebenmann, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin, 1992, 215 – 226. MR 94g:57016 10 L. MOSHER, “Examples of quasi-geodesic flows on hyperbolic 3-manifolds” in Topology ’90 (Columbus, Ohio, 1990), ed. B. N. Apanasov, W. D. Neumann, A. W. Reid, and L. C. Siebenmann, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin, 1992, 227 – 241. MR 93i:58120 11 W. D. NEUMANN and D. ZAGIER, Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307 – 332. MR 87j:57008 8, 9 K. P. SCANNELL, Infinitesimal deformations of some SO(3, 1) lattices, Pacific J. Math. 194 (2000), 455 – 464. MR 2001c:57018 1, 2, 3, 10 S. P. TAN, Deformations of flat conformal structures on a hyperbolic 3-manifold, J. Differential Geom. 37 (1993), 161 – 176. MR 94a:57029 1 W. P. THURSTON, The geometry and topology of three-manifolds, notes, Princeton University, Princeton, 1980, http://msri.org/publications/books/gt3m 1, 4 A. WEIL, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149 – 157. MR 30:199 10 H. WHITNEY, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545 – 556. MR 20:2342 9

Department of Mathematics and Computer Science, Saint Louis University, Saint Louis, Missouri 63103, USA; [email protected]


ON THE LOCALLY BRANCHED EUCLIDEAN METRIC GAUGE JUHA HEINONEN and DENNIS SULLIVAN

To Professor Yuri˘ı Grigor’evich Reshetnyak on his seventieth birthday Abstract A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to produce the branched coordinates for the gauge. 1. Introduction A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz; that is, locally the ratio d(x, y)/d 0 (x, y) of two metrics is bounded from above and below by positive constants independent of the points x and y. In this paper, we present a characterization for metric gauges that are locally “branched Euclidean” and discuss an obstruction to removing the branching. We consider n-dimensional gauges that are embeddable in a finite-dimensional Euclidean space and whose local cohomology groups in dimensions (n − 1) and higher are similar to those of an n-manifold. Our approach is to stipulate enough structure so that one can consider differential Whitney 1-forms on the gauge together with an orientation on the measurable cotangent bundle that is compatible with a chosen local topological orientation. We call an n-tuple ρ = (ρ1 , . . . , ρn ) of locally defined 1-forms on an n-dimensional gauge a (local) Cartan-Whitney presentation of the gauge if ess inf ∗ (ρ1 ∧ · · · ∧ ρn ) > 0. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 14 November 2000. Revision received 29 May 2001. 2000 Mathematics Subject Classification. Primary 57M12; Secondary 30C65. Heinonen’s work supported by National Science Foundation grant number DMS-9970427. Sullivan’s work supported by National Science Foundation grant number DMS-9975527.

15

(1.1)

16

HEINONEN and SULLIVAN

We prove that if the gauge supports, in addition, a Poincaré inequality, then each (local) Cartan-Whitney presentation ρ determines a positive integer-valued function Res(ρ, ·), the residue of the presentation, such that the metric gauge is locally Euclidean at a point p if and only if the residue (of some presentation) satisfies Res(ρ, p) = 1. Moreover, for each presentation ρ, the residue function Res(ρ, ·) assumes the value 1 on a dense open set of full measure with complement at most (n − 2)-dimensional. In particular, the existence of local Cartan-Whitney presentations implies that the gauge is locally Euclidean almost everywhere. The main ingredient of the proof is a general form of a theorem of Reshetnyak [Re1]. We show that the map Z x 7→ f (x) = (ρ1 , . . . , ρn ), (1.2) [ p,x]

defined through integration of the 1-forms ρ1 , . . . , ρn as in (1.1), defines a Lipschitz branched cover into Rn , with the property that lim inf

y→x, y6 =x

| f (x) − f (y)| ≥c >0 d(x, y)

(1.3)

for all x and for some c > 0 independent of x. The residue Res(ρ, p) is the local index of the map (1.2) at p. All this is made more precise in our main theorem, Theorem 4.2. To prove the theorem, we make use of the recent advances in differential analysis and nonlinear potential theory on metric measure spaces with Poincaré inequality. The metric gauges that admit local Cartan-Whitney presentations need not be manifolds in general, and even if they are manifolds they need not be locally Euclidean (see Examples 2.4). But they are always branched Euclidean. Indeed, our study leads to a characterization of a locally branched Euclidean metric gauge. Definition 1.4 A metric gauge is said to be locally branched Euclidean if it is n-dimensional, satisfies the local cohomology condition as in Axiom I, and admits local BLD-maps into Rn . To describe the terminology in Definition 1.4, let (X, d) be a locally compact, ndimensional metric space, n ≥ 2, with integral cohomology groups in degrees (n − 1) and higher locally equivalent to those of an n-manifold (as in Axiom I). We call X locally BLD-Euclidean if every point in X has an open neighborhood U and a finiteto-one, open and sense-preserving Lipschitz map f : U → Rn such that 1 length α ≤ length f ◦ α ≤ L length α L

(1.5)

for each path α in U , where the constant L ≥ 1 is independent of α. Such maps are called maps of bounded length distortion or B L D-maps. Note that the local coho-

ON THE LOCALLY BRANCHED EUCLIDEAN METRIC GAUGE

17

mology condition allows us to speak about sense-preserving maps. Finally, the BLDcondition is bi-Lipschitz invariant, and so it makes sense to speak about (local) BLDmaps of a metric gauge into Rn . In Euclidean spaces, BLD-maps form a subclass of more general quasi-regular mappings or mappings of bounded distortion, introduced by Reshetnyak in the 1960s (see [Re1], [Re2], [Ri], [MV], [Su]). In general spaces, BLD-maps are examples of regular maps in the terminology of [DS] (see [HR2, Theorem 4.5]). The local degree function for the map in (1.2) was studied in [Su] in the context of Lipschitz manifolds; in particular, condition (1.3) was proved in [Su] in this case. Note that condition (1.3) easily follows from the BLD-condition by the path-lifting property for discrete and open maps (see [HR2, Section 3.3]). We show that a locally branched Euclidean metric gauge is characterized by four axioms, Axioms I – IV presented in Section 2, provided that we also make the a priori assumption that the gauge is locally embeddable in a finite-dimensional Euclidean space with a metric orientation on its measurable tangent bundle (see Section 3.4 for the terminology). The axioms are a mixture of analysis, geometry, and topology. They stipulate local cohomological and measure-theoretic properties of the gauge, the existence of a Poincaré inequality, and the existence of local Cartan-Whitney presentations. It remains an interesting open problem to find an additional axiom that would remove the branching in the gauge. In our approach this amounts to an analytic characterization of local Cartan-Whitney presentations whose residue is everywhere 1 (see Remark 2.5 for a conjecture). Only a few nontrivial sufficient conditions for a locally Euclidean metric gauge are known: L. Siebenmann and D. Sullivan [SS] characterized the polyhedra in high dimensions that are Lipschitz manifolds, and T. Toro [T1], [T2] found positive answers in two other special cases. For related studies and examples, see [HR1], [HR2], [L], [Se2], and [Se3]. 2. The axioms Let M be a metric gauge on a set X , and let n ≥ 2 be an integer. We describe four axioms that are shown to be necessary and sufficient for X = (X, M ) to be locally branched Euclidean. The axioms are explained and analyzed more carefully in Section 3. Axiom I. X is locally compact and has integral cohomology modules in degrees (n − 1) and higher locally equivalent to those of an n-manifold. Axiom II. X is metrically n-dimensional, locally bi-Lipschitz embeddable in some Euclidean space, and locally metrically orientable.

18


Axiom III. X supports a Poincaré inequality. As is explained in Sections 3.21 and 3.23, Axioms I and II allow us to define gauge Whitney 1-forms on X . These are bounded measurable 1-forms on X with bounded exterior derivatives. A locally defined n-tuple ρ = (ρ1 , . . . , ρn ) of gauge Whitney 1-forms on X is called a (local) Cartan-Whitney presentation of the gauge if the associated volume form has constant sign and is uniformly bounded away from zero (in the almost everywhere sense). Axiom IV. Cartan-Whitney presentations exist locally on X . It is implicitly assumed that each of the above axioms includes the preceding ones to the extent its definition so requires. We have the following theorem. 2.1 Let X be a metric gauge that satisfies Axioms I and II. Then X is locally branched Euclidean if and only if it satisfies Axioms III and IV. THEOREM

The sufficiency part in Theorem 2.1 follows from our main theorem, Theorem 4.2, which is formulated in Section 4. For a further discussion of the axioms and for a proof of the necessity part in Theorem 2.1, see Section 5. The following theorem sums up some of the consequences of Theorem 4.2. THEOREM 2.2 If a metric gauge satisfies Axioms I – III, then each (local) Cartan-Whitney presentation ρ of the gauge determines (locally) a positive upper semicontinuous integer valued function, p 7→ Res(ρ, p), called the residue of the presentation, which for a fixed p is continuous in ρ in the L ∞ -topology, with the property that the gauge is locally Euclidean at a point p ∈ X if and only if a Cartan-Whitney presentation ρ can be found near p such that the residue satisfies Res(ρ, p) = 1. Moreover, for each (local) Cartan-Whitney presentation ρ, there is a closed set of zero measure and of topological dimension at most (n − 2) such that Res(ρ, p) = 1 for each p outside the closed set. COROLLARY 2.3 If a metric gauge satisfies Axioms I – IV, then it is locally Euclidean outside a closed set of zero measure and of topological dimension at most (n − 2).


19

Examples 2.4 If X is a compact polyhedron in some Euclidean space such that every point in X has a cone neighborhood with a link that is a homology (n − 1)-sphere, then X satisfies Axioms I – IV. The first two axioms are straightforward to verify, and the Poincaré inequality follows, for example, from [HeKo2, Section 6]. To establish Axiom IV, we observe that the method of J. Alexander [Al] (see also [BE]) can be used to map X locally to the standard (polyhedral) n-sphere by a sense-preserving piecewise linear branched cover. The pullback of the standard coframe on the sphere provides gauge Whitney 1-forms with property (1.1). It is known that the n-sphere Sn has polyhedral metrics as above that are not locally Euclidean if n ≥ 5 (see [Ca2], [Ca1], [E], [SS]). Also, on the lower-dimensional spheres S3 and S4 , there are metric gauges that are not locally Euclidean even though they satisfy Axioms I – IV (see [Se2], [Se3], [HR1], [HR2]). It is not known whether a 2-dimensional gauge is locally Euclidean if it satisfies Axioms I – III. Recently, T. Laakso [L] proved that a 2-dimensional gauge need not be locally Euclidean if it satisfies Axioms I and III, and is metrically 2-dimensional. Remark 2.5 It remains an interesting open problem to find a sharp analytic condition on a CartanWhitney presentation ρ that would assure that the residue of the presentation satisfies Res(ρ, p) = 1 for all p. We conjecture that this is the case for each presentation ρ in the Sobolev class H 1,2 (see Section 3.12 for the definition of Sobolev classes). This condition would be sharp, as shown by the pullback presentation in R2 under the map (r, θ) 7 → (r, 2θ) in polar coordinates. It was proved in [HeKi] that Res(ρ, ·) ≡ 1 for presentations ρ ∈ H 1,2 in Rn which are closed (i.e., dρ = 0). One can also ask if the membership of ρ in the space VMO (or perhaps in BMO with small norm) leads to the residue value 1. Here VMO and BMO stand for the spaces of vanishing mean oscillation and bounded mean oscillation, respectively. 3. Description of the axioms In this section, we describe the content of our axioms more carefully. 3.1. Cohomology manifolds Axiom I concerns only the local homology of the gauge. The second requirement means that for each x ∈ X there are arbitrarily small open neighborhoods U of x such p that Hcn (U ) = Z, that Hc (U ) = 0 for p = n − 1 and for p > n, and that the standard homomorphism Hcn (V ) → Hcn (U ) is a surjection if V ⊂ U is an open neighborhood of x. (Here Hc∗ denotes the Alexander-Spanier cohomology with compact supports.)

20


If X satisfies Axiom I and is finite-dimensional, then dim X = n (see [HW, p. 151]). 3.2. Metrically n-dimensional sets We call a metric space metrically n-dimensional if it can be expressed as a countable union of Lipschitz images of subsets of Rn plus a set of Hausdorff n-measure zero, and if for each compact set K in the space, there is a constant C K ≥ 1 such that C K−1r n ≤ Hn B(x, r ) ≤ C K r n (3.3) for all balls B(x, r ) of radius r < C K−1 centered at x ∈ K . Here and hereafter, Hn denotes the Hausdorff n-measure in a metric space. Thus, a space is metrically n-dimensional if it is (countably) n-rectifiable in the sense of geometric measure theory and if it satisfies an appropriate local version of the condition known as Ahlfors n-regularity. In particular, a metrically n-dimensional space has Hausdorff dimension n, and the Hausdorff n-measure is locally finite and positive. Being metrically n-dimensional is a bi-Lipschitz invariant condition, and it is now clear what the first requirement in Axiom II means. The second requirement, that X be locally embeddable in some Euclidean space, means that every point in X has a neighborhood that can be bi-Lipschitz embedded in some R N . 3.4. Metric orientation A gauge as in Axiom I is locally orientable in the sense that every point in it has a connected neighborhood U with Hcn (U ) = Z. A choice of a generator gU in Hcn (U ) is an orientation of U ; it canonically determines an orientation of each connected open subset V of U , for the canonical homomorphism Hcn (V ) → Hcn (U ) is an isomorphism. Assume now that U is an oriented open subset of X and that f : U → E is a continuous map into an oriented n-dimensional real vector space E. Then, for each open connected set D with compact closure in U and for each component A of E \ f (∂ D), the map f | f −1 (A) ∩ D : f −1 (A) ∩ D → A is proper, and the local degree µ(A, D, f ) is the integer that satisfies ξ E 7 → µ(A, D, f ) gU

(3.5)

≈ Hcn (E) ←−−−− Hcn (A) → Hcn f −1 (A) ∩ D → Hcn (U ),

(3.6)

under the map

where ξ E and gU denote the fixed orientations of E and U , respectively.


21

Next, assume that U ⊂ R N is a metrically n-dimensional embedded neighborhood of a point in X of finite Hausdorff n-measure. Then U has a unique (approximate) tangent n-plane Tx U at Hn -a.e. point x ∈ U (see [F, Theorem 3.2.19]). We view the collection of these planes as a measurable tangent bundle T U over U . The bundle T U is a measurable subbundle of T R N (with respect to the measure Hn bU ) and inherits a metric from R N. An orientation ξ = {ξx } of T U is a measurable choice of orientations on the approximate tangent planes: ξ = ξx : ξx ∈ ∧n Tx U is a simple (unit) n-vector . (3.7) To say that U is metrically oriented is to say that an orientation ξ on T U can be chosen so as to be compatible with a given local orientation gU on U ; such compatibility allows us to use a degree theory for Lipschitz mappings as in the case of a smooth manifold. To give a precise definition, let x ∈ U be a point such that the tangent space Tx = Tx U exists. Because U satisfies (3.3) locally, the set y ∈ R N : dist(y − x, Tx ) > |y − x| (3.8) does not meet U near the point x for each > 0. (Indeed, otherwise the Ahlfors regularity condition (3.3) would imply that U has positive n-density at x along a set as in (3.8), contradicting the definition for approximate tangent planes; see [F, Theorem 3.2.19].) Thus, if πx denotes the projection πx : R N → x + Tx to the affine n-plane x + Tx , the preimage πx−1 (x) does not meet U \ {x} near x. In particular, πx induces a map Hcn (Tx ) → Hcn (U )

(3.9)

as in (3.6). It is easy to see that this map does not depend on the choice of the domain D in (3.6) for D sufficiently small. Then we say that U is metrically orientable if U is orientable and if there is an orientation gU of U and an orientation ξ = {ξx } of T U such that ξx 7→ gU under the map in (3.9) for Hn -a.e. point x ∈ U . The pair (gU , ξ ) is called a metric orientation of U . Finally, we say that X is locally metrically orientable if every point in X has a neighborhood that is metrically orientable. Example 3.10 A metric space is locally linearly contractible if for each compact set K in the space

22


there is a constant C K > 0 such that for points x ∈ K and radii r < C K−1 , the metric balls B(x, r ) are contractible in concentric balls B(x, C K r ). It clearly makes sense to speak about a locally linearly contractible metric gauge. It is not hard to see that if X satisfies Axiom I, is metrically n-dimensional and locally embeddable in some Euclidean space, and is locally linearly contractible, then X is locally metrically orientable. Indeed, if U is as above, the local linear contractibility guarantees that there is a neighborhood G of U in R N and a retraction ψ : G → U such that, locally, ψ(y) − y ≤ C dist(y, U ) (3.11) with C ≥ 1 independent of y. It is then easy to see, with the above notation, that the map f = ψ ◦ πx : D → U is homotopic to the identity through maps D \ {x} → U \ {x} if D is a small enough connected open neighborhood of x in U . By using this and (3.11), one checks that πx induces an isomorphism in (3.9), and the metric orientation can be defined via this isomorphism. (See [Se1] for more discussion on local linear contractibility and related issues.) 3.12. Sobolev classes Assuming that X satisfies Axiom II, we define Sobolev spaces H 1, p (U ) in each embedded metrically n-dimensional neighborhood U in R N . (The metric orientation is not needed here.) Although the spaces to be defined depend on the chosen embedding, the membership in a space of a particular degree of integrability does not. Therefore, it makes sense to speak about (local) Sobolev classes of functions on X . Because of the rectifiability properties of U , the definition of the Sobolev space H 1, p (U ) is rather straightforward. In particular, we do not need the recent and more sophisticated (albeit equivalent) Sobolev space theories as in, for example, [Cr], [FHK], or [Sh]. Thus, let U be a metrically n-dimensional set in R N of finite Hausdorff nmeasure. There is a bounded linear operator d from Lipschitz functions defined on U to bounded measurable sections of T ∗ U which vanishes on (locally) constant functions and satisfies |du| ≤ Lip(u), (3.13) where Lip(u) is the Lipschitz constant of u, as well as d(uv) = v du + u dv

(3.14)

d( f ◦ u) = f 0 (u) du

(3.15)

and


23

for all f ∈ C 1 (R), where it is understood that (3.13) – (3.15) hold almost everywhere with respect to the Hausdorff measure Hn on U . Indeed, du is the approximate differential of u, as in [F, Theorem 3.2.19]. For 1 < p < ∞ we define the Sobolev space H 1, p (U ) as the closure of all Lipschitz functions on U in the norm Z 1/ p Z 1/ p p ||u||1, p = |u| d Hn + |du| p d Hn . (3.16) U

U

Thus, u ∈ H 1, p (U ) if and only if u ∈ L p (U ) and there are a measurable L p integrable section α of T ∗ U and a sequence (u j ) of Lipschitz functions on U such that u j → u in L p (U ) and du j → α in L p (U ). 3.17. Poincaré inequality We say that X supports a Poincaré inequality if X is locally pathwise connected and if every point in X has a metrically n-dimensional embedded neighborhood U , together with constants C ≥ 1 and τ ≥ 1, such that Z Z |u − u B |2 d Hn ≤ C (diam B)2 |du|2 d Hn (3.18) B

τB

for each metric ball B satisfying τ B ⊂ U and for each Lipschitz function u in τ B, where τ B denotes the ball that is concentric with B, but with radius τ times the radius of B, and u B denotes the integral average of u in B. The validity of a uniform Poincaré inequality of the type (3.18), together with mild assumptions on the Hausdorff measure, implies that the space possesses many strong geometric and analytic properties (see, e.g., [Cr], [HaKo], [HeKo2], [Se5], [Sh]). In particular, we require the following fact, proved in [FHK, Theorem 10]. PROPOSITION 3.19 If U is a metrically n-dimensional embedded neighborhood that supports a Poincaré inequality as in (3.18), then the operator d from Lipschitz functions on U to L 2 sections of T ∗ U is closable.

Thus, under the presence of a Poincaré inequality, the section α above is independent of the sequence (u j ); it is denoted by du and called the weak differential of u. Besides being used in Proposition 3.19, inequality (3.18) is used in Proposition 4.22, which in turn is crucial in the proof of our main theorem, Theorem 4.2. We note that there are variants of condition (3.18) that could equally well be used in Axiom II; we have chosen (3.18) for its relative simplicity (see Section 5.2 for a further discussion).

24


Example 3.20 If X satisfies Axioms I – II and in addition is locally linear contractible (as defined in Example 3.10), then X supports a Poincaré inequality as described above. This follows from work of S. Semmes [Se1]. As pointed out in Section 5.4, in Theorem 2.1 we cannot replace Axiom III by the requirement that X be linearly locally contractible. 3.21. Whitney forms Let U be an n-rectifiable subset of R N of finite Hausdorff n-measure, and let ξ = {ξx } be an orientation on T U as in (3.7). Then the pair (U, ξ ) defines an n-dimensional current by integration: for each smooth n-form ω in R N, the action Z

(U, ξ ), ω = ω(x), ξx d Hn (x) (3.22) U

is defined in the usual way by using the chosen a.e. defined orientation and the Hausdorff measure. The action (3.22) can be extended to a larger class of differential forms in R N, which we call Whitney forms. By definition, these are forms ω of bounded measurable coefficients whose distributional exterior differential dω also has bounded measurable coefficients. For Whitney forms we have ddω = 0 in the sense of distributions. One can also pull back Whitney forms by Lipschitz maps, and d F ∗ (ω) = F ∗ (dω) if F is Lipschitz and ω is a Whitney form. To briefly explain why (3.22) extends to Whitney forms, we recall that the space of Whitney m-forms as defined above can be identified as the dual of flat m-chains in R N in the flat norm (see [W2, Section IX.7], [F, Section 4.1.19]). Now every mdimensional rectifiable current in R N is a flat m-chain by [F, Section 4.1.24], and every oriented m-dimensional rectifiable set is an m-dimensional rectifiable current through formula (3.22) by [F, Section 4.1.28] (cf. Section 3.26). Although Whitney m-forms are a priori only a.e. defined with respect to Lebesgue measure of the ambient space R N , they have representatives such that the action on m-dimensional rectifiable currents makes sense by integration. This is a theorem of H. Whitney [W2, Theorem 9A, p. 303]. (Compare this with the special and better-known case m = 0 when the Whitney forms are nothing but Lipschitz functions; the differential of a Lipschitz function has a well-defined restriction to each rectifiable curve.) Whenever we are dealing with Whitney forms in this paper, we tacitly assume that the good representatives have been picked. Finally, Whitney forms can be defined and studied in any open set in R N . 3.23. Gauge Whitney 1-forms If X satisfies Axioms I and II, then we can define Whitney 1-forms locally on X via their action on rectifiable curves by using the local embeddings in Euclidean space


25

and the bi-Lipschitz invariance of line integrals. We call these invariantly defined forms gauge Whitney 1-forms. To be more precise, we let U be open in X ; we abuse notation and understand that U is already embedded in some R N . Then every Whitney 1-form ω defined in an open neighborhood G of U in R N determines a gauge Whitney 1-form: if U 0 is a different embedding of U in R M , then there is a bi-Lipschitz homeomorphism h from U 0 onto U ; the map h can be extended to a Lipschitz map H : R M → R N by Kirszbraun’s theorem (see [F, Theorem 2.10.43]), and the Whitney 1-form H ∗ (ω) is defined in the open neighborhood G 0 = H −1 (G) of U 0 and satisfies Z Z H ∗ (ω) = ω H −1 (γ )

γ

for all rectifiable curves γ in U . Axiom IV means that for each point in X there are a metrically oriented neighborhood U and an n-tuple of gauge Whitney 1-forms ρ1 , . . . , ρn defined in U such that ess inf ∗ (ρ1 ∧ · · · ∧ ρn ) > 0, (3.24) where the Hodge star operator ∗ : ∧n T U → R is determined by the given metric orientation. Note that ρ1 ∧ · · · ∧ ρn is a Whitney n-form by [W2, p. 277] or by [Re2, Lemma 4.4, p. 133]. Condition (3.24) is independent of the chosen embedding of U in Euclidean space. Remark 3.25 We could have defined gauge Whitney 1-forms in Section 3.23 more intrinsically (say, in the spirit of [W1, p. 4]) without the requirement of local extension in the ambient Euclidean space. However, such an extension is necessary for our proof of Theorem 4.2. If we strengthened Axiom II by requiring that X be locally embeddable in some Euclidean space as a (local) Lipschitz retract, then local extensions of intrinsically defined forms would exist. This new axiom would also imply the current Axiom III, via local linear contractibility as in Remark 3.20 but would not be necessary for the gauge to be locally branched Euclidean (see Section 5.4). It is not clear whether intrinsically defined gauge Whitney 1-forms can be extended (locally) to the ambient Euclidean space under the present axioms, nor whether our proof could be made to work without such extension. 3.26. Stokes cycles We now address the precise technical sense in which our objects define “abstract cycles” locally; namely, for each local embedding of the gauge in R N , we have the expected integration by parts formula. In more detail, recall that an n-dimensional rectifiable current in R N is a current with compact support which is representable

26


by integration over an oriented n-rectifiable set with integer multiplicities (see [F, Theorem 4.1.28]). Thus, each n-dimensional rectifiable current is associated with a triple (W, ξ, µ), where W is an n-rectifiable set, ξ = {ξx } is a measurable choice of unit n-vectors on ∧n T W , and µ is an integer-valued Hn -integrable (multiplicity) function on W . We call a current T in R N an n-dimensional Stokes current if it is an n-dimensional rectifiable current with compact support and if an associated triple can be chosen so that the set W is locally compact and satisfies W ∩ spt ∂ T = ∅.

(3.27)

In other words, each point in W should have a neighborhood such that h∂ T , ωi = hT, dωi = 0 for each smooth, and hence Whitney, (n − 1)-form ω with support in the neighborhood. If we start with an n-rectifiable, locally compact bounded set W , together with a choice of orientation ξ on T W , and if (3.27) holds for the current T = (W, ξ ), then we say that W represents an n-dimensional Stokes current in the orientation ξ . Stokes currents allow for localization: if (W, ξ ) is an n-dimensional Stokes current and W 0 ⊂ W is open in W , then (W 0 , ξ 0 ) is an n-dimensional Stokes current, where ξ 0 = ξ |W 0 is the restriction of the orientation ξ to W 0 . We call an n-dimensional metric gauge a (local) Stokes cycle if every point in the gauge has an embedded neighborhood in some Euclidean space which represents an n-dimensional Stokes current in a metric orientation; whether or not an embedded neighborhood has this property is independent of the choices. We learned the proof of the following proposition from Stephen Semmes, whose participation we thus gratefully acknowledge. 3.28 If X satisfies Axioms I and II, then X is a local Stokes cycle. PROPOSITION

Proof Let U be an embedded (in R N ) metrically oriented neighborhood of a point p in X , and denote by T = (U, ξ ) the corresponding n-current. We have to show that there is δ > 0 such that Z

hT, dωi = dω(x), ξx d Hn (x) = 0 (3.29) U

for each smooth (n − 1)-form ω with support in the N -ball B( p, δ). We first show that (3.29) is true for forms ω of the form ω = F ∗ (α),

(3.30)


27

where α is a smooth (n − 1)-form on Sn and F : R N → Sn is a smooth map, homotopic to a constant through maps Ft : R N → Sn such that Ft |U \ B( p, δ) ≡ σ , where σ ∈ Sn is independent of t. To this end, we use the pushforward current F# (T ), which satisfies

F# (T ), dα = T, d F ∗ (α) = hT, dωi. We have the integral expression

F# (T ), dα =

Z

Sn

dα(y), η(y) dy,

where dy = d Hn (y) on Sn , and η(y) = 0 or X η(y) = a(x)O y = O y x∈F −1 (y)∩U

X

a(x).

x∈F −1 (y)∩U

Here O = {O y } denotes the standard orientation on Sn and a(x) = 1 if the approximate differential d F(x) : (Tx , ξx ) → (Ty , O y ) is sense preserving, and a(x) = −1 in the opposite case (see [F, Section 4.1.30] for these facts). Now the sum on the right is the sum of the signs of the Jacobians of F, which equals the degree of F, thus zero, almost everywhere (cf. (4.29)). (The assumption on metric orientation is used here.) Therefore, (3.29) follows for forms as in (3.30). To prove the general case, we observe first that by linearity and by change of coordinates we may assume that ω is of the form ω(x) = u(x) dλn−1 , where dλn−1 = d x1 · · · d xn−1 and u is a smooth function with compact support. If b(x) is any smooth bump function with support on B = B( p, δ) and total integral 1, then the convolution ω = b ∗ ω satisfies h∂ T , ω i → h∂ T , ωi as → 0 (see [W2, (12), p. 176]). On the other hand, one easily computes that h∂ T , ω i = hT, dω i = hT, db ∗ ωi Z Z

= u(y) db (x − y)dλn−1 , ξx d x dy, spt ω

U

which is zero, provided Z U

db (x − y)dλn−1 , ξx d x = 0

for all y ∈ R N . This reduces the problem to the case where ω(x) = b(x) dλn−1

(3.31)

and b(x) is a bump function of our choice. It remains to find a form ω that is both of the form (3.30) and the form (3.31). To this end, let α be an (n − 1)-form on Sn−1 which is a volume form multiplied by a

28


nonnegative (but nonzero) function on Sn−1 so that α ≡ 0 in a small neighborhood of a point σ ∈ Sn−1 . Next, extend α to be an (n − 1)-form on Sn ⊃ Sn−1 as follows: first, pull back by the map π1 ◦ π2 , where π2 : Rn+1 \ {xn+1 − axis} → Rn \ {0} and π1 : Rn \ {0} → Sn−1 are projections; then multiply by a nonnegative function of |xn+1 | which vanishes if |xn+1 | ≥ 1/4 and is equal to 1 in a neighborhood of 0; and finally, restrict (and extend) to Sn ⊂ Rn+1 . Note that this extension α vanishes on definite neighborhoods of the south and the north poles of Sn , as well as on a neighborhood of a great circle that connects the poles through the point σ . We now describe a mapping F : R N → Sn as required in (3.30). First, let F1 : Rn−1 → Sn−1 be a map of degree 1 that assumes the value σ outside B(0, δ). (We may assume that p = 0 ∈ R N and that δ > 0 is small.) Then, using coordinates (λ, x 0 ) in R N = Rn−1 × R N −n+1 , we define F = ϕ ◦ F2 , where F2 (λ, x 0 ) = (F1 (λ), (1/π)arctan(|x 0 |/δ)), F2 : R N → Sn−1 × [0, 1/2), and ϕ is the projection of Sn−1 × [0, 1/2) onto an open subset of Sn . It is easy to see that ω = F ∗ (α) depends only on λ = (x1 , . . . , xn−1 ) and |x 0 | = |(xn , . . . , x N )|; indeed, it is easy to see from the definitions that ω is of the form ω(λ, x 0 ) = f (λ)h(|x 0 |) d x1 · · · d xn−1 for some nonnegative (but nonzero) functions f and h. This completes the proof of Proposition 3.28. 4. Locally branched Euclidean gauge We assume in this section that X satisfies Axioms I – IV. Pick a point p ∈ X and an embedded (open, connected) neighborhood U of p in some R N . We assume that U is metrically oriented by (gU , ξ ), that the current (U, ξ ) is an n-dimensional Stokes current, and that a Cartan-Whitney presentation ρ = (ρ1 , . . . , ρn ) is given on U . Thus, the Whitney 1-forms ρ1 , . . . , ρn are defined in a neighborhood G of U in R N , and they satisfy ess inf ∗ (ρ1 ∧ · · · ∧ ρn ) ≥ δ > 0 (4.1) in U . By shrinking U if necessary, we may also assume that each 2-simplex [x, y, z], generated by points x, y, z in U , lies in G. In particular, each line segment [ p, x] for x ∈ U lies in G. Under these assumptions, we prove the following theorem, which should be regarded as the main result of this paper. THEOREM 4.2 The neighborhood U can be chosen small enough so that the mapping Z f (x) = (ρ1 , . . . , ρn ), x ∈ U,

(4.3)

[ p,x]

is a sense-preserving, discrete, and open Lipschitz mapping from U to Rn which sat-


29

isfies (1.5). In particular, we have lim inf

y→x, y6 =x

| f (x) − f (y)| ≥c>0 |x − y|

(4.4)

for all x ∈ U and for some c > 0 independent of x. The mapping f is locally biLipschitz outside the branch set B f of measure zero and of topological dimension at most n − 2. A mapping f : U → Rn is sense-preserving if the local degree µ(A, D, f ) defined in (3.5) is positive for each domain D compactly contained in U and for each component A of Rn \ f (∂ D) that meets f (D); we assume that Rn is equipped with its standard orientation. A map is discrete if the preimage of each point is a discrete set, and the branch set B f is the closed point set in the domain of f where f does not define a local homeomorphism. For a discrete and open mapping f , the branch set B f always has topological dimension at most n − 2 by [Ch1], [Ch2], and [V2]. (The hypotheses in [Ch1], [Ch2], [V2] are somewhat different from what is required by Axiom I. However, the proof in [V2] in particular is valid in the present context.) What we show here is that the mapping f given in (4.3) is a discrete, open, and sense-preserving map with volume derivative uniformly bounded away from zero. The BLD-property (1.5) for such maps follows from [HR2, Theorem 6.18]. To be precise, our axioms are slightly weaker than the assumptions on the source space X in [HR2]. The axioms are sufficient, however, to run the proof in [HR2, Theorem 6.18], for the required analysis there needs only the validity of a Poincaré inequality. This is clear from the references used in the proof of [HR2, Theorem 6.18]. As an additional technical point, one needs to know here that a Poincaré inequality as in (3.18) implies quasiconvexity of the space (for this, see [Cr, Appendix] or [HaKo, Proposition 4.4]). Finally, observe that property (1.3) follows from (1.5) via a simple path-lifting argument (cf. [HR2, Section 3.3]) and that the sufficiency part of both Theorem 2.1 and Corollary 2.3 follows from Theorem 4.2. Next, we define the residue of ρ by Res(ρ, p) = the local degree at p of the map f given in (4.3).

(4.5)

Thus, Res(ρ, p) = 1 if and only if p lies outside the branch set of f , and we conclude that Theorem 2.2 follows from Theorem 4.2, except the claim about continuity in ρ, which is clear from the proof. Proof of Theorem 4.2 The proof is presented in several subsections.

30


4.6. f is Lipschitz with uniformly positive volume derivative One should compare the argument here to that given in [Su] in the context of Lipschitz manifolds. We have f (x) − f ( p) = f (x) ≤ ||ρ||∞ |x − p|, where ρ = (ρ1 , . . . , ρn ). If x ∈ U , x 6= p, and y is near x, then Z Z Z Z f (x) − f (y) = ρ− ρ+ ρ− ρ [ p,x] [ p,y] [x,y] [x,y] Z Z dρ + ρ ≤ [ p,x,y] [x,y] ≤ ||dρ||∞ [ p, x, y] + ||ρ||∞ |x − y|. The area |[ p, x, y]| of the 2-simplex [ p, x, y] is at most a constant times |x − y|, and we conclude that f is uniformly locally Lipschitz in U . In fact, f is uniformly locally Lipschitz in a small neighborhood of p in the ambient space R N by the same argument. We may therefore assume that f is Lipschitz in U . Because f is Lipschitz, its (approximate) differential d f = (d f 1 , . . . , d f n ) determines Whitney 1-forms in a neighborhood of U in R N . (See [F, Theorem 3.2.19] and [W2, Chapter X], and recall that we can extend f to a Lipschitz map R N → Rn by Kirszbraun’s theorem.) In particular, d f ∈ L ∞ , and for x, y ∈ U we have Z Z (ρ − d f ) = ρ − f (y) − f (x) [x,y] [x,y] Z Z Z Z = ρ− ρ+ ρ= dρ, [x,y]

[ p,y]

[ p,x]

[ p,x,y]

which gives Z

[x,y]

(ρ − d f ) ≤ ||dρ||∞ [ p, x, y] ≤ ||dρ||∞ | p − x||x − y|

if |x − y| |x − p|. This implies that the L ∞ -norm of ρ − d f satisfies ||ρ − d f ||∞ ≤ ||dρ||∞ | p − x|. By further shrinking U if necessary, we thus find that d f1 ∧ · · · ∧ d fn ≥ δ0 > 0 almost everywhere in U (cf. [W2, Theorem 7C, p. 265]).

(4.7)


31

4.8. Potential theory Here we follow the fundamental ideas of Reshetnyak [Re1]. For b ∈ Rn , the function u(y) = u b (y) = − log |y − b|

(4.9)

solves the quasi-linear elliptic equation − ∗ d ∗ |du|n−2 du = 0 in Rn \ {b}, where ∗ is the Hodge star operator in Rn . In particular, the (n − 1)-form α = ∗|du|n−2 du is closed in Rn \ {b}. Because α is smooth and f is Lipschitz, we have d f ∗ (α) = f ∗ (dα) = 0

(4.10)

in the nonempty (relatively) open set U \ f −1 (b). Note that f is not constant in U by (4.7). To justify equality (4.10) and the other upcoming differential calculus on Lipschitz forms, we refer the reader to the discussion in [W2, Section X.9] (see also [Re2, Sections II.4.3 and 4.4]). Recall that we can think of f as being defined in all of R N . Next, if ϕ is a compactly supported Lipschitz function in U \ f −1 (b), we calculate d f ∗ (α)ϕ = d f ∗ (α)ϕ + (−1)n−1 f ∗ (α) ∧ dϕ = (−1)n−1 f ∗ (α) ∧ dϕ, and recalling Proposition 3.28, we thus obtain Z

0 = ∂U , f ∗ (α)ϕ = d f ∗ (α)ϕ U Z = (−1)n−1 f ∗ (α) ∧ dϕ.

(4.11)

(4.12)

U

(We suppress the fixed orientation ξ from the notation here and below.) In conclusion, Z Z

∗ ∗ 0= f (α) ∧ dϕ = ∗ f (α), dϕ d Hn , (4.13) U

U

where the ∗-operator is determined by the fixed inner product and orientation on T ∗ U . Equality (4.13) means that the 1-form ∗ f ∗ ∗ |du|n−2 du is coclosed in U \ f −1 (b) in a weak sense; that is, Z

∗ ∗ f ∗ |du|n−2 du, dϕ d Hn = 0 (4.14) U

32


for all compactly supported Lipschitz functions ϕ in U \ f −1 (b). Let us reformulate (4.14) as follows. By (4.7), for almost every x in U we can define a linear map G(x) : 31 Tx U → 31 Tx U by the formula T

G(x) = det d f (x)2/n [d f (x)]−1 [d f (x)]−1 .

(4.15)

Here T denotes transpose, determined by the fixed inner products, and we use the natural fiberwise identification of the tangent and cotangent spaces. Now (4.14) states that Z

Ax (dh(x)), dϕ(x) d Hn (x) = 0 (4.16) U

for all compactly supported Lipschitz functions ϕ in U \ f −1 (b), where h = u ◦ f is a (locally) Lipschitz function in U \ f −1 (b) and

(n−2)/2 Ax (η) = G(x)η, η G(x)η. To see this, we observe first that ∗ f ∗ ∗ = (−1)n−1 det d f d f −1

(4.17)

on 1-forms by Laplace’s formula (see, e.g., [Ri, Chapter I.1]). A pointwise calculation now shows that dh = d f T du and hence that T

T

Ax (dh) = det d f (n−2)/n hd f −1 d f −1 dh, dhi(n−2)/2 det d f 2/n d f −1 d f −1 dh T

T

= det d f |d f −1 dh|n−2 d f −1 d f −1 dh = det d f |du|n−2 d f −1 du = (−1)n−1 ∗ f ∗ ∗ |du|n−2 du almost everywhere, as required. It follows that (4.14) and (4.16) are indeed reformulations of each other. Finally, we observe that because d f is bounded and because condition (4.7) holds, we have

n/2

Fx (η) = G(x)η, η = Ax (η), η ≈ |η|n (4.18) for all measurable sections η of T ∗ U . The constants in (4.18) are independent of η.


33

4.19. Quasi-continuous Sobolev functions Recall the definition for Sobolev space H 1, p (U ) from Section 3.12. In what follows, we only need the case where p = n. (An analogous discussion is valid for all 1 < p < ∞.) For a set E ⊂ U , we define its n-capacity to be the number Z Cn (E) = inf |u|n + |du|n d Hn , (4.20) U

where the infimum is taken over all u ∈ H 1,n (U ) such that u ≥ 1 almost everywhere in an open neighborhood of E. We also need the following variational counterpart of Cn . Assume that E is a compact subset of an open set V ⊂ U . Then the variational n-capacity of E in V is the number Z capn (E, V ) = inf |du|n d Hn , (4.21) V

where the infimum is taken over all compactly supported Lipschitz functions u in V such that u ≥ 1 on E. Note that (4.21) was defined by using Lipschitz test functions, whereas (4.20) used arbitrary Sobolev functions. These are the most natural ways to define the two capacities, although it is true (and important) that in both cases the pool of test functions can be altered without altering the value of the capacity. Also, note that the definition of capn (E, V ) can be extended to arbitrary subsets of V in a standard manner (see [HKM, p. 27]). A set E in U is said to be of n-capacity zero if Cn (E) = 0. One can show by using the Poincaré inequality (3.18) that for a compact set E, we have Cn (E) = 0 if and only if capn (E, V ) = 0 for every (equivalently, some) relatively compact open set V containing E (see the arguments in [HKM, pp. 49, 34]). The next result follows from [HeKo2, Theorem 5.9]; the Poincaré inequality (3.18) is crucial here. PROPOSITION 4.22 A compact set of zero n-capacity in U has Hausdorff dimension zero.

A real-valued function u defined on a set E ⊂ U is said to be n-quasi-continuous if for each > 0 there is an open set G with Cn (G) < such that u|E \G is continuous. A sequence (u j ) of functions on E is said to converge n-quasi-uniformly to a function u on E if for each > 0 there is an open set G with Cn (G) < such that u j → u uniformly in E \ G. If a property holds except on a set of zero n-capacity, we say it holds n-quasi-everywhere.

34


PROPOSITION 4.23 A function in the Sobolev space H 1,n (U ) has an n-quasi-continuous representative, and two such representatives agree n-quasi-everywhere. Every convergent sequence of n-quasi-continuous functions in H 1,n (U ) subconverges n-quasi-uniformly to an n-quasi-continuous function.

The proof for the existence part in Proposition 4.23 is standard. We leave its detailed verification to the reader following the presentation of [HKM, Chapters 2, 4] and using properties (3.13) – (3.15). The same holds true for the last assertion in Proposition 4.23. The proof of the uniqueness up to a set of zero capacity of the quasi-continuous representative in [HKM, Theorem 4.12] is somewhat complicated, relying on nontrivial results from the theory of quasilinear variational inequalities in Rn , and it is not clear if the argument can be used in the present setting. However, T. Kilpeläinen [K] has recently given a short, elementary proof for the uniqueness that applies very generally; in particular, it applies in our case, and Proposition 4.23 follows. 4.24. f is light We show that the Lipschitz map f : U → Rn given in (4.3) is light, that is, that the preimage of every point under f is a totally disconnected set. We show that the fiber f −1 (b) has zero n-capacity in U for each b ∈ Rn . This suffices by Proposition 4.22. Here we depart from Reshetnyak’s original argument, which used Harnack’s inequality for solutions to degenerate elliptic equations, and instead follow the proof in [HeKo1]. The idea in [HeKo1] (which avoids Harnack’s inequality) was to construct an n-quasi-continuous function in a neighborhood B of each point in f −1 (b) which takes only two values: 1 on f −1 (b) ∩ B, and 0 elsewhere. Then necessarily f −1 (b) has zero n-capacity. To this end, pick b ∈ Rn and consider the function u = u b as defined in (4.9). Denote, for each positive integer k ≥ 1, 1 B, 2 where h = u ◦ f = − log | f − b| as in Section 4.8, B is some fixed open ball (in U ) centered at a point x0 ∈ ∂ f −1 (b) ∩ U such that the closed ball B lies in U , and (1/2)B denotes the closed ball with the same center as B but half the radius. Note that the required point x0 exists because f is not constant and U is connected. It suffices to show that 1 E ∞ = f −1 (b) ∩ B 2 has zero n-capacity. With the discussion in Sections 4.19 and 4.8 understood, the argument is very similar to that in [HeKo1]. For convenience we repeat the main points. (In fact, the E k = {h ≥ k} ∩


35

situation here is easier than in [HeKo1] because the degeneracy of the equation is less severe.) First, we claim that the minimization problem Z Ik = inf Fx dv(x) d Hn (x), (4.25) Fk

B

where Fk = v ∈ H01,n (B) : v is n-quasi-continuous and v ≥ 1 n-quasi-everywhere on E k

and Fx (η) is given in (4.18), is solved by a unique (up to a set of zero capacity) minimizer vk ∈ Fk . (Here the Sobolev space H01,n (B) is the closure of compactly supported Lipschitz functions in B with respect to the norm (3.16).) The proof of the claim follows the standard arguments of the calculus of variations (see, e.g., [HKM, Chapter 5] or [Re2, Chapter III.3]). We equip H01,n (B) with the uniformly convex norm Z 1/n ||v|| F = ||v||n + Fx (dv) d Hn , B

which by (4.18) is equivalent to the norm in (3.16) (with p = n), so that a minimizing sequence has a weakly convergent subsequence from which one can extract a sequence of convex combinations that converges strongly to a function vk (by Mazur’s lemma). By the lower semicontinuity of norms, vk is a minimizer, and by Proposition 4.23, we may assume that vk is in Fk . Finally, the uniqueness follows from the strict convexity of F. Next, analogously to [HeKo1, Lemma 4.8], one can show that wk = kvk ≤ h

(4.26)

quasi-everywhere in B \ E k . The crucial fact in proving (4.26) is the validity of equation (4.16): because vk uniquely minimizes (4.25), we easily obtain Z Z Fx (dwk ) d Hn < Fx (dh) d Hn , {wk >h}

{wk >h}

while on the other hand, Z Z Fx (dwk ) − Fx (dh) d Hn ≥ {wk >h}

{wk >h}

∇η Fx (dh), dwk − dh d Hn

Z =n

{wk >h}

Ax (dh), dwk − dh d Hn = 0,

36


where the last equality follows from (4.16) by (Lipschitz) approximation. It follows that wk ≤ h almost everywhere, hence quasi-everywhere by Proposition 4.23. We have thus constructed a sequence (vk ) of quasi-continuous functions in 1,n H0 (B) such that (1) vk = 1 n-quasi-everywhere on E k ; (2) vk ≤ h/k n-quasi-everywhere on B \ E k . Moreover, the sequence (vk ) is bounded in H01,n (B), so that a sequence of convex combinations of vk ’s converges strongly to a quasi-continuous function v∞ in B. (This is, again, by Mazur’s lemma and Proposition 4.23). By (1), v∞ = 1 n-quasieverywhere on E ∞ , while by (2), v∞ = 0 n-quasi-everywhere on B \ E ∞ because h is finite outside E ∞ . This implies that E ∞ has zero n-capacity, as was to be proved. Therefore, f : U → Rn is a light map. 4.27. f is sense-preserving For almost every x in U , the approximate differential d f (x) exists (see [F, Section 3.2.16 and Theoreom 3.2.19]) and satisfies det d f (x) = (d f 1 ∧ · · · ∧ d f n )(x) ≥ δ 0 > 0

(4.28)

by condition (4.7). Let D be a relatively compact domain in U , and let A be a component of Rn \ f (∂ D) which meets f (D). Because f −1 (A) ∩ D is open and nonempty, and because the fiber f −1 (y) is finite for almost every y ∈ Rn by [F, Theorem 3.2.22], there is a point y in A whose preimage in D consists of finitely many points x such that d f (x) exists, (4.28) holds, and f is approximately differentiable at x. (We use here the fact that Lipschitz maps are absolutely continuous in measure.) It follows from an easy homotopy argument that X µ(A, D, f ) = sign det d f (x) > 0, (4.29) x∈ f −1 (y)∩D

as required. Thus, f is sense-preserving in U . 4.30. Conclusion It is not hard to see that a sense-preserving light map U → Rn is discrete and open (cf. [TY], [Re2, Section II.6.3], [Ri, Section VI.5]). We have thus shown that the mapping f given in (4.3) is a sense-preserving, discrete, and open Lipschitz mapping with a definite lower bound for the Jacobian determinant as in (4.28). As discussed right after the statement of Theorem 4.2, this suffices, and the proof of Theorem 4.2 is thereby complete.


37

5. Concluding remarks There are some interesting issues about Axioms I – IV presented in this paper, that remain poorly understood and that are well worth further study. In this last section we briefly discuss these issues as well as the necessity part of Theorem 2.1. 5.1 We made the a priori assumption in Theorem 2.1 that the gauge is locally embeddable in some finite-dimensional Euclidean space as a metrically n-dimensional, metrically orientable set (see Axiom II). We do not know whether Axiom II is a necessary condition for a locally branched Euclidean metric gauge as defined in Definition 1.4. It follows from [HR2, Proposition 6.3 and Remark 4.16(a), (b)] that if f : U → n R is a Lipschitz BLD-map, where U is open in X , then U is n-rectifiable and locally Ahlfors n-regular (as in (3.3)). Thus, in the necessity part of Theorem 2.1 we do not need to assume the full Axiom II. It is conjectured in [HR2, Remark 6.32(b)] that the branch set B f of f has measure zero. If the conjecture is true, one can delete Axiom II (except, possibly, the embeddability part) from the a priori assumptions in Theorem 2.1. As to the embeddability part of Axiom II, very little is known in general about the question of which metric spaces can be bi-Lipschitz embedded in finite-dimensional Euclidean spaces. There are metrics d in Rn , for each n ≥ 2, such that the metric space (Rn , d) is n-rectifiable and Ahlfors n-regular (see Section 3.2), as well as linearly locally contractible (see Example 3.10), but not bi-Lipschitz embeddable in a finite-dimensional Euclidean space. One can even choose the metric d to be smooth Riemannian outside a closed singular set of dimension less than n, and, at the same time, not to be locally BLD-Euclidean. Note that such a space satisfies our Axioms I – III except for the embeddability part of Axiom II. See [L], [Se3] for these facts. It is interesting to note that in contrast to the bi-Lipschitz case, the embeddability problem in the quasi-symmetric category is fully understood (see [As], [Se4]). 5.2 For Axiom III we could choose the following a priori weaker form of Poincaré inequality: Z Z 1/n 1 1 |u − u B | d Hn ≤ C diam B |du|n d Hn . (5.3) Hn (B) B Hn (τ B) τ B (Notation here is as in (3.18).) Namely, the results we quoted from [FHK] and [HeKo2] in our proof for the sufficiency of the axioms use only inequality (5.3). In general, Poincaré inequalities with different L p -norms for the gradient are not equivalent, but in the presence of our axioms this is the case for 1 ≤ p ≤ n. In fact, as pointed out in [HR2, Section 9.6], locally Ahlfors n-regular spaces that admit local

38


BLD-maps to Rn support (locally) a Poincaré inequality like the one in (5.3) but with L 1 -norm on the right-hand side instead of the L n -norm. Essentially, this follows from Semmes’s work [Se1]. The presence of a Poincaré inequality alone does not imply any rectifiability properties for the space. On the other hand, one can ask if this is the case under additional metric or topological assumptions. In this regard, see the conjecture in [Cr, Conjecture 4.65]. We refer to [Cr], [FHK], [HaKo], [HeKo2], [Se1], [Se5], and [Sh] for more information on Poincaré inequalities and their use in metric space analysis. 5.4 There are metric gauges that are locally branched Euclidean but not locally contractible (cf. Examples 3.10 and 3.20). For an example, one can glue infinitely many copies of the Poincaré homology sphere minus a 3-ball along the boundaries of round 3-balls in R3 which are mutually disjoint and converge to the origin (see [Wi, p. 245]). It is not hard to do the construction metrically so that the resulting space admits BLDmaps into R3 (see [HR2, Remark 6.32(c)] for more details). 5.5 Let us finally discuss the necessity of Axioms I – IV in Theorem 2.1. Thus, let X be a metric gauge that satisfies Axioms I and II, and admits locally BLD-maps into Rn . (Note that Axiom I is incorporated in the definition for a BLD-map.) Because Axiom II holds, Axiom IV is immediate, for one can pull back the standard coframe in Rn and use the fact that the Jacobian determinant of a BLD-map is bounded away from zero (cf. [F, Chapter 3], [HR2, Section 6.13]). It remains to have the Poincaré inequality, Axiom III. Semmes showed in [Se1] (see, in particular, [Se1, Theorem 12.5 and Appendix B]) that an Ahlfors n-regular metric space (X, d) admits a Poincaré inequality, as in (3.18), for example, if for every pair of points x, y in X there exists a Lipschitz map F : X × (0, d(x, y)) → Sn such that F(·, t) = Ft is homotopically nontrivial for all t with Lipschitz constant at most C/ for < t < d(x, y) − and assumes a constant value outside a ball B(x, Ct) when t < (1/2)d(x, y) and outside a ball B(y, Ct) when (1/2)d(x, y) < t < d(x, y). The situation can be localized, and by postcomposing our BLD-map with an appropriate Lipschitz family of maps Rn → Sn , we can find desired maps Ft locally (see [HR2, Section 9.6] for more details). Acknowledgments. We thank Seppo Rickman and Stephen Semmes for helpful discussions on the topics of this paper. We also thank the referee for carefully scrutinizing the manuscript.


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Heinonen Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA; [email protected] Sullivan Graduate School and University Center, City University of New York, New York, New York 10036, USA, and Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA; [email protected]


WAVELETS, TILING, AND SPECTRAL SETS YANG WANG

Abstract We consider a function φ ∈ L 2 (Rd ) such that {| det(D)|1/2 φ(Dx − λ) : D ∈ D , λ ∈ T } forms an orthogonal basis for L 2 (Rd ), where D ⊂ Md (R) and T ⊂ Rd . Such a function φ is called a wavelet with respect to the dilation set D and translation set T . We study the following question: Under what conditions can a D ⊂ Md (R) and a T ⊂ Rd be used as, respectively, the dilation set and the translation set of a wavelet? When restricted to wavelets of the form φ = χˇ , this question has a surprising tie to spectral sets and their spectra. 1. Introduction Let φ(x) ∈ L 2 (Rd ). We call φ(x) a wavelet if there exist a set of (d × d) real matrices D and a subset T of Rd such that | det(D)|1/2 φ(Dx − λ) : D ∈ D , λ ∈ T (1.1) forms an orthogonal basis for L 2 (Rd ). The sets D and T are called the dilation set and the translation set, respectively, for the wavelet φ(x). Problem Characterize the pairs (D , T ) that are dilation sets and translation sets, respectively, of some wavelets. Wavelets arise in many applications in both pure and applied mathematics. They play a key role in digital signal processing and scientific computations. The simplest wavelet is the Haar wavelet φ(x) = χ[0,1/2] (x) − χ[1/2,1) (x) for the dilation set D = {2n : n ∈ Z} and the translation set T = Z, constructed by A. Haar [Ha] in 1910. Later I. Daubechies [Da] constructed a family of compactly supported wavelets for the same dilation and translation sets, which can be made arbitrarily smooth. The methods in [Da] have been used to construct a wide variety of wavelets in Rd . However, all such wavelets have lattice translation sets and dilation sets {An : n ∈ Z} DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 8 January 2001. Revision received 4 July 2001. 2000 Mathematics Subject Classification. Primary 52C20; Secondary 52C22. Author’s research supported in part by National Science Foundation grant number DMS-00-70586. 43

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for some expanding integer matrix A with | det(A)| = 2. (For a dilation matrix with | det A| > 2, more than one wavelet function is needed.) In a different direction, several authors studied wavelets from their Fourier transforms (see, e.g., [HW] and the references therein). X. Fang and X. Wang [FW] introduce the minimally supported frequency wavelets (MSF wavelet), which were also studied in E. Hernández, X. Wang, and G. Weiss [HWW1], [HWW2] and by other authors. In particular, X. Dai and D. Larson [DL] considered a special kind of MSF b = χ for some measurable sets in Rd . They proved wavelet, φ, which satisfies φ that such a φ(x) is a wavelet with dilation set D = {2n : n ∈ Z} and translation set T = Z if and only if (i) the set { + λ : λ ∈ Z} is a tiling of R, (ii) the set {2n : n ∈ Z} is a tiling of R. In other words, must tile R both translationally and multiplicatively. Here we use the term tiling loosely. A collection of measurable sets { j } is a tiling of Rd if it is a measurewise disjoint partition of Rd . The result was later extended to higher dimensions in [DLS] for T = Zd and D = {An : n ∈ Z}, where A is any expanding (d × d)-matrix. Such an is referred to as a wavelet set (with respect to D and T ). All the studies on wavelets so far, whether from multiresolution analyses or from frequency constructions, have considered wavelets whose dilation sets consist of the powers of a single scalar (in dimension one) or the powers of a single expanding matrix (in higher dimensions). Furthermore, the translation sets are restricted to lattices. These dilation sets and translation sets are rather “regular,” and such restrictions have limited our ability to construct wavelets in a simple way, particularly in higher dimensions where few simple constructions are available. Naturally, we may ask whether there are other dilation and translation sets. In particular, we may ask the following question. Question Is it possible for a wavelet to have “irregular” dilation and translation sets D and T ? Can we have an aperiodic T and a noncommutative D ? We answer the above question in the affirmative in this paper. We demonstrate that by allowing multiple scaling matrices and nonlattice translations, very simple wavelets can be constructed (see Example 3 in §4). To do so, we consider wavelet sets in the most general setting. Let D ⊆ GL(d, R), the set of all nonsingular (d × d)-matrices, and let T ⊆ Rd . A measurable set ⊂ Rd with Lebesgue measure 0 < µ() < ∞ is called a wavelet set with respect to the dilation set D and the translation set T if φ(x) = χˇ (x) is a wavelet with respect to D and T . We study the following question: For which pairs of dilation sets D and translation sets T does there exist a wavelet

WAVELETS, TILING, AND SPECTRAL SETS

45

set ? One of the main contributions of this paper is to establish a surprising connection between the above question and the study of spectral sets and tiling. A set ⊂ Rd with 0 < µ() < ∞ is called a spectral set if there exists a T ⊆ Rd such that {e2πihλ,ξ i : λ ∈ T } is an orthogonal basis for L 2 (). In this case, we call T a spectrum of , and we call (, T ) a spectral pair. Spectral sets relate to tiling by the following conjecture, due to B. Fuglede [Fu]. SPECTRAL SET CONJECTURE A set ⊂ Rd with 0 < µ()

< ∞ is a spectral set if and only if tiles Rd by

translation. The Spectral Set Conjecture is not resolved in either direction, even in dimension one. There are many other questions concerning spectral sets, particularly related to tiling. Some of these questions are very much related to the theme of this paper. We refer readers to [JP3], [LW2], [LRW], [La2], and the references therein for results on spectral sets. We establish the following tie between wavelet sets and spectral sets. 1.1 Let D ⊂ GL(d, R), and let T ⊂ Rd . Let be a subset of Rd with positive and finite Lebesgue measure. If {D T () : D ∈ D } is a tiling of Rd and if (, T ) is a spectral pair, then φ = χˇ is a wavelet with respect to D and T . Conversely, if φ = χˇ is a wavelet with respect to D and T and if 0 ∈ T , then {D T () : D ∈ D } is a tiling of Rd and (, T ) is a spectral pair. THEOREM

It was shown by Fuglede [Fu] that tiles Rd by a lattice L if and only if (, L ∗ ) is a spectral pair, where L ∗ is the dual lattice of L . Therefore condition (i) in the result of Dai and Larson [DL] stated earlier can be more appropriately stated as (i0 ) (, Z) is a spectral pair. In fact, this is what the authors have shown. The condition 0 ∈ T in Theorem 1.1 cannot be dropped, as shown by Example 1 in §4. There are two main objectives in this paper: the study of the structure of the dilation set D for a wavelet set, and of the existence of wavelet sets for a given pair D and T . We show that by allowing more general dilation sets D we can obtain some very elegant wavelets through simple wavelet sets. This contrasts sharply with the more restricted notion of wavelet sets where the dilation set D must have the form {An : n ∈ Z} for some expanding matrix A, for which nonfractal-like wavelet sets

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are difficult to construct. As for the rest of the paper, we state our other main theorems in §2. We then prove the theorems in §3. b(ξ ) = Throughout this paper, the Fourier transform is defined as φ R 2πihξ,xi ˇ e φ(x) d x and the inverse Fourier transform is defined as ψ(x) = RR −2πihx,ξ i e ψ(ξ ) dξ . R 2. Statement of main theorems An important notion in the study of wavelet sets is multiplicative tiling. Definition Let D ⊆ GL(d, R). D is a multiplicative tiling set of Rd if there exists a bounded ⊂ Rd of positive Lebesgue measure, with dist(, 0) > 0, such that {D() : D ∈ D } is a tiling of Rd . The set is called a multiplicative D -tile. D is said to be A-invariant for some A ∈ GL(d, R) if D A = D . Note that in the case of a translational tile we often require that the tile be bounded. Since, in some sense, multiplicative tiling can be viewed as translational tiling after taking logarithm, the requirement dist(, 0) > 0 is, in fact, natural and necessary for {log |ξ | : ξ ∈ } to be bounded. We say that a multiplicative tiling set D satisfies the interior condition if there exists a multiplicative tile for D such that has nonempty interior. As with the interior condition for multiplicative tiling sets, we say that a spectrum T has interior condition if there exists a spectral set with spectrum T whose interior is nonempty. THEOREM 2.1 Let D ⊂ GL(d, R) such that D T := {D T : D ∈ D } is a multiplicative tiling set, and let T ⊂ Rd be a spectrum, with both D T and T satisfying the interior condition. Suppose that D T is A-invariant for some expanding matrix A, and suppose that T − T ⊆ L for some lattice L of Rd . Then there exists a wavelet set with respect to D and T .

The assumption T − T ⊆ L is equivalent to T ⊆ L + λ0 for some λ0 ∈ Rd . In dimension one, all known spectra have this property. Counterexamples exist in higher dimensions, the simplest of which being the spectra for the unit cube. They can be rather aperiodic (see [LRW] and [IP]). The assumption that D T and T have the interior condition is most likely unnecessary. All known examples of multiplicative tiling sets admit a tile having nonempty interior, and likewise for spectra.


47

COROLLARY 2.2 Let D ⊂ GL(d, R) such that D T is an A-invariant multiplicative tiling set with interior condition, where A ∈ Md (R) is expanding. Let T be a lattice in Rd . Then there exists a wavelet set with respect to D and T . COROLLARY 2.3 ([DLS]) Let D = {A j : j ∈ Z} for

some expanding A ∈ Md (R). Let T be a lattice in Rd . Then there exists a wavelet set with respect to D and T . In dimension one, the structure of positive multiplicative tiling sets follows from the study by J. Lagarias and Y. Wang [LW1] on translational tiles in R. A key notion here is a special type of subsets of Z called complementing sets. Suppose that A ⊂ Z, and let N > 1 be an integer. The set A is called a complementing set (mod N ) if there exists a set B ⊂ Z such that A + B is a direct sum and is a complete set of residues (mod N ). The set B is called a complement of A (mod N ). 2.4 Let D ⊂ R. Then we have the following. (i) Suppose that D ⊂ R+ . Then D is a multiplicative tiling set if and only if D = {as β : β ∈ E }, where a, s > 0, s 6 = 1, and E = A + N Z for some complementing set A (mod N ). (ii) Denote |D | := {|t| : t ∈ D } (counting multiplicity). Then D is a multiplicative tiling set for a centrally symmetric multiplicative tile if and only if |D | is a positive multiplicative tiling set, that is, if and only if |D | = {as β : β ∈ E }, where a, s > 0, s 6 = 1, and E = A + N Z for some complementing set A (mod N ). THEOREM

Centrally symmetric wavelet sets are important, as the resulting wavelets are real. Theorem 2.4 states that if we take a positive multiplicative tiling set D , then we may change the sign of any subset of D , and the resulting set is still a multiplicative tiling set for a symmetric tile. 2.5 Let D ⊂ R such that |D | is a multiplicative tiling set. Then we have the following. (i) For any spectrum T with the interior condition such that T − T ⊆ cZ for some c 6= 0, there exists a wavelet set with respect to D and T . (ii) For any lattice T there exists a centrally symmetric wavelet set with respect to D and T . THEOREM

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3. Proof of theorems Of all the theorems in this paper, Theorems 2.1 and 2.5 are the most difficult. They are heavily combinatorial. The other results are rather straightforward. Proof of Theorem 1.1 Let φ(x) = χˇ (x). Denote φ D,λ := | det D|1/2 φ(Dx − λ). Then {φ D,λ : D ∈ D , λ ∈ bD,λ : D ∈ D , λ ∈ T } is. Note T } is an orthogonal basis for L 2 (Rd ) if and only if {φ that bD,λ (ξ ) = | det D|−1/2 e D −1 λ (ξ )χ D T () (ξ ), φ (3.1) where eω (ξ ) := e2πihω,ξ i . Assume that is a multiplicative D T -tile and (, T ) is a spectral pair. Then the set of exponentials {eλ (ξ ) : λ ∈ T } is an orthogonal basis for L 2 (). It follows that {e D −1 λ (ξ ) : λ ∈ T } is an orthogonal basis for L 2 (D T ()). Now tiles Rd multiplicatively by {D T : D ∈ D }. Hence {e D −1 λ (ξ ) : D ∈ D , λ ∈ T } is an orthogonal set of functions in L 2 (Rd ). It is, in fact, a basis because {D T () : D ∈ D } is a partition of Rd . Conversely, assume that is a wavelet set with respect to T and D with 0 ∈ T . bD,λ : D ∈ D , λ ∈ T } is an orthogonal basis for L 2 (Rd ). Fix λ = 0 ∈ T . Then {φ bD,0 : D ∈ D } is orthogonal. By (3.1), Then {φ bD,0 (ξ ) = | det D|−1/2 χ D T () (ξ ). φ bD,λ ⊆ Hence {D T () : D ∈ D } must be disjoint measurewise. Furthermore, supp φ T b D (). It follows from the fact that {φ D,λ : D ∈ D , λ ∈ T } is a basis that {D T () : D ∈ D } must be a tiling of Rd . Now fix a D ∈ D . Since {D T () : D ∈ D } is a tiling, the set of exponentials {e D −1 λ (ξ ) : λ ∈ T } must be an orthogonal basis for L 2 (D T ()) by (3.1). This means that {eλ (ξ ) : λ ∈ T } must be an orthogonal basis for L 2 (). Hence (, T ) is a spectral pair. We give an example in §4 showing that the assumption 0 ∈ T cannot be removed. To prove the existence of wavelet sets, we first establish several lemmas concerning spectral sets. Let L be a full-rank lattice in Rd . The dual lattice L ∗ of L is defined as L ∗ := α ∈ Rd : hα, βi ∈ Z for all β ∈ L . The next two lemmas have been proved in [J] and [JP1], respectively. We include the proofs here for completeness.


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LEMMA 3.1 Let (, T ) be a spectral pair such that T − T ⊆ L for some full-rank lattice L in Rd . Let α ∈ L ∗ . Then and + α are measurewise disjoint.

Proof Without loss of generality, we may assume that 0 ∈ T . Therefore T ⊆ L . In particular, eλ (−α) = 1 for all λ ∈ T . Define 1 = ξ ∈ : ξ + α ∈ , and assume that µ(1 ) > 0. We derive a contradiction. Partition 1 into subsets whose diameters are all less than |α|. Let 0 be one of the subsets that has µ(0 ) > 0. Clearly, we have 0 ∩ (0 + α) = ∅. Let f = χ0 , and let g = χ0 +α . Now X X f = h f, eλ i L 2 () eλ , g= hg, eλ i L 2 () eλ . λ∈T

λ∈T

Observe that eλ (ξ ) = eλ (ξ )eλ (−α) = eλ (ξ − α) for all λ ∈ T . Hence Z Z h f, eλ i L 2 () eλ = eλ (ξ ) dξ = eλ (ξ − α) dξ 0 0 Z = eλ (ξ ) dξ = hg, eλ i L 2 () eλ . 0 +α

So f and g have the same Fourier series. But f and g are orthogonal. This is a contradiction. We say that two measurable sets and ∗ in Rd are L -congruent for some lattice L S S if has a partition = α∈L α such that ∗ = α∈L (α + α) with the union being measurewise disjoint. LEMMA 3.2 Let (, T ) be a spectral pair such that T − T ⊆ L for some full-rank lattice L in Rd . Let ∗ be L ∗ -congruent to . Then (∗ , T ) is also a spectral pair.

Proof S S Let ∗ = α∈L (α + α) where = α∈L α is a partition. We first prove that {eλ : λ ∈ T } is orthogonal in L 2 (∗ ). For any λ1 , λ2 ∈ T we have eλ1 −λ2 (ξ − α) =

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YANG WANG

eλ1 −λ2 (ξ ) for all α ∈ L ∗ . Hence Z X Z eλ1 −λ2 (ξ ) dξ =

α∈L ∗ α +α

∗

=

X Z α

α∈L ∗

=

X Z α∈L ∗

Z

α

eλ1 −λ2 (ξ ) dξ

eλ1 −λ2 (ξ − α) dξ eλ1 −λ2 (ξ ) dξ

eλ1 −λ2 (ξ ) dξ

=

= 0. It remains to prove that {eλ : λ ∈ T } is a basis for L 2 (∗ ). Without loss of generality, we assume that 0 ∈ T . Then T ⊆ L ∗ . If not, there exists a nonzero f ∈ L 2 (∗ ) such that h f, eλ i L 2 (∗ ) = 0 for all λ ∈ T . Define g ∈ L 2 (∗ ) by g(ξ ) = f (ξ + α) for ξ ∈ α . Clearly, g 6= 0. We have for each λ ∈ T , X Z hg, eλ i L 2 () eλ = f (ξ + α)eλ (ξ ) dξ α∈L ∗ α

=

X Z α∈L ∗

=

α

f (ξ + α)eλ (ξ + α) dξ

X Z α∈L ∗

Z = ∗

α +α

f (ξ )eλ (ξ ) dξ

f (ξ )eλ (ξ ) dξ

= 0. This is a contradiction. e be measurable sets in Rd , and let A ∈ Md (R). We say that is ALet and S S e if there exists a partition = j∈Z j such that j∈Z A j ( j ) = e congruent to e. Observe that A-congruence is an equivalent relation. Furthermore, is a partition of e. This fact if tiles Rd multiplicatively by some A-invariant D T, then so does together with Lemma 3.2 allows us to prove Theorem 2.1. The proof is combinatorial in nature, and it is quite technical. However, the main idea is quite simple. We start with a multiplicative D T -tile 0 that is bounded away from zero and contains in its interior a spectral set. Now we cut out the spectral set and contract the leftover set by A−n for a sufficiently large n. The new set is still a multiplicative tile but not a spectral set. Now cut out the part within the spectral set that is congruent (mod L ∗ )


51

to the contracted portion. The remaining set is again a spectral set. We then contract the small cutout portion using A−n . The new set is again a multiplicative tile. This process is repeated ad infinitum. The limit set is then both a multiplicative tile and a spectral set. Proof of Theorem 2.1 Let t0 be a multiplicative tile by D T with nonempty interior, and let dist(t0 , 0) > 0. Let s0 be a spectral set with spectrum T and nonempty interior. Since Ak (t0 ) is also a multiplicative D T -tile and A is expanding, we may, without loss of generality, assume that t0 contains a sufficiently large ball. Furthermore, since any translate of s0 is again a spectral set with spectrum T , we may, without loss of generality, further assume that there exists an α ∗ ∈ L ∗ such that α ∗ ∈ (s0 )o ,

s0 ⊆ t0 .

We construct a wavelet set with respect to D and T by constructing multiplicative tiles {tn } and spectral sets {sn } with the property that sn ⊆ tn and limn→∞ sn = limn→∞ tn = . The sequence of sets is constructed iteratively using congruences, a technique similar to the one used in J. Benedetto and M. Leon [BL]. First we fix a sufficiently large K > 0 such that A−K (t0 ) + α ∗ ⊆ s0 . Since ∗ α ∈ (s0 )o , this is always possible. Let s1 = s0 ,

t1 = s1 ∪ e1 ,

e1 = A−K (t0 \ s0 ).

Note that s1 ∪ (t0 \ s0 ) is a partition of t0 ; so t1 is A-congruent to t0 and hence is a D T -tile. (Here the letter e in e1 stands for “extra.” This is the extra piece by which t1 differs from s1 .) Observe that e1 + α ∗ ⊆ s1 since e1 ⊆ A−K (t0 ). Define s2 = e1 ∪ s1 \ (e1 + α ∗ ) , t2 = s2 ∪ e2 , e2 = A−K (e1 + α ∗ ). (3.2) Note that s2 is L ∗ -congruent to s1 ; so by Lemma 3.2 it is a spectral set with spectrum T . Note also that t2 is a D T -tile because it is A-congruent to t1 since t1 = s2 ∪ (e1 + α ∗ ). It is now easy to see how we construct sn and tn iteratively. We let sn+1 = en ∪ sn \ (en + α ∗ ) , tn+1 = sn+1 ∪ en+1 ,

en+1 = A−K (en + α ∗ ).

(3.3)

The set sn+1 is L ∗ -congruent to sn , and tn+1 is A-congruent to tn . So all sn are spectral sets with spectrum T , and all tn are multiplicative D T -tiles. Furthermore,

52

YANG WANG

by induction the Lebesgue measure of the extra set en satisfies µ(en ) = µ(tn \ sn ) = | det A|−K µ(en−1 ) ≤ | det A|−n K µ(t0 ).

(3.4)

Finally, we show that limn→∞ sn = limn→∞ tn = up to a measure zero set. This can be seen from the fact that sn+1 4 sn = en ∪ (en + α ∗ ),

and so

µ(sn+1 4 sn ) = c r n ,

(3.5)

where c = 2µ(t0 ) and r = | det A|−K < 1. In other words, the difference between successive terms decays exponentially measurewise. Now we have \ [ \ sk ⊆ (sk \ sk+1 ). sk = sn \ sn 4 k≥n

k≥n

Hence

k≥n

\ µ sn 4 sk ≤ cr n + cr n+1 + · · · = c1r n . k≥n

Similarly, [ µ sn 4 sk ≤ c2r n . k≥n s It follows that lim supn n = lim infn sn up to a measure zero set. Let be the limit. By (3.5), = limn tn up to a measurable set. Thus is a wavelet set with respect

to D and T .

Proof of Corollaries 2.2 and 2.3 For Corollary 2.2 we only need to note that by [Fu] any lattice T is a spectrum of any fundamental domain of the dual lattice T ∗ , which can obviously be chosen to have nonempty interior. For Corollary 2.3 we note that D T is A T -invariant, and we need to show that it is a multiplicative tiling set with interior condition. Let B1 (0) be the unit T ball centered at zero, and let 0 = n≥0 (A T )n (B1 (0)). The intersection is finite since (A T )n (B1 (0)) ⊇ B1 (0) for sufficiently large n. So 0 is open. Set = A T (0 ) \ 0 . Then is a multiplicative tile with interior, and dist(, 0) > 0. Proof of Theorem 2.4 (i) Write = + ∪ − , where + = ∩ R+ and − = ∩ R− . Since D ⊂ R+ , D (+ ) := {d + : d ∈ D } is a tiling of R+ . Taking logarithm, we see that log D := {log d : d ∈ D } is a tiling set of R, with tile log + := {log x : x ∈ + }. Observe that log + is bounded because dist(, 0) > 0. The structure of log D is classified in Lagarias and Wang [LW1]. Part (i) of the theorem follows directly from [LW1, Th. 3]. (ii) By central symmetry of , we can write as = + ∪ (−+ ), where + = ∩ R+ . Observe that d = |d| for any d ∈ D . So D is a multiplicative tiling set for if and only if |D | is.


53

The proof of Theorem 2.5 is very similar to that of Theorem 2.1. It is combinatorial in nature. Proof of Theorem 2.5 First we note that D = D T . By the structure result in Theorem 2.4, D is s N -invariant, where s, N are as in Theorem 2.4. It is also s −N -invariant. One of s N and s −N is greater than 1. Furthermore, D must satisfy the interior condition (see [LW1]). Theorem 2.1 now immediately implies (i). The proof of (ii) is essentially identical to the proof of Theorem 2.1. Only minor modifications are needed. Without loss of generality, we assume that T = Z. By Theorem 2.4, |D | is a-invariant for some a > 1. Let t0 be a centrally symmetric multiplicative tile by D with nonempty interior, and let dist(t0 , 0) > 0. Since a k (t0 ) is also a centrally symmetric multiplicative D -tile, we may, without loss of generality, assume that [−m 0 − 1/2, −m 0 ] ∪ [m 0 , m 0 + 1/2] ⊆ s0 for some positive integer m 0 . Set s0 = [−m 0 − 1/2, −m 0 ] ∪ [m 0 , m 0 + 1/2]. Then s0 is a spectral set with spectrum T = Z. As in the proof of Theorem 2.1, we construct a centrally symmetric wavelet set with respect to D and T by constructing centrally symmetric multiplicative tiles {tn } and centrally symmetric spectral sets {sn } with the property that sn ⊆ tn and limn→∞ sn = limn→∞ tn = . First we fix a sufficiently large K > 0 such that a −K (t0 ) ⊆ (−1/2, 1/2). This is always possible because a > 1. Let s1 = s0 ,

t1 = s1 ∪ e1 ,

e1 = a −K (t0 \ s0 ).

As in the proof of Theorem 2.1, t1 is a-congruent to t0 and hence is a multiplicative |D |-tile. Furthermore, it is centrally symmetric, so it is also a multiplicative D -tile. Note that s1 , t1 , and e1 are all centrally symmetric. For any set S in R, we define p+ (S) := S ∩ R+ and p− (S) := S ∩ R− . Now a −K (t0 ) ⊆ (−1/2, 1/2) implies that p+ (e1 ) + m 0 ⊂ p+ (s1 ) = [m 0 , m 0 + 1/2] and p− (e1 ) − m 0 ⊂ p− (s1 ) = [−m 0 − 1/2, −m 0 ]. Let τm 0 (S) := ( p+ (S) + m 0 ) ∪ ( p− (S) − m 0 ). We have s2 = e1 ∪ s1 \ τm 0 (e1 ) , t2 = s1 ∪ e2 , e2 = a −K τm 0 (e1 ). (3.6) Note that s2 is T ∗ -congruent to s1 (T ∗ = Z), so by Lemma 3.2 it is a spectral set with spectrum T . Note also that t2 is a |D |-tile because it is a-congruent to t1 . Furthermore, s2 , t2 , and e2 are all centrally symmetric. Thus t2 is also a multiplicative D -tile.

54

YANG WANG

It is now easy to see how we construct sn and tn iteratively. We let sn+1 = en ∪ sn \ τm 0 (en ) , tn+1 = sn+1 ∪ en+1 ,

en+1 = a −K τm 0 (en ).

(3.7)

All these sets are centrally symmetric. The set sn+1 is T ∗ -congruent to sn , and tn+1 is a-congruent to tn . So all sn are spectral sets with spectrum T , and all tn are multiplicative |D |-tiles. Furthermore, by induction the Lebesgue measure of the extra set en satisfies µ(en ) = µ(tn \ sn ) = a −K µ(en−1 ) ≤ a −K n µ(t0 ).

(3.8)

Finally, we show that limn→∞ sn = limn→∞ tn = up to a measure zero set. This can be seen from the fact that sn+1 4 sn = en ∪ τm 0 (en ),

and so

µ(sn+1 4 sn ) = c a −K n ,

(3.9)

where c = 2µ(t0 ). In other words, the difference between successive terms decays exponentially measurewise. Now we have [ \ \ sk ⊆ (sk \ sk+1 ). sn 4 sk = sn \ k≥n

Hence

k≥n

k≥n

\ µ sn 4 sk ≤ ca −K n + ca −K (n+1) + · · · = c1 a −K n . k≥n

Similarly, [ µ sn 4 sk ≤ c2 a −K n . k≥n

It follows that lim supn sn = lim infn sn up to a measure zero set. Let be the limit. By (3.7), = limn tn up to a measurable set. Thus is a wavelet set with respect to D and T .

4. Examples In this section we present several examples of wavelet sets to emphasize various aspects of the complexity of the problem. Example 1 The condition 0 ∈ T in Theorem 1.1 cannot be dropped. Let = [−1, −1/2] ∪ [1/2, 1]. Let D = {2n : n ∈ Z} ∪ {−2n : n ∈ Z}, and let T = 2Z + 1/6. Then is a wavelet set with respect to D and T . But D is not a multiplicative tiling set of , nor is T a spectrum of .


55

To prove that is indeed a wavelet set with respect to D and T , we observe that the corresponding wavelet φ = χˇ =

sin(π x) − sin(2π x) πx

(4.1)

is real and even. So φ(−2n x − λ) = φ(2n x + λ). Therefore φ is a wavelet with respect to D and T if and only if it is a wavelet with respect to D 0 = {2n : n ∈ Z} and T 0 = T ∪ (−T ). Now tiles Rd multiplicatively by D 0 . Furthermore, T 0 = 2Z + {1/6, −1/6} satisfies T 0 − T 0 ⊂ Z ∪ {0}, where Z φ denotes the set of zeros of χ b = φ(−x). Hence (, T 0 ) is a spectral pair (cf. [P] or [LW2]). By Theorem 1.1, φ is a wavelet with respect to D 0 and T 0 , so it is a wavelet with respect to D and T . Example 2 A multiplicative tiling set D ⊂ Rd does not have to be A-invariant for some matrix A 6= I , even in dimension one. For the in Example 1, the set D = {n 2n : n ∈ Z}, where n ∈ {1, −1} is always a multiplicative tiling set for . If we choose (n : n ∈ Z) to be aperiodic, then D is not a-invariant for any a 6 = 1. Example 3 The translation set T for a wavelet set need not be periodic nor satisfy T −T ⊆ L for some lattice L in Rd . Furthermore, the dilation matrices in D need not commute. Let be the unit square centered at (0, 3/2); that is, let = [−1/2, 1/2]2 + (0, 3/2). Let D = {±2n C1 , ±2n C2 : n ∈ Z}, where C1 and C2 are given by 2 0 C1 = , 0 1

0 C2 = 2

1 2 .

0

Let R denote the rectangle [−1, 1] × [−2, 2]. Note that C1 ∪ (−C1 ) ∪ C2 ∪ (−C2 ) = R \

1 R. 2

So tiles R2 multiplicatively by D . It is well known that all cubes are spectral sets. The spectra for cubes in Rd have been completely classified by several authors (see [JP3] for d = 2, 3, and see [IP] or [LRW] for general d). A set T is a spectrum for the unit cube if and only if T is a tiling set for the unit cube. (This unit cube should be viewed as the dual of .) For example, we may take T = {(n, m + en ) : m, n ∈ Z}.

56

YANG WANG

Then (, T ) is a spectral pair. So is a wavelet set with respect to D and T . Observe that T neither is periodic nor satisfies T − T ⊆ L for any lattice L . Furthermore, not all matrices in D commute, C1 C2 6 = C2 C1 . The corresponding wavelet is φ(x1 , x2 ) = e−3πi x2 ·

sin(π x1 ) sin(π x2 ) . π 2 x1 x2

Acknowledgments. The author wishes to thank David Larson; this paper is partly inspired by a conversation with him during a 1999 American Mathematical Society meeting in Charlotte, North Carolina. Many thanks should particularly go to Jeff Lagarias, who made several valuable comments for improving the paper. Finally, the author wishes to thank Steen Pedersen for pointing to several references. References [BL]

J. J. BENEDETTO and M. T. LEON, “The construction of multiple dyadic minimally supported frequency wavelets on Rd ” in The Functional and Harmonic Analysis

of Wavelets and Frames (San Antonio, Tex., 1999), Contemp. Math. 247, Amer. Math. Soc., Providence, 1999, 43 – 74. MR 2001a:42034 51 [Bo] M. BOWNIK, The construction of r -regular wavelets for arbitrary dilations, J. Fourier Anal. Appl. 7 (2001), 489 – 506. CMP 1 845 100 [BS] M. BOWNIK and D. SPEEGLE, Meyer type wavelet bases in R2 , preprint, 2000, http://euler.slu.edu/Dept/Faculty/speegled/speegled.html [DL] X. DAI and D. R. LARSON, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640. MR 98m:47067 44, 45 [DLS] X. DAI, D. R. LARSON, and D. M. SPEEGLE, Wavelet sets in Rn , J. Fourier Anal. Appl. 3 (1997), 451 – 456. MR 98m:42048 44, 47 [Da] I. DAUBECHIES, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909 – 996. MR 90m:42039 43 [FW] X. FANG and X. WANG, Construction of minimally supported frequency wavelets, J. Fourier Anal. Appl. 2 (1996), 315 – 327. MR 97d:42030 44 [Fu] B. FUGLEDE, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101 – 121. MR 57:10500 45, 52 [Ha] A. HAAR, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 (1910), 331 – 371, http://134.76.163.65/agora docs/28599TABLE OF CONTENTS.html 43 ´ [HWW1] E. HERNANDEZ, X. WANG, and G. WEISS, Smoothing minimally supported frequency wavelets, I, J. Fourier. Anal. Appl. 2 (1996), 329 – 340. MR 97h:42015 44 [HWW2] , Smoothing minimally supported frequency wavelets, II, J. Fourier. Anal. Appl. 3 (1997), 23 – 41. MR 98b:42049 44 ´ [HW] E. HERNANDEZ and G. WEISS, A First Course on Wavelets, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1996. MR 97i:42015 44


57

[ILP]

E. J. IONASCU, D. R. LARSON, and C. M. PEARCY, On wavelet sets, J. Fourier Anal.

[IP]

E. J. IONASCU and C. M. PEARCY, On subwavelet sets, Proc. Amer. Math. Soc. 126

[IKT]

A. IOSEVICH, N. H. KATZ, and T. TAO, Convex bodies with a point of curvature do not

[IP]

A. IOSEVICH and S. PEDERSEN, Spectral and tiling properties of the unit cube,

[J]

P. E. T. JØRGENSEN, Spectral theory of finite volume domains in Rn , Adv. Math. 44

[JP1]

P. E. T. JØRGENSEN and S. PEDERSEN, Spectral theory for Borel sets in Rn of finite

Appl. 4 (1998), 711 – 721. MR 2000b:42029 (1998), 3549 – 3552. MR 99b:42040 have Fourier bases, Amer. J. Math. 123 (2001), 115 – 120. CMP 1 827 279 Internat. Math. Res. Notices 1998, 819 – 828. MR 2000d:52015 46, 55 (1982), 105 – 120. MR 84k:47024 48

[JP2] [JP3] [K1] [K2] [La1] [La2] [LRW] [LS] [LW1] [LW2] [P] [PW] [SW]

measure, J. Funct. Anal. 107 (1992), 72 – 104. MR 93k:47005 48 , Group-theoretic and geometric properties of multivariable Fourier series, Exposition. Math. 11 (1993), 309 – 329. MR 94k:42050 , Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), 285 – 302. MR 2002d:42027 45, 55 M. N. KOLOUNTZAKIS, Packing, tiling, orthogonality and completeness, Bull. London Math. Soc. 32 (2000), 589 – 599. MR 2001g:52030 , Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), 542 – 550. MR 2001h:52019 I. ŁABA, Fuglede’s conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001), 2965 – 2972, MR 2002d:42007 , The spectral set conjecture and multiplicative properties of roots of polynomials, preprint, arXiv:math.CA/0010169 45 J. C. LAGARIAS, J. A. REEDS, and Y. WANG, Orthonormal bases of exponentials for the n-cube, Duke Math. J. 103 (2000), 25 – 37. MR 2001h:11104 45, 46, 55 J. C. LAGARIAS and P. W. SHOR, Keller’s cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 279 – 283. MR 93e:52040 J. C. LAGARIAS and Y. WANG, Tiling the line with translates of one tile, Invent. Math. 124 (1996), 341 – 365. MR 96i:05040 47, 52, 53 , Spectral sets and factorizations of finite abelian groups, J. Funct. Anal. 145 (1997), 73 – 98. MR 98b:47011b 45, 55 S. PEDERSEN, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), 496 – 509. MR 98b:47011a 55 S. PEDERSEN and Y. WANG, Universal spectra, universal tiling sets and the spectral set conjecture, Math. Scand. 88 (2001), 246 – 256. CMP 1 839 575 P. M. SOARDI and D. WEILAND, Single wavelets in n-dimensions, J. Fourier Anal. Appl. 4 (1998), 299 – 315. MR 99k:42067

School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA; [email protected]; http://math.gatech.edu˜wang


LOCALIZATION FOR ONE-DIMENSIONAL, CONTINUUM, BERNOULLI-ANDERSON MODELS ¨ DAVID DAMANIK, ROBERT SIMS, and GUNTER STOLZ

Abstract We use scattering theoretic methods to prove strong dynamical and exponential localization for one-dimensional, continuum, Anderson-type models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single-site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1. Introduction Perhaps the most studied and best understood type of random operators used to describe spectral and transport properties of disordered media are the Anderson models, whether one considers their original discrete version in `2 (Zd ) or their continuum analogs. They describe materials of alloy type, that is, with a structure in which single-site potentials are centered at the points of a regular lattice and then multiplied by random coupling constants, modelling the differing nuclear charges of the alloy’s component materials. It is therefore physically most relevant to study the case where the coupling constants take on only finitely many values. The special case of a two-component alloy, that is, one with coupling constants given by Bernoulli random variables, has been dubbed the Bernoulli-Anderson model. Unfortunately, most of the rigorous results on Anderson models require more regularity of the probability distributions governing the coupling constants. Results on exponential localization and, DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 16 October 2000. Revision received 3 August 2001. 2000 Mathematics Subject Classification. Primary 82B44. Damanik’s work partially supported by German Academic Exchange Service through Hochschulsonderprogramm III (Postdoktoranden). Sim’s work partially supported by National Science Foundation grant number DMS-9706076. Stolz’s work partially supported by National Science Foundation grant numbers DMS-9706076 and DMS0070343. 59

60

DAMANIK, SIMS, and STOLZ

more recently, dynamical localization have usually been proven under the assumption that the distribution is absolutely continuous or at least has an absolutely continuous component (see [12] and [41] for results from the 1980s, and see [1], [23], [14], [21], [31], [34], [50] for some of the more recent contributions). In dimension d > 1 the best results with respect to weak regularity are due to R. Carmona, A. Klein, and F. Martinelli [11] for the discrete case and to P. Stollmann [47] and to D. Damanik and Stollmann [15] for continuum models, where localization properties are shown under merely the assumption of Hölder-continuity of the distribution. Discrete distributions, and in particular the Bernoulli-Anderson model, are currently accessible only in d = 1. For the discrete, one-dimensional Anderson model (h ω u)(n) = u(n + 1) + u(n − 1) + qn (ω)u(n),

n ∈ Z,

(1.1)

exponential localization has been proven at all energies for arbitrary, nontrivial distribution (i.e., support containing more than one point) of the independent, identically distributed (i.i.d.) random variables qn . This was first proven by Carmona, Klein, and Martinelli [11] and later by C. Shubin, R. Vakilian, and T. Wolff [43] with a different approach that, in turn, is based on results from [8] and [33]. Results on localization for Anderson models with a singularly distributed potential on the discrete strip are shown in [32]. An interesting recent paper by S. De Bièvre and F. Germinet [17] studies the so-called random dimer model. This is the case where in (1.1) one chooses q2n+1 (ω) = q2n (ω) and the q2n , n ∈ Z, as i.i.d. random variables with only two values ±q for some q > 0. The paper [17] establishes dynamical localization for the random dimer model in compact intervals away from certain critical energies, where the Lyapunov exponent vanishes and delocalized states exist. In this paper we prove localization, exponential and dynamical, for continuous one-dimensional Anderson models with arbitrary nontrivial distribution. Let Hω = −

d2 + Vper + Vω , dx2

ω ∈ ,

(1.2)

with Vω (x) =

X

qn (ω) f (x − n).

(1.3)

n∈Z

We assume that the background potential Vper has period 1 and is real valued and locally in L 1 . The single-site potential f ∈ L 1 is real valued, supported in [−1/2, 1/2], and not zero (in the L 1 -sense). Note that we do not need to assume that f has fixed sign, as is frequently done in other works on continuum Anderson models. The coupling constants qn are i.i.d. random variables on a complete probability space . We assume that the support of their common distribution µ is bounded and nontrivial, that is, contains at least two points. Note that without restriction we may

BERNOULLI-ANDERSON LOCALIZATION

61

assume that supp(µ) contains 0 and 1. (If {a, b} ⊂ supp(µ), then replace Vper by P Vper + a n f (· − n) and f by (b − a) f .) We do this in all our proofs. Under these assumptions the operators Hω can be defined either by form methods or via Sturm-Liouville theory and are self-adjoint for every ω. Representing, as usual, as an infinite product and qn (ω) as ωn , one easily sees that Hθk ω f = τk∗ Hω τk f

for f ∈ D(Hθk ω )

with τk f = f (· − k) and the ergodic shifts (θk ω)n = ωn+k . Thus (see, e.g., [12]), Hω has nonrandom almost sure spectrum 6 and spectral types 6ac , 6sc , and 6 pp . Our first main result is the following. 1.1 (Exponential localization) Almost surely, the operator Hω has pure point spectrum; that is, 6ac = 6sc = ∅, and all eigenfunctions decay exponentially at ±∞. THEOREM

Our method to prove Theorem 1.1 is basically to adapt the approach of [11] to continuum models. We do, however, use a variable-energy multiscale analysis, which was introduced in [18] for discrete models and later adapted to the continuum (see, e.g., [21], [23], [48]). Recently, it has been demonstrated by Damanik and Stollmann [15] that variableenergy multiscale analysis implies strong dynamical localization. Thus our method of proof also yields the following. THEOREM 1.2 (Strong dynamical localization) Let Hω be defined as above. There exists a discrete set M ⊂ R such that for every compact interval I ⊂ R \ M, every compact set K ⊂ R, every p > 0,

E sup |X | p e−it Hω PI (Hω )χ K < ∞,

(1.4)

t>0

where PI is the spectral projection onto I . We note that the “strong” in Theorem 1.2 refers to the ability to show finite expectation in (1.4). This is stronger than showing that the supremum in (1.4) is almost surely finite, which has also been used to describe dynamical localization. For more details on dynamical localization, we refer to [42], [23], [2], [4], and [15]. Based on our results, one can also use the bootstrap multiscale analysis recently introduced by Germinet and Klein [24] to prove localization. In fact, their work yields strong dynamical localization in the Hilbert-Schmidt norm (see [24]). Another advantage of bootstrap multiscale analysis is that it can be started with a very weak form of the initial length scale estimate (ILSE), discussed here in Section 6. This weak form

62


of the ILSE follows by a simple argument from positivity of the Lyapunov exponent (see [22]). The more involved large deviation arguments used in [11] and in Section 6 have the advantage of yielding decay of the resolvent and of eigenfunctions at the rate of the Lyapunov exponent. A crucial tool in [11] as well as in our work is positivity of the Lyapunov exponent. To define it in the continuous case, let gλ (n, ω) denote the transfer matrix from n − 1/2 to n + 1/2 of −u 00 + Vω u = λu; (1.5) that is, for any solution u of (1.5), u(n + 1/2) u(n − 1/2) = g (n, ω) . λ u 0 (n + 1/2) u 0 (n − 1/2)

(1.6)

For n ∈ N, let Uλ (n, ω) = gλ (n, ω) · . . . · gλ (1, ω). The subadditive ergodic theorem (see, e.g., [12]) guarantees the existence of the Lyapunov exponent, that is, the limit 1 E log kUλ (n, ω)k . n→∞ n

γ (λ) = lim

Kotani theory implies that γ (λ) is positive for almost every λ ∈ R since Hω is nondeterministic in the sense of S. Kotani [37]. But this is insufficient for a proof of localization due to the fact that the singular distribution of µ prevents spectral averaging techniques from being used to prove absence of singular continuous spectrum. For the discrete model (1.1), one can instead use Furstenberg’s theorem on products of independent random matrices to show that the Lyapunov exponent is positive for all energies, which is the starting point of the approach in [11]. We use the same idea, but there is an additional, significant difficulty to overcome. For the continuous models (1.2) and (1.3) there may exist a set of critical energies at which Furstenberg’s theorem does not apply, and in most cases, γ (λ) vanishes. This gives rise to the exceptional set M in Theorem 1.2 and is a continuum analog of the critical energies observed in [17] for the random dimer model. We manage to prove discreteness of the set of critical energies M for arbitrary single-site potentials f with support in [−1/2, 1/2] and arbitrary periodic background Vper by using methods from scattering theory. More precisely, we consider scattering at f with respect to the periodic background Vper . For energies λ in the interior of spectral bands of the periodic operator H0 = −d 2 /d x 2 + Vper , we introduce the reflection and transmission coefficients b(λ) and a(λ) relative to Vper . The roots of b(λ) give rise to critical energies where the Lyapunov exponent vanishes and extended states exist. Since f 6= 0, it can be shown that this set of energies is discrete. Other


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discrete exceptional sets are included in M, among them the band edges of H0 and roots of the analytic continuations of b(λ) and a(λ) to the spectral gaps of H0 . Away from M, positivity of γ follows from Furstenberg’s theorem. The fact that extended states exist at some critical energies does not affect Theorem 1.1 since the discrete exceptional set cannot support continuous spectrum. It does, however, affect dynamical localization in that the interval I in Theorem 1.2 specifically excludes these critical values. The existence of extended states at critical energies in certain one-dimensional tight-binding models or, equivalently, in disordered harmonic chains is well known in the physics literature. Various aspects of this are discussed in [38, Section 10], which also includes continuum models. Additional references can be found through [16] and [17]. These works also note the vanishing of single-site reflection coefficients as the basic mechanism for obtaining critical energies and discuss the anomalous transport behavior near these energies. In several of these models, the discrete random potential is of the form X Vω (n) = qk (ω) f (n − k`), (1.7) k∈Z

where the single-site potential f : Z → R is supported on {0, . . . , ` − 1}. For example, in the dimer model as studied in [17], one sets ` = 2, f = χ{0,1} . While in our work we focus on a systematic study of exceptional energies in continuum Anderson models, we expect that our methods can be adapted to yield exponential and dynamical localization for general discrete Anderson models of the form (1.7). While we exclude neighborhoods of critical energies in our discussion of dynamical localization, as does [17], it is an interesting open problem to describe rigorously the transport behavior near critical energies. A review of results on positivity of the Lyapunov exponent which have been obtained from Furstenberg’s theorem is given in [41, Section 14.A]. Most of these results were obtained for discrete Schrödinger operators and Jacobi matrices (see, e.g., [27], [26], [20]). Results for continuum Anderson models (1.2) and (1.3) had been restricted to special exactly solvable cases that could be reduced to discrete models. For example, the set where γ vanishes was characterized for the cases where Vper = 0 and either f is a δ-point-interaction (see [26]) or f = χ[−1/2,1/2] (see [5]). For the latter, if it is assumed that supp(µ) = {0, ξ } for some ξ > 0, then one gets the critical energies {λ : γ (λ) = 0} = {ξ + π 2 n 2 , n ∈ Z}. The first paper where reflection coefficients are used rigorously in combination with Furstenberg’s theorem to characterize the exceptional set for general f is [35]. Many of the ideas used in Section 2 can be found there. They assume Vper = 0 and thus can work with the classical reflection and transmission coefficients for scattering at f . Also, [35] assumes that the distribution µ of qn has continuous density, but the methods used can be modified to include some examples with singular distributions.

64


We also mention that the recent paper [36] contains bounds for the Lyapunov exponent, which might be useful in studying transport properties near critical energies. As our methods of proof show, critical energies are most likely to exist if qn takes on only a small number of different values. We follow [35] in conjecturing that no critical energies exist (and therefore the Lyapunov exponent is positive for all energies) if the support of µ is sufficiently rich. Since analyticity is involved, this should already be the case if the support of µ has an accumulation point. That energies with vanishing reflection coefficient at a single site have zero Lyapunov exponent was also observed for random displacement models in [46]. That the latter model shows properties similar to the Bernoulli-Anderson model is not surprising: both models display identical single-site potentials at random distances. To our knowledge, the only previous work on localization for continuum Bernoulli-Anderson models is [9]. This work, in combination with remarks in [10], looks at the example Vper = 0, f = χ[−1/2,1/2] , supp(µ) = {0, 1} and states exponential localization for all energies. Unfortunately, it seems that some details of the proof, which is sketched in [9] and is quite different from ours, have never appeared in print. We now outline the contents of the remaining sections. In Section 2 we prove positivity of the Lyapunov exponent away from a discrete set using Furstenberg’s theorem. A key tool is to show that the identical vanishing of b(λ) on a band of H0 implies that f = 0. This generalizes the well-known fact that there are no compactly supported solitons to scattering at periodic background. Sections 3 and 4 establish that the Lyapunov exponent and the integrated density of states are Hölder continuous away from the exceptional set. Both sections rely heavily on known results for the discrete case, which can all be found in [12]. We outline these results and discuss the necessary changes for continuum models. In particular, we make use of a version of the Thouless formula by Kotani [37]. The next two sections provide the main ingredients for the start of a multiscale analysis. In Section 5 a Wegner estimate is proven, again away from the exceptional set. Here we can closely follow the argument of [11]. Section 6 provides an initial length scale estimate for the same energies. Here our argument differs somewhat from the approach in [11]. While the latter uses both positivity of the Lyapunov exponent and the Wegner estimate to get an ILSE, we prefer to use an approach that shows that ILSE follows directly from positivity of γ . This uses some results on large deviations for γ , which can all be found in [7]. We note that this also gives a new, more natural proof of an ILSE for discrete models. Having established all that is necessary for a multiscale analysis, we can then prove Theorems 1.1 and 1.2 in Section 7 by merely referring to well-known results, for example, [48] and [15]. Appendix A contains some a priori estimates for solutions of the Schrödinger


65

equation which we use frequently. In Appendix B some basic facts about cocycles and the existence and uniqueness of invariant measures for group actions are listed. They are used in Section 3.

2. Positivity of the Lyapunov exponent 2.1. The Floquet solutions We start by collecting some facts from Floquet theory for the periodic operator H0 := −d 2 /d x 2 + Vper , (see, e.g., [19]). For any z ∈ C, let u N (·, z) and u D (·, z) denote the solutions of −u 00 + Vper u = zu (2.1) with u N (−1/2) = u 0D (−1/2) = 1 and u 0N (−1/2) = u D (−1/2) = 0. The transfer matrix of (2.1) from −1/2 to 1/2 is the matrix u N (1/2, z) u D (1/2, z) g0 (z) = , u 0N (1/2, z) u 0D (1/2, z) which is entire in z as solutions of (2.1) (resp., their derivatives) are, for each fixed x, entire in z. The eigenvalues of g0 (z) are the roots of ρ 2 − D(z)ρ + 1 = 0;

(2.2)

that is, p

D(z)2 − 4 , (2.3) 2 where D(z) = Tr[g0 (z)]. As roots of (2.3), the functions ρ± are algebraic with singularities at points with D(z) = ±2. The spectrum of H0 , σ (H0 ), consists of bands that are given by the sets of real energies λ for which D(λ) ≤ 2. Let (a, b) be a stability interval of H0 , that is, a maximal interval such that D(λ) < 2 for every λ ∈ (a, b). As both u N (·, λ) and u D (·, λ) are real for λ ∈ (a, b), one has ρ± (z) =

ρ± (λ) =

D(z) ±

p 1 D(λ) ± i 4 − D(λ)2 , 2

|ρ± (λ)| = 1, and ρ− (λ) = ρ+ (λ). Let S := z ∈ C : z = λ + iη where a < λ < b and η ∈ R be the vertical strip in the complex plane containing (a, b). For z = λ + iη ∈ S, all of the following are equivalent:

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(i) |ρ± (z)| =1, (ii) η = 0, (iii) D(z) ∈ (−2, 2). That (ii) ⇒ (iii) and (iii) ⇒ (i) is clear. To see that (i) ⇒ (ii), assume that (i) is true for some z = λ + iη, where η 6= 0. As both ρ± have modulus 1, then all solutions of (2.1) are bounded. By Weyl’s alternative, however, if η 6 = 0, then there exists a solution in L 2 near +∞, the Weyl solution, while all other solutions are unbounded. This is a contradiction. The arguments above imply that ρ± have analytic continuations to all of S, and that the only possible algebraic singularities occur at a and b. In addition, as ρ+ (z)ρ− (z) = det[g0 (z)] = W [u N , u D ] = 1, we have by continuity of | · | that exactly one of ρ+ and ρ− satisfies |ρ(z)| < 1 in the upper half-plane. Without loss of generality, let us denote by ρ+ the eigenvalue for which |ρ+ (λ + iη)| < 1 for all η > 0 and λ ∈ (a, b). This corresponds to choosing a branch of the square root in (2.3). Then ρ− satisfies |ρ− (λ + iη)| > 1 for all η > 0 and λ ∈ (a, b). Since we also have |ρ± (λ)| = 1 for λ ∈ (a, b), it follows from the Schwarz reflection principle (apply a linear fractional transformation) that |ρ+ (λ + iη)| > 1 and |ρ− (λ + iη)| < 1 for all η < 0 and λ ∈ (a, b). For z ∈ S, let v± (z) be the eigenvectors of g0 (z) corresponding to ρ± (z) with first component normalized to be 1; that is, 1 v± (z) = . (2.4) c± (z) One may easily calculate c± (z) =

ρ± (z) − u N (1/2, z) . u D (1/2, z)

(2.5)

As u D (1/2, z) is never zero in S (Dirichlet eigenvalues of H0 are either in the gaps of σ (H0 ) or at the band edges), v± are analytic in S. In particular, as u −1 D has at most a pole at a and b, then c± , and hence v± , have at worst algebraic singularities at a and b. (More precisely, c± are branches of a multivalued analytic function with algebraic singularities at the boundaries of stability intervals.) Let φ± (·, z) be the Floquet solutions of (2.1), that is, the solutions satisfying cφ± (−1/2, z) = v± (z). (2.6) 0 (−1/2, z) φ± We first note that φ± (·, λ + iη) ∈ L 2 near ±∞ if η > 0, and φ± (·, λ + iη) ∈ L 2 near ∓∞ if η < 0. Thus in this setting the Floquet solutions are the Weyl solutions. Secondly, for fixed x, φ± (x, ·) are analytic in S with at worst algebraic


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singularities at a and b arising from the singularities in the initial conditions v± . Lastly, {φ+ (·, z), φ− (·, z)} are a fundamental system of (2.1) for every z ∈ S, as ρ+ (z) 6 = ρ− (z) on S. 2.2. Scattering with respect to a periodic background 1 (R) be real valued with supp( f ) ⊂ [−1/2, 1/2]. Consider the Let 0 6 = f ∈ L loc operator d2 H = − 2 + Vper + f dx 2 in L (R). Take z ∈ S, and let u + be the solution of −u 00 + (Vper + f )u = zu

(2.7)

satisfying u + (x, z) =

( φ+ (x, z)

for x ≤ −1/2,

a(z)φ+ (x, z) + b(z)φ− (x, z) for x ≥ 1/2.

(2.8)

Since φ± are linearly independent for z ∈ S, this defines a(z) and b(z) uniquely. If Vper = 0, then a and b are related to the classical transmission and reflection coefficients by a = t −1 and b = r t −1 . In particular, vanishing of b is equivalent to vanishing of the reflection coefficient. Thus b and u + take the role of a (modified) reflection coefficient and Jost solution relative to the periodic background Vper . Since for λ ∈ (a, b) we know that φ− (x, λ) = φ+ (x, λ) from (2.4), (2.5), and (2.6), by taking u − to be the solution of (2.7) with u − (x, λ) = φ− (x, λ) for x ≤ −1/2, we get for x ≥ 1/2, u − (x, λ) = a(λ)φ− (x, λ) + b(λ)φ+ (x, λ).

(2.9)

Using constancy of the nonzero Wronskian, we arrive at the familiar relation |a(λ)|2 − |b(λ)|2 = 1

(2.10)

for λ ∈ (a, b), corresponding to |r |2 + |t|2 = 1. PROPOSITION 2.1 The coefficients a(·) and b(·), defined on S as above, are branches of multivalued analytic functions with at most algebraic singularities at boundaries of stability intervals.

Proof Recall that u + is the solution of (2.7) with u + (−1/2, z) φ+ (−1/2, z) = 0 (−1/2, z) = v+ (z). u 0+ (−1/2, z) φ+

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t Thus u + (1/2, z), u 0+ (1/2, z) is analytic in S with at worst algebraic singularities at a and b, that is, at the boundaries of the stability interval. (Here and henceforth, we denote by (x, y)t the column vector corresponding to the row vector (x, y).) But, as 0 (1/2, z))t have identical analytic propwas determined before, both (φ± (1/2, z), φ± erties, and by the definition of a(z) and b(z), −1 a(z) φ+ (1/2, z) φ− (1/2, z) u + (1/2, z) = , 0 (1/2, z) φ 0 (1/2, z) b(z) φ+ u 0+ (1/2, z) −

(2.11)

so we are done. 2.2 If b(λ) = 0 for all λ ∈ (a, b), then f is identically zero. LEMMA

Proof Suppose that b(λ) = 0 for all λ ∈ (a, b) and hence all z ∈ S by analyticity. We know then that for every λ ∈ (a, b) and η > 0, the Jost solution u + (x, λ + iη) is a(λ + iη)φ+ (x, λ + iη) for all x ≥ 1/2; that is, u + is exponentially decaying in this region of the upper half-plane. Thus u + is the Weyl solution for the perturbed equation (2.7). We may therefore calculate the Weyl-Titchmarsh M-function (see, e.g., [13]), MVper + f , for (2.7) on the half-line (−1/2, ∞): MVper + f (λ + iη) =

u 0+ (−1/2, λ + iη) φ 0 (−1/2, λ + iη) = + = MVper (λ + iη), u + (−1/2, λ + iη) φ+ (−1/2, λ + iη)

where the latter is the M-function of (2.1) on (−1/2, ∞). As the M-functions are analytic in the entire upper half-plane, we conclude that MVper + f (z) = MVper (z) for all z ∈ C+ . Thus, by the recent results of B. Simon [44] and of F. Gesztesy and Simon [25], which provide generalizations (applicable in our setting) to the classical results of [6], [39], and [40], this implies that f is identically zero. As we assume throughout that f is not identically zero, we know that {λ ∈ (a, b) : b(λ) = 0} is in fact finite as accumulations at algebraic singularities are not possible. Now we consider a gap. Take α such that −∞ ≤ α < a < b and (α, a) is a maximal, nontrivial gap in the spectrum of H0 . (If a = inf σ (H0 ), then α = −∞.) Consider the following split strip: S 0 := {z = λ + iη : α < λ < b, η ∈ R} \ [a, b). For i = 1, 2, let ρi (z) be the branches of (2.2) with |ρ1 (z)| < 1 and |ρ2 (z)| > 1 for all z ∈ S 0 , which are well defined since [a, b) is excluded. We first note that ρ1 = ρ+ on the upper half of S, but ρ1 = ρ− on the lower half of S. Secondly, as before, it can


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be seen that ρi are analytic in S 0 and have at worst algebraic singularities at α, a, and b, and therefore they may be continued analytically across (a, b). In particular, ρi is the analytic continuation of ρ j , where i, j ∈ {1, 2} with i 6 = j. For z ∈ S 0 , choose analytic eigenvectors vi (z) of g0 (z) to corresponding ρi (z) (see [28, Chapter II.4]). Taking φi to be the solutions of (2.1) with (φi (−1/2, z), φi0 (−1/2, z))t = vi (z) for z ∈ S 0 , we see again that φ1 (·, z) (resp., φ2 (·, z)) is in L 2 near +∞ (resp., −∞); that is, they are the Weyl solutions. Set u i to be the Jost solutions of (2.7) satisfying ( φi (x, z), x ≤ −1/2, u i (·, z) = (2.12) ai (z)φi (x, z) + bi (z)φ j (x, z), x ≥ 1/2, for z ∈ S 0 and the same i, j convention used above. As in (2.11) above, one sees that a1 (z) and b1 (z) are analytic in S 0 . In fact, in the upper half S+ of S, they coincide with a(z) and b(z) since for z ∈ S+ we have that v+ (z) and v1 (z) coincide up to a constant multiple, which cancels in (2.11) and in the corresponding expression for a1 (z) and b1 (z). Thus a1 (z) and b1 (z) are analytic continuations of the restrictions of a(z) and b(z) to S+ . Similarly, it is seen that a2 (z) and b2 (z) are analytic continuations of the restrictions of a(z) and b(z) to the lower half S− of S. Thus a1 (z), b1 (z), a2 (z), and b2 (z) have at most algebraic singularities at α, a, and b. In particular, since they cannot vanish identically by f 6 = 0, the set λ ∈ (α, b) : a1 (λ)b1 (λ)a2 (λ)b2 (λ) = 0 (2.13) is discrete and, if α 6= −∞, finite. 2.3. Transfer matrices and positivity of γ To prove positivity of γ (λ), we investigate properties of the transfer matrices. As our minimal assumption is that the support of the distribution contains the points 0 and 1, we know that there are at least two nontrivial transfer matrices: the free matrix g0 (λ) corresponding to (2.1) and the perturbed matrix g1 (λ) corresponding to (2.7). Set G(λ) to be the closed subgroup of SL(2, R) generated by {gq (λ) : q ∈ supp(µ)}. Let P(R2 ) be the projective space, that is, the set of the directions in R2 , and let v be the direction of v ∈ R2 \ {0}. Note that SL(2, R) acts on P(R2 ) by gv = gv. We say that G ⊂ SL(2, R) is strongly irreducible if and only if there is no finite G-invariant set in P(R2 ). THEOREM 2.3 Given a family of operators {Hω }, as in (1.2) and (1.3), with f 6= 0 and 0, 1 ∈ supp(µ), there exists a discrete set M ⊂ R such that G(λ) is noncompact and strongly irreducible for all λ ∈ R \ M. In particular, γ (λ) > 0 for all λ ∈ R \ M.

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We first prove that G(λ) is not compact by showing that a sequence of elements has unbounded norm. This argument is valid for all but a discrete set of λ’s. Once noncompact, the group G is then known to be strongly irreducible if and only if for each v ∈ P(R2 ), #{gv : g ∈ G} ≥ 3 (2.14) (see [7]). We prove that this condition is also satisfied for all but a slightly larger, yet still discrete, set of λ’s. The conclusion concerning positivity of γ follows from a general theorem of Furstenberg which in this context states that if the group G(λ) is noncompact and strongly irreducible, then γ (λ) > 0 (see also [7]). Note that noncompactness of G(λ) and (2.14) are both properties that are preserved if supp(µ) is increased. Thus we assume without loss of generality that supp(µ) = {0, 1}, that is, that G(λ) is generated by g0 (λ) and g1 (λ). In the construction of M we treat stability intervals and spectral gaps of H0 separately. We first show that in any given stability interval (a, b), there is at most a finite number of λ’s such that Furstenberg’s theorem does not apply. We then show the same for finite spectral gaps (α, a) of H0 . For the spectral gap (−∞, inf σ (H0 )), we allow for accumulation of exceptional energies at −∞. Finally, we join all these exceptional sets and the endpoints of stability intervals to get the discrete set M. LEMMA 2.4 Let λ be in a stability interval (a, b) of H0 . Then

g1 (λ) = C(λ)g˜ 0 (λ)s(λ)C(λ)−1 ,

(2.15)

where

1 0 Re[ρ(λ)] Im[ρ(λ)] , g˜ 0 (λ) = , (2.16) Re[c(λ)] Im[c(λ)] − Im[ρ(λ)] Re[ρ(λ)] Re[a(λ) + b(λ)] Im[a(λ) + b(λ)] s(λ) = , (2.17) − Im[a(λ) − b(λ)] Re[a(λ) − b(λ)]

C(λ) =

and c(λ) = c+ (λ) = c− (λ), ρ(λ) = ρ+ (λ) = ρ− (λ), as given in (2.3) and (2.5). Proof Since λ ∈ (a, b), we have c+ (λ) = c− (λ), ρ+ (λ) = ρ− (λ), and Im[c(λ)] 6= 0 from (2.3) and (2.5). Thus C(λ) is invertible. For the rest of the proof we drop the fixed parameter λ. First, note that expressing the solutions u N and u D in terms of the Jost solutions u ± yields u + (1/2) u − (1/2) 1 1 g1 (λ) = . u 0+ (1/2) u 0− (1/2) c c


71

Clearly, u + (1/2) u − (1/2) φ+ (1/2) φ− (1/2) a = 0 (1/2) φ 0 (1/2) u 0+ (1/2) u 0− (1/2) b φ+ −

b , a

and using the fact that φ± are the Floquet solutions, we have φ+ (1/2) φ− (1/2) 1 1 ρ 0 . 0 (1/2) φ 0 (1/2) = c c φ+ 0 ρ − Hence we see that 1 1 ρ g1 (λ) = c c 0

0 ρ

a b

b a

−1 1 1 . c c

(2.18)

Let 1 1 −i Q := ; 2 1 i then it is easily checked that a b Q = s, Q −1 b a

Q −1

ρ 0

0 Q = g˜ 0 , ρ

1 1 Q = C, c c

with C, g˜ 0 , s as defined in (2.16) and (2.17). Thus (2.15) follows from (2.18). Since g0 (λ) corresponds to trivial scattering, it follows from (2.15) that g0 (λ) = C(λ)g˜ 0 (λ)C(λ)−1 .

(2.19)

˜ Let G(λ) be the subgroup of SL(2, R) generated by g˜ 0 (λ) and g˜ 0 (λ)s(λ) or, equiva˜ lently, by g˜ 0 (λ) and s(λ). G(λ) is conjugate to G(λ), and thus G(λ) is noncompact ˜ and strongly irreducible if and only if G(λ) is noncompact and strongly irreducible. Before proving Theorem 2.3, we note that for every λ in a stability interval (a, b), ρ(λ) = eiω , where ω = ω(λ) ∈ (0, π ). Thus cos(ω) sin(ω) g˜ 0 (λ) = (2.20) − sin(ω) cos(ω) is merely a rotation. In particular, for any λ ∈ (a, b) satisfying b(λ) = 0, one has a(λ) = eiα by (2.10), and therefore s(λ) is reduced to a rotation matrix as well. For these finitely many λ’s (see the paragraph immediately following the proof of ˜ Lemma 2.2), G(λ) is a group of rotations and thereby compact. Moreover, by (2.15) and (2.19) we know that at such energies the norm of products of the transfer matrices does not grow; hence γ (λ) = 0. Let M˜ (a,b) := {λ ∈ (a, b) : b(λ) = 0}. It is clear then that only away from this set, M˜ (a,b), can we hope to apply Furstenberg’s theorem and prove positivity of γ .

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Proof of Theorem 2.3 The stability intervals. We first consider energies in a stability interval (a, b). Let λ ∈ (a, b) \ M˜ (a,b) , that is, such that b(λ) 6 = 0. Set a(λ) = Aeiα , b(λ) = Beiβ , and see that (2.10) implies A2 − B 2 = 1 with A > B > 0, A + B > 1, and A − B < 1. In this notation one may recalculate cos(α) sin(α) cos(β) sin(β) s(λ) = A +B . − sin(α) cos(α) sin(β) − cos(β) ˜ We wish to show that a sequence of elements in G(λ) has unbounded norm. To 2 do so, consider an arbitrary element of P(R ), represented by cos(θ) v(θ) := . sin(θ) One sees that the relation s(λ)v(θ) = Av(θ − α) + Bv(β − θ) suggests the choice θ 0 := (1/2)(α + β), yielding s(λ)v(θ 0 ) = (A + B)v [β − α]/2 . Defining R(θ) := ks(λ)v(θ)k2 , a short calculation using A2 − B 2 = 1 shows that R(θ) − 1 = 2B [B + A cos(α + β − 2θ)] . As a consequence, we see that R(θ) − 1 is π -periodic and has exactly two roots in [0, π ). In particular, R(θ) = 1 if and only if cos(α + β − 2θ) = −B/A, which shows that the distance between the zeros of R(θ) − 1 is not equal to π/2: recall that B 6 = 0 (and B 6 = A). Similarly, R(θ) > 1 if and only if cos(α + β − 2θ) > −B/A, and hence |{θ ∈ [0, π) : R(θ) > 1}| > π/2. As a result, there exists a compact interval K (not necessarily in [0, π)) with |K | > π/2 and ks(λ)v(θ )k > c > 1 for all θ ∈ K . Now, applying s(λ) once to v(θ 0 ), θ 0 as above, produces a new vector with norm greater than one, but the direction, initially θ 0 , is possibly altered. A vector with this new direction may not increase in norm by directly applying s(λ) again. We may, however, apply the modified free matrix (2.20) and rotate this new direction by ω. As ω ∈ (0, π), finitely many applications of g˜ 0 (λ) produce a vector v with direction in K . Once in K , a direct application of s(λ) does increase the norm size uniformly by c > 1, as indicated above. In this manner we can produce a sequence of elements in ˜ G(λ) with unbounded norm. To finish the proof of Theorem 2.3 in the stability interval (a, b), we choose M (a,b) := λ ∈ (a, b) : λ ∈ M˜ (a,b) or D(λ) = 0 .


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˜ Clearly, M (a,b) is finite, and by the above, G(λ) is not compact for λ ∈ (a, b)\ M (a,b) . It remains to check (2.14) for these values of λ. But this is trivial as D(λ) 6= 0 implies ω 6= π/2, and therefore the free transfer matrix produces three distinct elements in projective space. The spectral gaps. For all energies λ ∈ (α, a), a maximal spectral gap of H0 , G(λ) is noncompact. This is easily seen as λ ∈ (α, a) implies |D(λ)| > 2, and hence the free transfer matrix, g0 , has an eigenvalue |ρ| = |ρ2 | > 1. Thus repeated iteration of g0 on v 2 , the direction of v2 , produces an unbounded sequence of elements. It remains to check (2.14), which we do for λ ∈ (α, a) \ M (α,a) , where M (α,a) := λ ∈ (α, a) : a1 (λ)b1 (λ)a2 (λ)b2 (λ) = 0 . As was clear in (2.13), M (α,a) is discrete, and finite if α 6 = −∞. Let λ ∈ (α, a) \ M (α,a) ; that is, ai (λ) 6= 0 and bi (λ) 6 = 0 for i = 1, 2. By (2.12), this is equivalent to s(λ)v i ∈ / {v 1 , v 2 }, (2.21) where v i are the directions of the eigenvectors vi of g0 (λ), again for i = 1, 2. If v∈ / {v 1 , v 2 }, then #{g0 (λ)n v : n ∈ Z} = ∞. If, on the other hand, v ∈ {v 1 , v 2 }, then we use (2.21) to conclude that an initial application of s(λ) followed by iteration of g0 (λ) gives an infinite orbit. This shows (2.14) and completes the proof for energies in a spectral gap. We can now complete the proof by taking M := M1 ∪ M2 ∪ M3 , where [ M1 := M (a,b) , (a,b)

the union being taken over all stability intervals in σ (H0 ), [ M2 := M (α,a) , (α,a)

the union being taken over all maximal gaps of H0 , and M3 is defined to be the set containing all the endpoints of the stability intervals. 3. Hölder continuity of the Lyapunov exponent In this section we prove that the Lyapunov exponent for the family of random operators defined by (1.2) and (1.3) is Hölder continuous on every compact interval of R which does not contain energies in the exceptional set M from Theorem 2.3. A result of this type is proven in [12] for an analogue of (1.1) defined on a discrete strip. We give an outline of the proof, indicating what changes are necessary in order to see that

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the argument from [12] can be adapted to our setting. The main changes are due to less explicit forms of the continuum transfer matrices. Fix a compact interval I ⊂ R \ M, where M is the discrete set from Theorem 2.3. 3.1 The Lyapunov exponent for (1.2) and (1.3) is uniformly Hölder continuous on I ; that is, there exists a number α > 0 and a constant C for which THEOREM

|γ (λ) − γ (λ0 )| ≤ C|λ − λ0 |α for all λ, λ0 ∈ I . Proof The rest of this section is used to prove Theorem 3.1, where we rely on general facts from Appendix B. As in the introduction, let gλ (n, ω) be the transfer matrix of (1.5) from n − 1/2 to n + 1/2. If u N (·, λ, ω) and u D (·, λ, ω) are the solutions of (1.5) with u N (n − 1/2) = u 0D (n − 1/2) = 1, u 0N (n − 1/2) = u D (n − 1/2) = 0, then u N (n + 1/2, λ, ω) u D (n + 1/2, λ, ω) gλ (n, ω) = . (3.1) u 0N (n + 1/2, λ, ω) u 0D (n + 1/2, λ, ω) This shows that for fixed n and λ, gλ (n, ·) : → SL(2, R) is measurable, and for fixed λ, gλ (n, ω) are real-valued, i.i.d. random matrices. Here we have used supp( f ) ⊂ [−1/2, 1/2]. Let µλ denote the distribution of gλ in SL(2, R). Using (3.1) and the boundedness of the distribution µ of qn , we get from Lemma A.1, kgλ (n, ω)k2 ≤ exp C1 + |λ| + C2 |qn (ω)| ≤ C3 (3.2) and from Lemma A.2,

gλ (n, ω) − gλ0 (n, ω) ≤ C|λ − λ0 |,

(3.3)

where all constants are uniform in ω, n, and λ, λ0 ∈ I . Take G = SL(2, R), B = P(R2 ), and the action gv = gv as described in Section 2. The projective distance δ(x, y) =

|x1 y2 − x2 y1 | kxk · kyk

for x, y ∈ R2

defines a metric on P(R2 ). (Note that δ(x, y) = | sin(ψ)|, where ψ is the angle between x and y.) As in Example B.1, set B˜ := B × B \ {(b, b) : b ∈ B}, and define


75

a cocycle σ1 : G × B˜ → (0, ∞) by δ(gx, g y) σ1 g, (x, y) := . δ(x, y)

(3.4)

A short calculation shows that kxk · kyk σ1 g, (x, y) = kgxk · kgyk and hence that σ 1 (g) := sup σ1 (g, b); ≤ kgk2 ; b∈ B˜ C+|λ| e for

thus (3.2) implies that σ 1 (g) ≤ µλ -a.e. g. In particular, this shows that for every t ∈ R, Z sup σ 1 (g)t dµλ (g) < ∞. λ∈I

This proves that for (3.4), Lemma B.2(i) is satisfied uniformly with respect to λ ∈ I . Proving that Lemma B.2(ii) also holds for (3.4), again uniformly with respect to λ ∈ I , requires more work. To this end, for any v ∈ P(R2 ), define n kgλ vk o 8λ (v) := E log . kvk The next lemma corresponds to [12, Lemma V.4.7]. 3.2 The mapping 8λ : P(R2 ) → R is continuous. The mapping 8 : I × P(R2 ) → R with 8(λ, v) = 8λ (v) satisfies sup 8λ (v) − 8λ0 (v) ≤ C|λ − λ0 |

LEMMA

(i) (ii)

(3.5)

v∈P(R2 )

for some constant C = C(I ). In particular, 8 is continuous. Proof (i) By (3.2), log kgλ vk/kvk ≤ log kgλ k ≤ C uniformly in ω and v. Thus by dominated convergence, limn→∞ 8λ (v n ) = 8λ (v) if v n → v. (ii) We have o n 8λ (v) − 8λ0 (v) = E log kgλ vk ≤ E{log kgλ g −1 λ0 k} kgλ0 vk 0 ≤ E log(kgλ − gλ0 k · kgλ−1 0 k + 1) ≤ C|λ − λ |, 0 where the last inequality follows as kgλ − gλ0 k · kgλ−1 0 k ≤ C|λ − λ | for all λ, λ0 ∈ I by (3.2) and (3.3).

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As we know that for each fixed λ ∈ I , G(λ) is noncompact and strongly irreducible, then combining [12, Proposition IV.4.11 and Theorem IV.4.14], we have that there exists a unique µλ -invariant probability measure νλ on P(R2 ) for which Z kgvk γ (λ) = log dµλ (g) dνλ (v). (3.6) kvk 3.3 The Lyapunov exponent γ is continuous on I ;

COROLLARY

(i) (ii)

kUλ (n)vk o 1 n E log = γ (λ), n→∞ n kvk lim

uniformly with respect to λ ∈ I and v ∈ P(R2 ). Proof (i) We first show that νλ0 → νλ weakly as λ0 → λ. Note that in general if µλn → µλ0 weakly and νλn → νλ0 weakly, then µλn ∗ νλn → µλ0 ∗ νλ0 weakly (for the definition of the convolution, see Appendix B). By (3.3) we have that µλ0 → µλ weakly, and an application of the Arzela-Ascoli theorem implies that every subsequence of {νλ0 } contains a weakly convergent subsequence. Using this, the convergence of the convolutions, and the uniqueness of the invariant measures, we see that νλ0 → νλ . Now, continuity follows from (3.6), that is, from noting that γ (λ) = νλ (8λ ), and estimate (3.5) (see [12, Corollary V.4.8]). (ii) This follows as in the proof of (3.5) with (3.2) and (3.3) replaced by corresponding estimates for Uλ (n, 1), which again follow from Lemmas A.1 and A.2 (see [12, Proposition V.4.9]). 3.4 There exists an integer N for which Z sup log σ1 (g, b) dµλN (g) < 0. LEMMA

λ∈I,b∈ B˜

Proof One calculates that Z Z Z 1 kgxk 1 kgyk 1 δ(gx, g y) n n dµλ (g) = − log dµλ (g) − log dµnλ (g) log δ(x, y) n kxk n kyk n 1 n kUλ (n)xk o 1 n kUλ (n)yk o = − E log − E log . n kxk n kyk The result follows by letting n → ∞ and noting both Corollary 3.3(i) and (ii).


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We have thus shown that Lemma B.2 is applicable to the cocycle σ1 defined in (3.4). Following the lines of [12], this leads to Hölder continuity properties with respect to λ of the invariant measures νλ and the operators Tλ defined by (Tλ f )(x) = R f (gx) dµλ (g) for f in the Hölder spaces Lα (P(R2 )) (see Appendix B). 3.5 There exists an α0 > 0 such that for 0 < α ≤ α0 there exist ρα < 1 and Cα < ∞ such that n δ(U (n)x, U (n)y)α o λ λ sup E ≤ Cα ραn α δ(x, y) 2 λ∈I, x6= y∈P(R ) LEMMA

and sup kTλn f − νλ ( f )k ≤ Cα k f kα ραn λ∈I

for n = 1, 2, . . . . Proof Lemma 3.4 confirms that both assumptions of Lemma 7 hold uniformly with respect to λ ∈ I . That this implies the above estimates, again uniformly with respect to λ ∈ I , follows from observing that the proofs of [12, Propositions IV.3.5 and IV.3.15] yield uniform results under uniform assumptions. 3.6 There exists an α0 > 0 such that for 0 < α ≤ α0 there exists a Cα < ∞ such that LEMMA

and for all

λ, λ0

kTλ f − Tλ0 f kα/2 ≤ Cα k f kα |λ − λ0 |α/2

(3.7)

|νλ ( f ) − νλ0 ( f )| ≤ Cα k f kα |λ − λ0 |α/2

(3.8)

∈ I and f ∈ Lα .

Proof The bound (3.7) follows as in [12, Proposition V.4.13], using ((gλ − gλ0 )x)1 (gλ0 x)2 + (gλ0 x)1 ((gλ0 − gλ )x)2 δ(gλ x, gλ0 x) = kgλ xkkgλ0 xk k(gλ − gλ0 )xk ≤2· ≤ 2 · kgλ − gλ0 k · kgλ−1 k ≤ C|λ − λ0 |. kgλ xk The bound (3.8) follows as in [12, Proposition V.4.14], using Lemma B.2(c) as done therein. Compiling these results, Theorem 3.1 follows as in [12, Theorem V.4.15].

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4. Hölder continuity of the integrated density of states In this section we prove an analogue of Theorem 3.1 for the integrated density of states. We begin with its definition. For L ∈ N and 3 L (0) = [−L/2, L/2], let H3 L (0) (ω) := Hω on L 2 3 L (0) (4.1) with Hω as in (1.2) and (1.3) and Dirichlet boundary conditions set at ±L/2. Fix λ ∈ R, and let N L ,ω (λ) denote the number of eigenvalues of H3 L (0) (ω) less than or equal to λ. By Kingman’s super-additive ergodic theorem, we have that the limit N (λ) := lim

L→∞

N L ,ω (λ) L

(4.2)

exists for almost every ω and in expectation. In particular, N (λ) = sup L≥1

1 E N L ,ω (λ) . L

(4.3)

The limit N (λ) in (4.2) does not depend on the particular choice of the boundary conditions one sets on 3 L (0); however, the monotonicity of the limit in expectation (4.3) may be changed. For example, with Neumann boundary conditions one gets an “inf ” rather than a “sup.” N (λ) is called the integrated density of states. As in Section 3, fix a compact interval I ⊂ R \ M with M as in Theorem 2.3. 4.1 The integrated density of states for (1.2) and (1.3) is uniformly Hölder continuous on I. THEOREM

The proof of this theorem follows easily from Theorem 3.1 once we have a means of relating N and γ . For our model, one has the following. PROPOSITION 4.2 (The Thouless formula) Let N and γ be, respectively, the integrated density of states and Lyapunov exponent for (1.2) and (1.3). Then there exists α ∈ R such that, for every λ ∈ R, Z λ − t γ (λ) = −α + log (4.4) d N (t). t − i R

Remarks. (i) The above version of the Thouless formula arises in Kotani’s work [37]. It differs from the discrete version (see [12, Proposition VI.4.3]) essentially by the normalization term t − i in (4.4), which compensates for the noncompact support of d N . Alternatively, one can work with the normalized Lyapunov exponent γ − γ0 and integrated density of states N − N0 , where γ0 and N0 are the corresponding quantities


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for the free Laplacian (see [3]). We decided to work with (4.4) since it was shown in [30, Appendix 2] that Kotani theory applies to singular potentials. While [30] works with L 2 -assumptions on the potential, it has been pointed out that all that is needed to make Kotani theory work are two basic estimates for the m-function m(z, ω) on (0, ∞) of an ergodic random operator: if K is a compact subset of C+ , the complex upper half-plane, then there are constants C(K ) < ∞ and δ(K ) > 0 such that |m(z, ω)| ≤ C(K )

(4.5)

Im[m(z, ω)] ≥ δ(K )

(4.6)

and uniformly for z ∈ K and ω ∈ . In our models (1.2) and (1.3) we have only used L 1loc -assumptions on the potentials. That (4.5) is satisfied in this case follows from results on m-function asymptotics in [25]. To get (4.6), we use Z ∞ Im[m(z)] = Im[z] · |u|2 d x, (4.7) 0

−u 00 + (Vper + Vω )u

where u is the solution of = zu with u(0) = 1 and u 0 (0) = m(z) (see, e.g., [13]). Combining Lemmas A.1 and A.3, we get Z 2 |u|2 d x ≥ C1 |u(1)|2 + |u 0 (1)|2 ≥ C1 C2 1 + |m(z)|2 . 0

Thus (4.7) yields (4.6). (ii) To apply Kotani’s results from [37], one must have an R-ergodic system; that is, there must exist a group {θt : t ∈ R} of measure-preserving transformations for which the dynamical system (, F , θt , µ) is ergodic and the random potential vω satisfies vθt ω (x) = vω (x + t). W. Kirsch’s result in [29] shows how a Z-ergodic system, that is, one for which the transformations θt are parametrized by t ∈ Z, can be associated with an R-ergodic system embedded in a larger probability space. The associated system is constructed in such a way that the corresponding integrated density of states and Lyapunov exponent for both systems are equal. Our model is Z-ergodic when equipped with translations. Thus we must apply Kotani’s results to the corresponding R-ergodic system after using Kirsch’s suspension procedure. (iii) In Kotani’s work [37], N and γ arise as the real and imaginary parts of the nontangential limit of a specific Herglotz function (the w-function). To see that the integrated density of states and the Lyapunov exponent actually coincide with these nontangential limits, see [10, Propositions V.12 and VI.1]. Based on (4.4), the proof of Theorem 4.1 is very similar to the proof of the corresponding result in the discrete case (see [12, Proposition VI.3.9]). Some small changes arise

80


from the use of the slightly different Thouless formula (4.4) and the necessity to exclude the set M. We use some basic facts about Hilbert transforms, which for a square integrable function ψ are defined by Z 1 ψ(t) (T ψ)(x) = lim dt. + π ε→0 |x−t|>ε x − t One has two basic results, as stated in [12]. (a) T ψ is a square integrable function, and T 2 ψ = −ψ almost everywhere with respect to Lebesgue measure. (b) If ψ is Hölder continuous on some interval [−a, a], then T ψ is Hölder continuous on [−a/2, a/2]. Note that if ψa (t) := ψ(t − a), then T ψa = (T ψ)a . Thus if ψ is Hölder continuous on some interval [x0 − a, x0 + a], then T ψ is Hölder continuous on [x0 − a/2, x0 + a/2]. Proof of Theorem 4.1 First, observe that d N -integrability of log |(λ − t)/(t − i)| yields Z λ+ε λ − t log lim d N (t) = 0, t −i ε→0+ λ−ε which easily implies lim | log(ε)| N (λ + ε) − N (λ − ε) = 0,

ε→0+

(4.8)

that is, implies log-Hölder continuity and, in particular, continuity of N . Let λ0 ∈ I , and pick a > 0 such that [λ0 − 4a, λ0 + 4a] ⊂ R \ M and thus γ is Hölder continuous in [λ0 − 4a, λ0 + 4a]. Take ψ(t) := N (t)χ{t:|t−λ0 |≤4a} , and note that (T 2 ψ)(t) = −N (t) for almost every t with |t − λ0 | ≤ 4a by (a). For |λ − λ0 | < 4a, calculate Z λ0 +4a Z λ − t λ − t d N (t) = log γ (λ) + α− log d N (t) t − i t − i |t−λ0 |>4a λ0 −4a Z λ−ε Z λ0 +4a = lim log |λ − t| d N (t) + log |λ − t| d N (t) ε→0+

−

1 2

Z

λ+ε λ0 +4a

λ0 −4a

λ0 −4a

log(t 2 + 1) d N (t).


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Integrating the first two integrals above by parts and rearranging yields Z λ − t π(T ψ)(λ) = γ (λ) + α − log d N (t) t − i |t−λ0 |>4a Z 1 λ0 +4a log(t 2 + 1) d N (t) − log(λ0 − λ + 4a)N (λ0 + 4a) + 2 λ0 −4a + log(λ − λ0 + 4a)N (λ0 − 4a),

(4.9)

where (4.8) has been used. From observing (4.9), we see that T ψ is Hölder continuous in [λ0 − 2a, λ0 + 2a], and thus T 2 ψ is Hölder continuous in [λ0 − a, λ0 + a] by (b). Since N is continuous, we get (T 2 ψ)(t) = −N (t) for all t ∈ [λ0 − a, λ0 + a]. Thus N is Hölder continuous in [λ0 − a, λ0 + a]. A compactness argument yields uniform Hölder continuity over all of I . 5. The Wegner estimate In this section we prove a Wegner estimate that constitutes one of the two ingredients that enable us to start the multiscale induction. Given Hölder continuity of the integrated density of states, as was proven in the preceding section, our proof of the Wegner estimate can be carried out in a way analogous to [11]. Since a few modifications of the arguments in [11] are required, we present sufficiently many details for the reader’s convenience. Fix throughout this section a compact interval I = [a, b] ⊂ R \ M, where M is the discrete set found in Theorem 2.3. Our goal is to prove estimates uniformly in I . It is therefore convenient to have these two properties in a ball of fixed radius around each point in I . By our above results we know that there is some ξ > 0 such that for every λ ∈ Iξ = [a − ξ, b + ξ ], G(λ) is noncompact and strongly irreducible. In other words, for every λ ∈ I , we have that G(λ0 ) is noncompact and strongly irreducible for every λ0 ∈ [λ − ξ, λ + ξ ]. For odd L ∈ N and 3 = 3 L (0) = [−L/2, L/2], let H3 (ω) be the restriction of Hω to 3 with Dirichlet boundary conditions at −L/2 and L/2. We prove the following theorem. THEOREM 5.1 (Wegner estimate) For every β ∈ (0, 1) and every σ > 0, there exist L 0 ∈ N and α > 0 such that β β P dist(λ, σ (H3 (ω))) ≤ e−σ L ≤ e−αL (5.1)

for all λ ∈ I and L ≥ L 0 . To prove Theorem 5.1, we need Lemmas 5.2 and 5.3.

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LEMMA 5.2 There exist α1 > 0, δ > 0, n 0 ∈ N such that for all λ ∈ I, n ≥ n 0 , and x normalized, we have E kUλ (n)xk−δ ≤ e−α1 n .

Proof This lemma can be proven in exactly the same way as [11, Lemma 5.1]. For the reader’s convenience, we sketch the argument briefly. By our choice of the interval I and the results from Sections 2 and 3, in particular Corollary 3.3(i), we have γ := inf{γ (λ) : λ ∈ I } > 0. Using the inequality e y ≤ 1 + y + y 2 e|y| and Hölder’s inequality, one shows as in [11] that for every λ ∈ I and every δ > 0, we have E kUλ (n, ω)xk−δ = E kgλ (n, ω) · . . . · gλ (1, ω)xk−δ ≤ 1 − δE log kgλ (n, ω) · . . . · gλ (1, ω)xk 1/2 n/2 + δ 2 n 2 E (log kgλ (1, ω)k)4 E kgλ (1, ω)k2δ ≤ 1 − δE log kgλ (n, ω) · . . . · gλ (1, ω)xk + δ 2 n 2 C1 C2n for some finite constants C1 = C1 (I ), C2 = C2 (I ). Hence, by Corollary 3.3(ii), we have for some n 0 = n 0 (I ) uniformly in λ ∈ I and x in the unit sphere, 1 n E kUλ (n 0 , ω)xk−δ ≤ 1 − n 0 δγ + δ 2 n 20 C1 C2 0 ≤ 1 − ε 2

(5.2)

for some ε > 0, provided δ is small enough. Iterating (5.2) as in [11] yields E kUλ (n, ω)xk−δ ≤ C(1 − ε)bn/n 0 c ≤ e−α1 n for all n ≥ n 1 and λ ∈ I , for some α1 = α1 (I ) > 0 and n 1 = n 1 (I ), where bn/n 0 c is the largest integer less than or equal to n/n 0 . 5.3 There exist ρ > 0 and C < ∞ such that for every λ ∈ I and every ε > 0, we have for L ≥ L 0 , LEMMA

P There exist λ0 ∈ (λ − ε, λ + ε) and φ ∈ D(H3 ), kφk = 1, such that (H3 (ω) − λ0 )φ = 0, |φ 0 (−L/2)|2 + |φ 0 (L/2)|2 ≤ ε2 ≤ C Lερ . (5.3) Proof We follow the same strategy as Carmona, Klein, and Martinelli in their proof of [11, Lemma 5.2]; that is, we use Hölder continuity of the integrated density of states to derive estimate (5.3). The only difficulty that arises is that the cutoff of eigenfunctions


83

as performed by Carmona, Klein, and Martinelli in the discrete case may produce elements outside the domain of the local Hamiltonian. We therefore use a smooth cutoff procedure and show that the argument still goes through. Note first that it suffices to prove (5.3) for small ε > 0. It follows from Theorem 4.1 that there are constants ρ = ρ(Iξ ) > 0 and C1 = C1 (Iξ ) < ∞ such that for every λ, λ0 ∈ Iξ , N (λ) − N (λ0 ) ≤ C1 |λ − λ0 |ρ . (5.4) Now fix λ ∈ I , ε > 0, and L. For k ∈ Z, let 3k be the interval [k L − L/2, k L + L/2], and denote by H3k (ω) the operator H (ω) restricted to 3k with Dirichlet boundary conditions. Let Ak be the event Ak = Ak (λ, ε, L) = {ω ∈ : H3k (ω) has an eigenvalue λk ∈ (λ − ε, λ + ε) such that the corresponding normalized eigenfunction φk satisfies |φk0 (k L − L/2)|2 + |φk0 (k L + L/2)|2 ≤ ε2 }. Let p = p(λ, ε, L) = P{Ak }. Clearly, p is independent of k and equals the left-hand side of (5.3). Fix some n ∈ N, and let Hn (ω) be the operator H (ω) restricted to 3(n) := Sn k=−n 3k = [−n L − L/2, n L + L/2] with Dirichlet boundary conditions at −n L − L/2 and n L + L/2. Let k1 , . . . , k j ∈ {−n, . . . , n} be distinct and such that the event Akl occurs. For each such l we construct a normalized function φ˜l in the domain of Hn (ω) which is supported on 3kl (hence {φ˜ 1 , . . . , φ˜ j } form an orthonormal set) such that for every l,

(Hn (ω) − λ)φ˜l ≤ C2 ε, (5.5) where C is a constant that depends only on the single-site potential f and the singlesite distribution µ. By disjointness of supports, we also have

φ˜l , Hn (ω)φ˜l 0 = 0 = Hn (ω)φ˜l , Hn (ω)φ˜l 0 for l 6 = l 0 . (5.6) By [45, Lemma A.3.2], (5.5) and (5.6) imply that the number of eigenvalues of Hn (ω) (counted with multiplicity) in [λ − C2 ε, λ + C2 ε] is bounded from below by j. In other words, # l ∈ {−n, . . . , n} : Al occurs ≤ # eigenvalues of Hn (ω) in [λ − C2 ε, λ + C2 ε] . Thus if ε is small enough (namely, such that [λ − C2 ε, λ + C2 ε] ⊆ Iξ or, equivalently, ε ≤ ξ/C2 ), we have

84


1 # l ∈ {−n, . . . , n} : Al occurs 2n + 1 1 ≤ lim # eigenvalues of Hn (ω) in [λ − C2 ε, λ + C2 ε] n→∞ 2n + 1 1 = L lim # eigenvalues of Hn (ω) in [λ − C2 ε, λ + C2 ε] n→∞ (2n + 1)L = L N (λ + C2 ε) − N (λ − C2 ε)

p = lim

n→∞

≤ LC1 (2C2 ε)ρ =: C Lερ , where the intermediate steps hold true for almost every ω. It remains to construct φ˜l with the desired properties. Fix some l, and consider the function φl which is defined on 3kl and vanishes at the boundary points. In principle we would like to extend φl by zero on 3(n) \ 3kl . However, this function does not in general belong to the domain of Hn , so we could not even evaluate (Hn − λ) applied to this function. Instead we use a smooth extension of φl to 3(n). Fix once and for all a smooth function χ that obeys 0 ≤ χ ≤ 1, χ (x) = 0 for x ≤ 0, and χ (x) = 1 for x ≥ 1. Let xlL = kl L − L/2 and xlR = kl L + L/2, and define φˆl for x ∈ 3(n) by   0 if x 6∈ 3kl ,    χ (x − x L )φ (x) if xlL ≤ x ≤ xlL + 1, l l φˆl (x) =  φl (x) if xlL + 1 ≤ x ≤ xlR − 1,     χ (−x + xlR )φl (x) if xlR − 1 ≤ x ≤ xlR . Then φˆl clearly belongs to the domain of Hn , and it has norm bounded by kφl k = 1. We want to estimate k(Hn − λ)φˆl k. Now φl is an eigenfunction corresponding to the eigenvalue λkl ∈ (λ − ε, λ + ε), so we write k(Hn − λ)φˆl k ≤ k(Hn − λkl )φˆl k + ε. To estimate k(Hn − λkl )φˆl k, we consider (H − λkl )φˆl (x) = −φˆl00 (x) + Vω (x)φˆl (x) − λkl φˆl (x)


85

for x ∈ 3(n). Hence we have k(Hn − λkl )φˆl k2 =

xlL +1

Z

χ 00 (x − x L )φl (x) + 2χ 0 (x − x L )φ 0 (x) 2 d x l l l

xlL

Z

xlR

+ xlR −1

00 χ (−x + x R )φl (x) + 2χ 0 (−x + x R )φ 0 (x) 2 d x l l l

2 Z x L +1 00

χ (x − x L ) 2

φl l l

· ≤

2χ 0 (x − x L ) d x

φ0 xlL l l L ∞ (xlL , xlL +1)

2 Z x R 00

φl

χ (−x + x R ) 2 l l

+ ·

φ0

2χ 0 (−x + x R ) d x xlR −1 l l L ∞ (xlR −1, xlR )

φl (x L ) 2 φl (x R ) 2 l l

+

≤ C3

φ 0 (x L )

φ 0 (x R ) l l l l = C3 |φl0 (xlL )|2 + |φl0 (xlR )|2 ≤ C3 ε2 , where the constant C3 depends on the single-site potential, the single-site distribution, and the function χ . Here we have used Lemma A.1 in the second-to-last estimate. Let us define φ˜l := φˆl /kφˆl k. By construction and by Lemma A.1 we have, for ε sufficiently small, kφˆl k ≥ 1/2, and hence (5.5) holds true with a suitable C2 that depends only on the single-site potential, the single-site distribution, and the function χ. Moreover, by construction {φ˜ 1 , . . . , φ˜ j } form an orthonormal set and obey (5.6). This concludes the proof of the lemma. We are now in position to give the following. Proof of Theorem 5.1 We closely follow the proof of [11, Theorem 4.1] and make the necessary modifications. Let β, σ, I be as above, and for each odd L ∈ N, set n L = bτ (L/2)β c + 1 with some τ > 0 to be chosen later. For every λ ∈ I and θ > 0, we define the events

β (λ,L) Aθ = gλ (−(L + 1)/2 + n L ) · . . . · gλ (−(L − 1)/2)(0, 1)t > eθ(L/2) ,

β (λ,L) Bθ = gλ ((L + 1)/2 − n L )−1 · . . . · gλ ((L − 1)/2)−1 (0, 1)t > eθ(L/2) .

86


Let κ = τ α1 /2δ with α1 and δ from Lemma 5.2. Then β P dist(λ, σ (H3 (ω))) ≤ e−σ L n \ β ≤ P dist(λ, σ (H3 (ω))) ≤ e−σ L ∩

(λ0 ,L)

|λ−λ0 |≤e−σ L

n

+ P A(λ,L) ∩ Bκ(λ,L) ∩ κ

[ |λ−λ0 |≤e−σ L

n

+ P A(λ,L) ∩ Bκ(λ,L) ∩ κ

β

[ |λ−λ0 |≤e−σ L

β

(Aκ/2 )c

(λ0 ,L)

o

(λ0 ,L)

o

(Bκ/2 )c

β

(Aκ/2

o (λ0 ,L) ∩ Bκ/2 )

)c + P (Bκ(λ,L) )c + P (A(λ,L) κ = (i) + (ii) + (iii) + (iv), and Lemma 5.2 immediately implies β

(iv) ≤ 2e−(1/2)τ α1 (L/2) ,

(5.7)

provided L is large enough. Together with Lemma A.3, Lemma 5.3 yields, for L large enough, β β (i) ≤ C˜ L max{e−σ L ρ , e−(1/2)κ(L/2) ρ }

(5.8)

with C˜ independent of λ and L. Finally, using (3.3), for L large enough one proves, similarly to [11], (ii) + (iii) ≤ 2e−α2 (L/2)

β

(5.9)

for some suitable α2 > 0 if τ is chosen small enough. The assertion now follows from (5.7) – (5.9). 6. The initial length scale estimate Fix λ ∈ R \ M, let νλ be the unique µλ -invariant measure on P(R2 ) (see (3.6)), and let δ(x, y) be the projective distance of x, y ∈ P(R2 ). The measure νλ is Hölder continuous. LEMMA 6.1 There exist ρ > 0 and C > 0 such that for all x ∈ P(R2 ) and ε > 0, one has νλ {y : δ(x, y) ≤ ε} ≤ Cερ .

Proof This follows from [7, Corollary VI.4.2]. Note that in the case when G(λ) ⊂ SL(2, R)


87

R the assumptions required there are equivalent to kgkτ dµλ (g) < ∞ for some τ > 0, and note that G(λ) is noncompact and strongly irreducible. In particular, noncompactness is equivalent to the contractivity required in [7] (see [12, Proposition IV.4.11]). Integrability of kgkτ with respect to µλ for all τ > 0 follows from (3.2) and boundedness of the distribution of qn . We know that (1/n) log kUλ (n)k converges to γ (λ) in expectation. We need a large deviation result for this limit. In fact, the following result on the asymptotics of kUλ (n)xk for any initial vector x 6 = 0 holds. 6.2 There exists α > 0 such that for every ε > 0 and x 6 = 0, one has LEMMA

lim sup n→∞

1 log P | log kUλ (n)xk − nγ (λ)| > nε < −α. n

Proof This follows from [7, Theorem V.6.2], whose assumptions that G(λ) is strongly irreR ducible and kgkτ dµλ (g) < ∞ for some τ > 0 are satisfied. An immediate consequence of Lemma 6.2 is that for every ε > 0 and x 6 = 0, there exists n 0 ∈ N such that P e(γ (λ)−ε)n ≤ kUλ (n)xk ≤ e(γ (λ)+ε)n ≥ 1 − e−αn for n ≥ n 0 . (6.1) Next, we establish a large deviation result for |hUλ (n)x, yi|, that is, in particular for the matrix elements of the transfer matrices. LEMMA 6.3 Fix y with kyk = 1. For all ε > 0 there are n 0 ∈ N and δ0 > 0 such that |hU (n)x, yi| λ sup P < e−εn < e−δ0 n for n ≥ n 0 . kUλ (n)xk x6 =0

Proof We closely follow the proof of [7, Proposition VI.2.2]. Define f n : [0, 1] → R by   if 0 ≤ t ≤ e−εn ,  1 f n (t) = 2 − teεn if e−εn ≤ t ≤ 2e−εn ,   0 if 2e−εn ≤ t ≤ 1. Then

f n (t) − f n (t 0 ) ≤ |t − t 0 |eεn

for all t, t 0 ∈ [0, 1].

(6.2)

88


Define 8n : P(R2 ) → R by E D z 8n (z) = f n ,y . kzk For z, z 0 ∈ P(R2 ), choose representatives z and z 0 such that kzk = kz 0 k = 1 and the angle between z and z 0 is at most π/2. Then by (6.2), √

8n (z) − 8n (z 0 ) ≤ hz, yi| − |hz 0 , yi eεn ≤ kz − z 0 keεn ≤ 2δ(z, z 0 )eεn . (6.3) Recall that δ(z, z 0 ) = | sin(angle(z, z 0 ))|. This implies, using notation from Section 3, that √ k8n kα = k8n k∞ + m α (8n ) ≤ 2eεn + 1 for all 0 < α < 1. (6.4) The definition of 8n shows that |hU (n)x, yi| λ P < e−εn ≤ E 8n (Uλ (n)x) kUλ (n)xk Z Z ≤ E 8n (Uλ (n)x) − 8n dνλ + 8n dνλ .

(6.5)

Noting that E 8n (Uλ (n)x) = (Tλn 8n )(x), we may use Lemma 3.5 to conclude that for 0 < α ≤ α0 there exists ρ < 1 and n 0 ∈ N such that for n ≥ n 0 and all x, Z

√ 1

E 8n (Uλ (n)x) − 8n dνλ ≤ Cα ρ n k8n kα ≤ ρ n ( 2eεn + 1) ≤ e−δ1 n , (6.6) 2 where in the last step ε is assumed to be sufficiently small, that is, such that log ρ+ε < 0, and δ1 := (1/2)| log ρ + ε|. Next, choose the unit vector w = (w1 , w2 )t = (y2 , −y1 )t . We therefore have |hu/kuk, yi| = δ(u, w) for all u 6 = 0. It then follows from Lemma 6.1 that there exist β > 0 and C > 0 such that Z 8n dνλ ≤ νλ u : |hu/kuk, yi| ≤ 2e−εn = νλ u : δ(u, w) ≤ 2e−εn ≤ 2β Ce−εβn . (6.7) Inserting (6.6) and (6.7) into (6.5) completes the proof of Lemma 6.3 with 0 < δ0 < min{δ1 , εβ}. We have assumed that ε is sufficiently small, but the result extends to large ε with unchanged δ0 . Note that results corresponding to Lemmas 6.2 and 6.3 also hold for n → −∞ since the result from [7] can be applied in the same way to the products of random matrices gλ−1 (n) × · · · × gλ−1 (−1) for n < 0 and the Lyapunov exponent can equivalently be defined as 1 γ (λ) = lim E kgλ−1 (n) · . . . · gλ−1 (−1)k . n→−∞ |n|


89

We can now combine Lemmas 6.2 and 6.3 to show that with large probability the matrix elements |hUλ (n)x, yi| grow exponentially at almost the rate of the Lyapunov exponent. COROLLARY 6.4 Let kxk = kyk = 1. Then for every ε > 0 there exist δ > 0 and n 0 ∈ N such that P |hUλ (n)x, yi| ≥ e(γ (λ)−ε)n ≥ 1 − e−δn for n ≥ n 0 .

Proof From (6.1) and Lemma 6.3, we get for all ε > 0, hUλ (n)x, yi ≥ e−εn kUλ (n)xk ≥ e(γ (λ)−2ε)n

for n ≥ n 0

with probability at least 1 − e−αn − e−δ0 n . This yields the assertion. We are now ready to state and prove the main result of this section, an initial length scale estimate at energy λ. Let L ∈ 3Z \ 6Z and 3 = 3 L (0) = [−L/2, L/2]. Let H3 (ω) be the restriction of Hω to 3 with Dirichlet boundary conditions at −L/2 and L/2. For λ 6∈ σ (H3 (ω)), let R3 (λ) = (H3 (ω) − λ)−1 . Define the characteristic functions χ int = χ3 L/3 (0) = χ[−L/6,L/6] and χ out = χ3 L (0)\3 L−2 (0) = χ[−L/2,−L/2+1]∪[L/2−1,L/2] . For γ , λ ∈ R and ω ∈ , let us call the cube 3 (γ , λ)-good for ω if λ 6∈ σ (H3 (ω)) and kχ out R3 (λ)χ int k ≤ e−γ L/3 . We have the following theorem. THEOREM 6.5 For every ε > 0, there exist δ > 0 and L 0 ∈ N such that for L ≥ L 0 with L ∈ 3Z\6Z, we have P 3 is (γ (λ) − ε, λ)-good for ω ≥ 1 − e−δL . (6.8)

Proof Let u ± be the solutions of Hω u = λu with Dirichlet boundary conditions at ±L/2; that is, let u + (L/2) = u − (−L/2) = 0, u 0+ (L/2) = u 0− (−L/2) = 1. Then the Green function G 3 (λ, x, y) (i.e., the kernel of R3 (λ)) is given by ( u + (x)u − (y) for x ≥ y, 1 G 3 (λ, x, y) = (6.9) W (u + , u − ) u − (x)u + (y) for x < y, where the Wronskian W (u + , u − )(x) = u + (x)u 0− (x) − u 0+ (x)u − (x)

90


is constant in x. Setting x = L/2, we get

t W (u + , u − ) = u − (L/2), u 0− (L/2) , (−1, 0)t

= Aλ (L/2, −L/2)(0, 1)t , (−1, 0)t , where Aλ (x, y) denotes the transfer matrix from y to x. By stationarity we can use Corollary 6.4 to conclude that P |W (u + , u − )| ≥ e(γ (λ)−ε)L ≥ 1 − e−δL for L ≥ L 0 . (6.10) Note that λ ∈ σ (H3 (ω)) if and only if W (u + , u − ) = 0. Thus the event in (6.10) implies that λ 6∈ σ (H3 (ω)). Let x ∈ [L/2 − 1, L/2] and y ∈ [−L/6, L/6]. Then |u + (x)| ≤ C uniformly in ω and L

(6.11)

by Lemma A.1. Also,

|u − (y)| ≤ (u − (y), u 0− (y))t = Aλ (y, −L/2)(0, 1)t

≤ Aλ (y, by + 1/2c − 1/2) · Aλ (by + 1/2c − 1/2, −L/2)(0, 1)t . Again by Lemma A.1,

Aλ (y, by + 1/2c − 1/2) ≤ C,

(6.12)

and by stationarity and (6.1) (note that −L/6 ≤ by + 1/2c − 1/2 ≤ L/6), P kAλ (by + 1/2c − 1/2, −L/2)(0, 1)t k ≤ e(γ (λ)+ε)2L/3

≥ 1 − e−αL/3

if L/3 ≥ L 0 . (6.13)

Combining (6.10) – (6.13) and using G 3 (λ, x, y) = u − (y)u + (x)/W (u + , u − ), we get P |G λ (λ, x, y)| ≤ Ce−(γ (λ)−5ε)L/3 ≥ 1 − e−δL − e−αL/3 for L sufficiently large. In a completely analogous way, the same estimate is found if x ∈ [−L/2, −L/2 + 1], y ∈ [−L/6, L/6]. From this it can now be seen easily that for every ε > 0 there exist δ > 0 and L 0 ∈ N such that for L ≥ L 0 and L ∈ 3Z \ 6Z, we have Z sup χ out (x)G 3 (λ, x, y)χ int (y) dy ≤ e−(γ (λ)−ε)L/3 x

and

Z sup y

out χ (x)G 3 (λ, x, y)χ int (y) d x ≤ e−(γ (λ)−ε)L/3

with probability at least 1 − e−δL . The theorem now follows by Schur’s test.


91

Theorem 6.5 can be regarded as a fixed-energy initial length scale estimate. Our goal is to start a variable-energy multiscale induction that ultimately yields both Theorem 1.1 and Theorem 1.2. We therefore need an estimate of the form (6.8), where the energy is not fixed but rather varying over an interval. A result of this kind is established, using the Wegner estimate and an argument from [18], in the following corollary. COROLLARY 6.6 For every λ ∈ I and every β ∈ (0, 1), σ, ε > 0, let α > 0, δ > 0, and L 0 ∈ N be as in Theorems 5.1 and 6.5, respectively. For every 0 < ε < ε 0 and every L ≥ L 0 , let

κL =

1 −2σ L β −(γ (λ)−ε0 )L/3 e e − e−(γ (λ)−ε)L/3 . 2

Then we have for every L ≥ L 0 , β P ∀λ0 ∈ (λ − κ L , λ + κ L ) : 3 is (γ (λ) − ε0 , λ0 )-good ≥ 1 − e−δL − e−αL . (6.14) Proof β With probability 1 − e−δL − e−αL we have that both the event in (6.8) and the complementary event in (5.1) hold. Thus by assumption we have that for every λ0 ∈ (λ − κ L , λ + κ L ), we have λ0 6 ∈ σ (H3 (ω)) and, moreover, by the resolvent equation,

out

χ R3 (λ0 )χ int = χ out R3 (λ) + (λ − λ0 )R3 (λ0 )R3 (λ) χ int

≤ χ out R3 (λ)χ int + |λ − λ0 | · χ out R3 (λ0 )R3 (λ)χ int

≤ χ out R3 (λ)χ int + |λ − λ0 | · R3 (λ0 ) · R3 (λ) ≤ e−(γ (λ)−ε)L/3 + 2 · |λ − λ0 | · e2σ L

β

≤ e−(γ (λ)−ε )L/3 . 0

Thus for these ω’s the cube 3 is (γ (λ) − ε0 , λ0 )-good. 7. Proof of the main theorems In the preceding two sections we have established the two ingredients, namely, a Wegner estimate and an initial length scale estimate, that are necessary to start the multiscale induction that, by known results, implies both Theorem 1.1 and Theorem 1.2. In this section we briefly show how to reduce these two theorems to known results, given Theorem 5.1 and Corollary 6.6. Let M be the discrete set found in Theorem 2.3. Proof of Theorem 1.1 Fix an arbitrary compact interval I ⊂ R \ M. It follows from Theorems 5.1 and 6.5

92


that we have both a Wegner estimate and a fixed-energy initial length scale estimate for every λ ∈ I . Corollary 6.6 shows that these two results imply a variable-energy initial length scale estimate for a ball B(λ) of explicit radius around λ. The variableenergy multiscale analysis as presented, for example, in [48] then establishes variableenergy resolvent decay estimates on a sequence (L k )k∈N of length scales for energies in B(λ). These estimates, together with the existence of polynomially bounded eigenfunctions for spectrally almost every energy, yield pure point spectrum in B(λ) with exponentially decaying eigenfunctions for almost every ω ∈ (see, e.g., [48] for details). Thus we have exponential localization in R \ M for almost every ω ∈ . Finally, since by general principles (see [12]) the set M carries almost surely no spectral measure, we have exponential localization in R for almost every ω ∈ and hence Theorem 1.1. Proof of Theorem 1.2 It essentially follows from [15] that the variable-energy resolvent decay estimates, as given by the output of the the variable-energy multiscale analysis, imply strong dynamical localization in the sense of Theorem 1.2. For the curious reader we briefly sketch the argument, referring to [15] for necessary notation. Given a compact interval I ⊂ R \ M, a compact set K ⊂ R, and p > 0, we first let γ = min{γ (λ) : λ ∈ I } > 0. Next, we choose L 1 large enough so that, for every λ ∈ I , Theorem 5.1 and Corollary 6.6 imply both W (I, L , 2, q), L ≥ L 1 , and G(B(λ), L 1 , γ − ε0 , ξ ) of [15] with parameters sufficient to cover the desired p. Having this length scale fixed, we decompose the interval I into a finite disjoint union of intervals I1 , . . . , Im , each of them having length bounded by κ L 1 . We split the projection PI (Hω ) in (1.4) into the Pm finite sum i=1 PIi (Hω ) and treat each term separately. For every i, we can apply [15, Theorem 3.1], with initial length scale L 1 , since all the other conditions (e.g., the assumptions regarding independence (INDY), a geometric resolvent inequality (GRI), a Weyl-type estimate (WEYL), an eigenfunction decay inequality (EDI), as well as the assumption in [15, Theorem 3.1(i)]) are known to hold for the concrete operators Hω under consideration (see [48]). This allows us to establish (1.4), with I replaced by Ii . By summing over i, we get Theorem 1.2.

Appendices A. A priori solution estimates Here we provide several a priori estimates for solutions of the Schrödinger equation which are used repeatedly in the main text. For convenience we include proofs of


93

these standard facts. A.1 Let q ∈ L 1loc (R), let u be a solution of −u 00 + qu = 0, and let x, y ∈ R. Then n Z max(x,y) o |u(x)|2 + |u 0 (x)|2 ≤ |u(y)|2 + |u 0 (y)|2 exp |q(t) + 1| dt . LEMMA

min(x,y)

Proof For R(t) := |u(t)|2 + |u 0 (t)|2 , one has |R 0 (t)| = 2(q(t) + 1) Re u 0 (t)u(t) ≤ |q(t) + 1|R(t); that is, |(ln R(t))0 | ≤ |q(t) + 1|, which implies the lemma. A.2 For i = 1, 2, let qi ∈ L 1loc (R), let u i be solutions of −u i00 + qi u i = 0 with u 1 (y) = u 2 (y), and let u 01 (y) = u 02 (y) for some y ∈ R. Then for any x ∈ R, LEMMA

1/2 |u 1 (x) − u 2 (x)|2 + |u 01 (x) − u 02 (x)|2 n Z max(x,y) 1/2 o |q1 (t)| + |q2 (t)| + 2 dt ≤ |u 1 (y)|2 + |u 01 (y)|2 exp min(x,y)

Z

max(x,y)

×

|q1 (t) − q2 (t)| dt.

min(x,y)

Proof Without restriction, let y ≤ x. The solutions u 1 and u 2 satisfy Z x u 1 (x) − u 2 (x) 0 = dt u 01 (x) − u 02 (x) (q1 (t) − q2 (t))u 1 (t) y Z x 0 1 u 1 (t) − u 2 (t) dt, + q2 (t) 0 u 01 (t) − u 02 (t) y and thus

Z

u 1 (x) − u 2 (x)

u 0 (x) − u 0 (x) ≤ 1

2

x

|q1 (t) − q2 (t)||u 1 (t)| dt

y x

Z + y

u 1 (t) − u 2 (t)

|q2 (t)| + 1

u 0 (t) − u 0 (t) dt. 1

(A.1)

2

Gronwall’s lemma (see, e.g., [51]) yields

Z x nZ x

u 1 (x) − u 2 (x) o

≤ |q (t) − q (t)||u (t)| dt exp |q (t)| + 1 dt . 1 2 1 2

u 0 (x) − u 0 (x) y y 1 2

94


By Lemma A.1 we have for all t ∈ [y, x], o n1 Z x 1/2 |u 1 (t)| ≤ exp |q1 (s) + 1| ds |u 1 (y)|2 + |u 01 (y)|2 . 2 y Inserting this into (A.1) yields the result. A.3 R x+1 Let q ∈ L 1loc,unif (R); that is, let kqk1,unif := supx x |q(t)| dt < ∞. Then there is a C > 0 depending only on kqk1,unif such that for all solutions u of −u 00 + qu = 0 and all x ∈ R, Z x+1 |u(t)|2 dt ≥ C |u(x)|2 + |u 0 (x)|2 . (A.2) LEMMA

x−1

Proof By Lemma A.1 there are constants 0 < C1 , C2 < ∞ depending only on kqk1,unif such that for all t ∈ [x − 1, x + 1], C1 |u(x)|2 + |u 0 (x)|2 ≤ |u(t)|2 + |u 0 (t)|2 ≤ C2 |u(x)|2 + |u 0 (x)|2 . With C3 := (C1 /2)1/2 and C4 := (2C2 )1/2 , we get C3 |u(x)| + |u 0 (x)| ≤ |u(t)| + |u 0 (t)| ≤ C4 |u(x)| + |u 0 (x)| . It now follows from elementary geometric considerations (see, e.g., [49, p. 218]) that [x − 1, x + 1] contains an interval of length min(2, C3 /4C4 ) on which |u| ≥ C3 (|u(x)| + |u 0 (x)|)/4. This yields (A.2). B. Cocycles and invariant measures In this appendix we collect some basic facts about cocycles and the existence and uniqueness of invariant measures for group actions, which are used in Section 3 to prove Hölder continuity of the Lyapunov exponent. All this can be found in [12, Chapter IV]. Let G be a metric group, with unit e, that is both locally compact and σ -compact. Let B be a metrizable topological space such that G acts on B; that is, to each (g, b) ∈ G × B one can continuously associate an element gb ∈ B for which (g1 g2 ) · b = g1 · (g2 · b) for g1 , g2 ∈ G and b ∈ B, and e · b = b for b ∈ B. A continuous map σ : G × B → (0, ∞) is called a cocycle if for all g1 , g2 ∈ G and b ∈ B, one has σ (g1 g2 , b) = σ (g1 , g2 b)σ (g2 , b). Note that if σ is a cocycle, then clearly σ t is also a cocycle for all t ∈ R. There is one particular example of primary importance.


95

Example B.1 Let B be a compact metric space, and let G be a group acting on B. Take B˜ := ˜ that is, g·(a, b) := B×B\{(b, b) : b ∈ B}, and consider the induced action of G on B; ˜ (ga, gb) for all g ∈ G and (a, b) ∈ B. If δ is the metric on B, then δ(ga, gb) σ1 g, (a, b) := δ(a, b) ˜ defines a cocycle on G × B. A crucial property of cocycles is that they satisfy certain integral estimates. For this reason the “pseudo-convolution” of a probability measure µ on G and a measure ν on B is introduced by Z (µ ∗ ν)( f ) := f (gb) dµ(g) dν(b) for all f ∈ B (B), the bounded measurable functions on B. Here if B = G, the above definition coincides with that of the ordinary convolution on G, and µn denotes the n-fold convolution µ ∗ µ ∗ · · · ∗ µ. We note that a cocycle is said to be µ-integrable if σ (g) := supb∈B σ (g, b) is µ-integrable. The concept of invariance is also important. A measure ν on B is said to be µinvariant if µ ∗ ν = ν. For B compact, the existence of an invariant measure is trivial. P In particular, any weak limit of the sequence (1/n) nj=1 µn ∗ m, where m is an arbitrary probability measure on B, is a µ-invariant probability measure. In addition, an operator T : B (B) → B (B) is defined by Z (T f )(b) := f (gb) dµ(g). Note that the operator T n is given by the formula above with µn replacing µ. The relationship between these operators and µ-invariant probability measures is illustrated in Lemma B.2. Lastly, if B is a compact metric space with metric δ, then for any real number α the space of α-Hölder continuous functions, Lα (B), is defined to be Lα (B) := f ∈ C(B) : m α ( f ) < ∞ , where m α ( f ) := sup

(a,b)∈ B˜

| f (a) − f (b)| . δ α (a, b)

Equipped with the norm k f kα = k f k∞ + m α ( f ), Lα (B) is a Banach space.

96


LEMMA B.2 ˜ Suppose that Consider the cocycle σ1 , as defined in Example B.1, on G × B. t (i) there exists a positive number τ such that σ1 is µ-integrable for |t| ≤ τ , and (ii) there exists an integer N such that Z sup log[σ1 (g, b)] dµ N (g) < 0. b∈ B˜

Then there exists a real number α0 such that for any α with 0 < α ≤ α0 there exist constants Cα < ∞ and ρα < 1 for which the following are true. (a) Z sup b∈B

(b)

σ1 (g, b)α dµn (g) ≤ Cα ραn

for n = 1, 2, . . . . T is a bounded operator on Lα satisfying kT n f − ν( f )kα ≤ k f kα Cα ραn

(c)

for n = 1, 2, . . . and f ∈ Lα , where ν is any µ-invariant probability measure. In particular, this proves uniqueness of the invariant measure. The operator T on Lα has eigenvalue 1, and the rest of the spectrum is contained in a disk of radius strictly less than 1. Moreover, T admits the following decomposition: T n f = ν( f ) + Q n f for f ∈ Lα , where ν is the invariant probability measure and Q is an operator on Lα of spectral radius strictly less than 1.

Proof See [12, Propositions IV.3.5 and IV.3.15 and Corollary IV.3.16]. Acknowledgments. We are grateful to B. Simon for pointing us to the simple proof of Lemma 2.2, that is, that there are no compactly supported solitons for scattering at periodic background. After this we learned a different proof of this fact from E. Korotyaev. We also acknowledge useful discussions with L. Pastur and R. Schrader, as well as helpful remarks by the referee. References [1]

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Damanik Department of Mathematics, California Institute of Technology, Pasadena, California 91125, USA; [email protected] Sims Department of Mathematics, University of California at Irvine, Irvine, California 92697-3875, USA; [email protected] Stolz Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35394-1170, USA; [email protected]


SINGULAR INTEGRALS ON SYMMETRIC SPACES OF REAL RANK ONE ALEXANDRU D. IONESCU

Abstract In this paper we prove a new variant of the Herz majorizing principle for operators defined by K-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove L p -boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hörmander-Michlin type. We also find sharp bounds on the L p -norm of large imaginary powers of the critical L p -Laplacian. 1. Introduction Classical singular integrals on Euclidean spaces are operators defined by Fourier multipliers m : Rn → C which satisfy the Hörmander-Michlin differential inequalities j

∂ξ m(ξ ) ≤ A j |ξ |−| j|

(1.1)

for all ξ ∈ Rn \ {0} and any index j = ( j1 , j2 , . . . , jn ). The operator Tm associated to the multiplier m is defined by Td f for any Schwartz function f , where m f = m · b b f denotes the Fourier transform of f . Alternatively, one can define singular integrals as convolution operators T f = f ∗ K , where the distributions K are appropriate generalizations of the classical Calderon-Zygmund kernels. The operator Tm extends to a bounded operator on L p (Rn ) for any p ∈ (1, ∞). The theory of singular integrals extends naturally to the setting of nilpotent Lie groups equipped with nonisotropic dilations (see [22, Chap. 13] and the reference given there). In this paper we introduce and study an analogue of the singular integral operators in the setting of noncompact symmetric spaces. Let G be a noncompact connected semisimple Lie group with finite center, let K be a maximal compact subgroup of G, and let X = G/K be an associated symmetric space. Our notation is standard and is recalled in Section 2. By Plancherel theorem, any bounded W -invariant function m on a∗ defines a bounded operator on L 2 (X) given by Tg f . Here e f denotes the m f = m· e DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 10 May 2000. Revision received 15 July 2001. 2000 Mathematics Subject Classification. Primary 43A85; Secondary 43A32. Author’s work supported by National Science Foundation grant number DMS 97-29992. 101

102

ALEXANDRU D. IONESCU

Fourier transform of the function f : X → C, and W is the Weyl group associated to the pair (g, a). The question of L p -boundedness of operators defined by multipliers is more delicate if p 6= 2. A necessary condition for boundedness on L p of the operator Tm is that the multiplier m extend to a bounded W -invariant holomorphic function in the interior of the tube T p = a∗ + ico(W · ρ p ). Here ρ p = |2/ p − 1|ρ, and co(W · ρ p ) denotes the interior of the convex hull in a∗ of the set of points {w · ρ p : w ∈ W }. This necessary condition was noticed by J.-L. Clerc and E. Stein [5], who also proved a sufficient condition when the group G is complex. By analogy with the Euclidean case, a natural theorem is the following: assume that p ∈ (1, 2) ∪ (2, ∞) and that the multiplier m extends to a holomorphic function in the interior of the tube T p . Assume, in addition, that m satisfies differential inequalities of the form −| j| j ∂ξ m(ξ ) ≤ A j 1 + |ξ | (1.2) for all ξ ∈ T p and any index j. Then the operator Tm extends to a bounded operator on L p (X) (see [5], [20], [3], [24], [1]). More refined statements (i.e., considering multipliers m with boundary values in certain Sobolev spaces or subject to minimal regularity assumptions) can be found in [1] and [11]. Notice, however, that there is an important difference between the differential inequalities (1.1) and (1.2). The multiplier in (1.1) is not assumed to be smooth at the origin, and this translates into a singularity of the corresponding kernel at infinity (in the sense that this kernel is not absolutely integrable at infinity and its large-scale cancellation plays an essential role). On the other hand, if a multiplier m on a symmetric space satisfies the differential inequalities (1.2), then the large-scale cancellation of the corresponding kernel is irrelevant. In this case, one uses the Herz majorizing principle in [13] or the Kunze-Stein phenomenon to control the L p -norm of the induced operator. One of our main theorems (Theorem 8) addresses this problem. We consider a certain class of multipliers m that have singularities at the points w · ρ p , w ∈ W , and we prove that the induced operators are still bounded on L p (X). (This possibility was noticed in [1, Sec. 5].) Our symbols satisfy the differential inequalities (4.1) (for groups G of real rank one), which are the natural analogues on symmetric spaces of the Hörmander-Michlin differential inequalities (1.1). As an application, we prove sharp estimates on the L p -norm of large imaginary powers of the critical Laplacian (Theorem 9). The only previously known estimates of this type were established in [9, Cor. 4.2] by a transference technique; we improve slightly on the polynomial power of the exponent and show that this improvement yields the sharp estimate. This paper is organized as follows. In Section 2 we recall some basic facts related to harmonic analysis on semisimple Lie groups and symmetric spaces. In Section 3 we prove a new transference principle (Theorem 1) and use it to establish a new variant of the Herz criterion for operators defined by certain K-bi-invariant kernels with large scale cancellation (Corollary 2). These kernels can be thought of as analogues of the

SINGULAR INTEGRALS ON SYMMETRIC SPACES

103

Calderon-Zygmund kernels on symmetric spaces. In Sections 4 and 5 we state and prove the two theorems described in the previous paragraph (Theorems 8 and 9). For simplicity we assume in this paper that the group G has real rank one. Suitable generalizations of the theorems to symmetric spaces of arbitrary real rank will be discussed in a future work. 2. Preliminaries Most of our notation related to semisimple Lie groups and symmetric spaces is standard and can be found, for example, in [12]. Let G be a noncompact connected semisimple Lie group with finite center, let g be the Lie algebra of G, let θ be a Cartan involution of g, and let g = k ⊕ p be the associated Cartan decomposition. Let K = exp k be a maximal compact subgroup of G, and let X = G/K be an associated symmetric space. Let a be a maximal abelian subspace of p; recall that we assumed that the group G has real rank one, that is, that dim a = 1. Let M be the centralizer of exp a in K, let 6 be the set of nonzero roots of the pair (g, a), and let W be the associated Weyl group. Since G has real rank one, it is well known that 6 is of the form 6 = {−α, α} or of the form 6 = {−2α, −α, α, 2α}. Let a+ = {H ∈ a : α(H ) > 0} be a positive Weyl chamber, and let 6 + be the corresponding set of positive roots. (In our case, 6 + = {α} or 6 + = {α, 2α}.) For any root β ∈ 6, let gβ be the root space P associated to β, let n = β∈6 + gβ , and let n = θ(n). Finally, let N = exp n, and let N = exp n. The group G has an Iwasawa decomposition G = K(exp a)N and a Cartan decomposition G = K(exp a+ )K. For each g ∈ G, denote by H (g) ∈ a and g + ∈ a+ the middle components of g in these decompositions. We also use the Iwasawa decomposition G = N(exp a)K. Let H0 be the unique element of a with the property that α(H0 ) = 1, and normalize the Killing form on g such that |H0 | = B(H0 , H0 )1/2 = 1. To simplify the notation, we often identify the Lie subgroup exp a with the real line R using the map r → exp(r H0 ). Notice that R+ is identified with exp a+ . Let ρ = (1/2)(m 1 · α + m 2 · 2α) and |ρ| = (1/2)(m 1 + 2m 2 ), where m 1 and m 2 are the dimensions of the root spaces g−α and g−2α , respectively. Let dg, dk, and dn be Haar measures on the groups G, R K, and N. Normalize dk such that K 1 dk = 1, and let dz be the translation invariant measure on the symmetric space X induced by the measure dg on the group G. After suitable normalizations of the Haar measures dg and dn, one has the following integral formulae: Z Z Z Z F(g) dg = c1 F k1 exp(t H0 )k2 (sinh t)m 1 (sinh 2t)m 2 dt dk1 dk2 G

K K R+

(2.1)

104


and

Z

Z Z Z F(g) dg = G

F n exp(t H0 )k e2|ρ|t dk dt dn,

N R K

for any continuous compactly supported function F : G → C. 3. A transference theorem Let φ : R → C be a function supported in R+ , and let A(r ) =

sup |φ(r ) − φ(r 0 )|.

(3.1)

|r 0 −r |≤1

Assume that φ satisfies the following basic assumptions: lim φ(r ) = 0

(3.2)

A(r ) = A < ∞.

(3.3)

r →∞

and

Z R

One should think of φ as a Calderon-Zygmund kernel on R with the singularity at the origin removed. Let p be a fixed exponent in the interval (1, 2), and let K p,φ : G → C be the K-bi-invariant kernel given by K p,φ k1 exp(r H0 )k2 = e−2|ρ|r/ p φ(r ) (3.4) for any r ≥ 0 and k1 , k2 ∈ K. Let |||∗ K p,φ ||| L p (X) denote the norm of the convolution operator defined by the kernel K p,φ on L p (X), and let ||| ∗ φ||| L p (R) denote the norm of the convolution operator defined by the kernel φ on L p (R). Our first main theorem is the following. 1 There is a constant C p such that

THEOREM

(i)

||| ∗ K p,φ ||| L p (X) ≤ C p A + ||| ∗ φ||| L p (R) . (ii)

Conversely, one has ||| ∗ φ||| L p (R) ≤ C p A + ||| ∗ K p,φ ||| L p (X) .

Remark. The point of this theorem is to be able to conclude that the two operator norms ||| ∗ K p,φ ||| L p (X) and ||| ∗ φ||| L p (R) are essentially proportional. This theorem is sharper than the classical transference principle of R. Coifman and G. Weiss [6, Th. 8.7] because the factor that makes the transition between the kernels φ and K p,φ is e2ρ(H )/ p . Notice that the transition factor in [6, Th. 8.7] is 1(H ), which is proportional to e2ρ(H ) for ρ(H ) ≥ 1.


105

Throughout this section the constants C p depend only on p and the group G. The letters c and C are used to denote universal constants depending only on the group G. As an application, we have a new variant of the Herz majorizing principle for operators defined by K-bi-invariant kernels. Assume that the function φ : R → C is supported in R+ and satisfies the differential inequalities ( |φ(r )| ≤ A0 (1 + r )−1 for any r ≥ 0, (3.5) |φ 0 (r )| ≤ A0 (1 + r )−2 for any r ≥ 0 and the cancellation condition Z

N 0

φ(r ) dr ≤ A0

(3.6)

for any N > 0. In this case, the function φ is a kernel of the Calderon-Zygmund type on the real line, and one has the following consequence of Theorem 1. COROLLARY 2 If p ∈ (1, 2) and if φ satisfies (3.5) and (3.6), then

||| ∗ K p,φ ||| L p (X) ≤ C p · A0 . This corollary should be compared with the following important criterion known as the Herz majorizing principle (see [13]). Assume that 1 ≤ p < 2 and that p 0 = p/( p − 1) is the conjugate exponent of p. Let K be a K-bi-invariant kernel on G. Then Z K exp(r H0 ) δ(r )e−2|ρ|r/ p0 dr, (3.7) ||| ∗ K ||| L p (G) = ||| ∗ K ||| L p (X) ≤ C p R+

where δ(r ) = (sinh r )m 1 (sinh 2r )m 2 is the factor that appears in the integral formula (2.1). For comparison, the L 1 -norm of the kernel K is Z K exp(r H0 ) δ(r ) dr. ||K || L 1 (G) = c1 R+

The inequality (3.7) is the best possible if the kernel K is positive. On the other hand, Corollary 2 cannot be obtained as a consequence of this inequality since the large-scale cancellation of the kernel φ plays a crucial role. We also remark that both Theorem 1 and Corollary 2 are false if p = 2. The rest of this section is devoted to proving Theorem 1. Proof of Theorem 1 Recall that for any locally integrable K-bi-invariant function K and any smooth compactly supported function f : X → C, the convolution f ∗ K is defined by the

106


formula f ∗ K (z) =

Z

f (h · 0)K (h −1 · z) dh, G

where 0 = G/K is the origin of the symmetric space X. Notice that Z Z f (h · 0)K p,φ (h −1 h 0 )g(h 0 · 0) dh dh 0 , ||| ∗ K p,φ ||| L p (X) = sup || f || p =||g|| p0 =1

G G

where the supremum is taken over all smooth compactly supported functions f, g : X → C. As usual, p 0 = p/( p − 1) is the conjugate exponent of p. We identify the group G with N × R × K using the Iwasawa decomposition G = N(exp a)K and the identification of exp a with R described in Section 2. This identifies the symmetric space X with N × R. The relevant measure on N × R corresponding to this identification is dµ = e2|ρ|s dn ds. To prove part (i) of the theorem, it suffices to prove that for any smooth compactly supported functions f, g : N × R → C one has Z Z Z Z f (m, s) N N R R

· K p,φ δ−s (m −1 n) exp((t − s)H0 ) g(n, t)e2|ρ|(s+t) ds dt dm dn ≤ C p A + ||| ∗ φ||| L p (R) || f || L p (N×R,dµ) ||g|| L p0 (N×R,dµ) . (3.8) By definition, δr (v) = (exp(r H0 ))v(exp(−r H0 )) for any r ∈ R and v ∈ N. It is well known that δr is a dilation of the group N. Denote by I p,φ ( f, g) the integral on the left-hand side of (3.8); in order to estimate |I p,φ ( f, g)|, we need to understand the connection between the Iwasawa decomposition and the Cartan decomposition of the group G. This idea was used by J.-O. Strömberg in [23]. 3 If v ∈ N and r ≥ 0, then + v exp(r H0 ) = r H0 + H (v) + E(v, r )H0 , LEMMA

(3.9)

where 0 ≤ E(v, r ) ≤ 2e−2r .

(3.10)

Proof Recall that g + and H (g) denote the a-components of the element g ∈ G in the Cartan decomposition and the Iwasawa decomposition of the group G. It follows from Kostant’s convexity theorem that ρ [v exp(r H0 )]+ − r H0 − H (v) ≥ 0.


107

Thus E(v, r ) ≥ 0. To prove the upper estimate in (3.10), one can use the explicit formulae in [12, Chap. II, Th. 6.1]: i2 h c0 cosh2 α((v exp(r H0 ))+ ) = cosh r + er |X |2 + c0 e2r |Y |2 2 and

2 e2α(H (v)) = 1 + c0 |X |2 + 4c0 |Y |2 ,

where X and Y are the coordinates of v in N corresponding to the root spaces g−α and g−2α , and c0 is a constant. The upper estimate in (3.10) follows from these two equalities and the observation that e2[r +α(H (v))+E(v,r )] /4 ≤ cosh2 α((v exp(r H0 ))+ ) . We use this lemma to estimate the function (v, r ) → K p,φ (v exp(r H0 )) for any v ∈ N and r ∈ R+ . Let P(v) = e−ρ(H (v)) ; it is well known that for any ε > 0, Z P(v)(1+ε) dv = Cε < ∞. (3.11) N

For any r, x ≥ 0, let Z Dφ(r, x) =

r +x

A(y) dy, r

where A is the function defined in (3.1). 4 If r ≥ 0 and v ∈ N, then K p,φ v(exp(r H0 )) ≤ C Ae−2|ρ|r/ p P(v)2/ p .

PROPOSITION

(i)

(ii)

(3.12)

If r ≥ 0 and v ∈ N, then K p,φ v(exp(r H0 )) = e−2|ρ|r/ p φ(r )P(v)2/ p + E p,φ (v, r ),

(3.13)

where |E p,φ (v, r )| ≤ Ce−2|ρ|r/ p Ae−2r + Dφ r, 2 + α(H (v)) P(v)2/ p . (3.14) Proof Part (i) follows immediately from definition (3.4) of the kernel K p,φ , Lemma 3, and the observation that |φ(r 0 )| ≤ 2A for any r 0 ≥ 0. To prove part (ii) of the proposition, notice that it follows from Lemma 3 that the difference between K p,φ v(exp(r H0 )) and e−2|ρ|r/ p φ(r )P(v)2/ p is equal to e−2|ρ|r/ p P(v)2/ p e−2|ρ|E(v,r )/ p φ r + α(H (v)) + E(v, r ) − φ(r ) .

108


Estimate (3.14) on this error term follows from estimate (3.10) and the definition of the function Dφ. Let χ be the characteristic function of the interval [0, ∞). Recall that φ is supported in the interval [0, ∞), and decompose the kernel K p,φ δ−s (m −1 n) exp((t − s)H0 ) in (3.8) into three parts: 1 − χ(t − s) K p,φ δ−s (m −1 n) exp((t − s)H0 ) + χ(t − s)E p,φ δ−s (m −1 n), (t − s) 2/ p −2|ρ|(t−s)/ p + φ(t − s)P δ−s (m −1 n) e . This induces a decomposition of the integral I p,φ ( f, g) on the left-hand side of (3.8) 1 ( f, g) + I 2 ( f, g) + I 3 ( f, g), where as I p,φ ( f, g) = I p,φ p,φ p,φ Z Z Z Z 1 ( f, g) = f (m, s)K p,φ δ−s (m −1 n) exp((t − s)H0 ) g(n, t) I p,φ N N R R

· e2|ρ|(s+t) 1 − χ(t − s) ds dt dm dn, Z Z Z Z 2 f (m, s)E p,φ δ−s (m −1 n), (t − s) g(n, t) I p,φ ( f, g) = N N R R

· e2|ρ|(s+t) χ(t − s) ds dt dm dn, and 3 I p,φ ( f, g)

Z Z Z Z =

2/ p −2|ρ|(t−s)/ p f (m, s)P δ−s (m −1 n) e φ(t − s)g(n, t)

N N R R 2|ρ|(s+t)

·e LEMMA

ds dt dm dn.

(3.15)

5

One has 1 |I p,φ ( f, g)| ≤ C p · A|| f || L p (N×R,dµ) ||g|| L p0 (N×R,dµ) .

Proof Let

(

F1 (s) = G 1 (t) =

1/ p p , N | f (m, s)| dm R 1/ p0 0 p . N |g(n, t)| dn 1 ( f, g) is dominated value of I p,φ

R

By Hölder inequality, the absolute by Z Z F1 (s)G 1 (t)e2|ρ|(s+t) 1 − χ(t − s) R R Z K p,φ δ−s (v) exp((t − s)H0 ) dv ds dt. · N

(3.16)

(3.17)


109

Recall that the map n 1 → n 2 = δr (n 1 ) is a dilation of N with dn 2 = e−2|ρ|r dn 1 . In addition, the Abel transform Z A K (H ) = eρ(H ) K v(exp H ) dv N

takes K-bi-invariant functions K to W -invariant functions on a (see [12, p. 396]). Thus if t ≤ s, one has Z K p,φ δ−s (v) exp((t − s)H0 ) dv N Z −2|ρ|s K p,φ v exp((t − s)H0 ) dv =e ZN −2|ρ|t K p,φ v exp((s − t)H0 ) dv ≤ C p · Ae−2|ρ|t e−2|ρ|(s−t)/ p . =e N

The last estimate is a consequence of (3.11) and (3.12). Therefore the absolute value of the expression in (3.17) is dominated by Z Z Cp · A F1 (s)G 1 (t)e2|ρ|s e−2|ρ|(s−t)/ p 1 − χ (t − s) ds dt, R R

which is equal to Z ∞Z Cp · A 0

0 0 F1 (s)e2|ρ|s/ p G 1 (s − r )e2|ρ|(s−r )/ p ds e2|ρ|r (1/ p −1/ p) dr. R

By Hölder inequality, the inner integral is dominated by Z 1/ p Z 1/ p0 0 |F1 (s)| p e2|ρ|s ds |G 1 (t)| p e2|ρ|t dt . R

R

Since p < 2 < p 0 , the estimate (3.16) follows. LEMMA

6

One has 2 |I p,φ ( f, g)| ≤ C p · A|| f || L p (N×R,dµ) ||g|| L p0 (N×R,dµ) .

(3.18)

Proof 2 ( f, g) is dominated by By Hölder inequality, the absolute value of I p,φ Z Z Z E p,φ δ−s (v), (t − s) dv ds dt. (3.19) F1 (s)G 1 (t)e2|ρ|(s+t) χ(t − s) R R

N

It follows from (3.11) and (3.14) that Z E p δ−s (v),(t − s) dv N

∞

Z h ≤ C p e−2|ρ|s e−2|ρ|(t−s)/ p Ae−2(t−s) + 0

i A(t − s + x)k p (x) d x ,

110


where k p (x) =

Z

P(v)2/ p χ 2 + α(H (v) − x) dv. N

We substitute this last estimate into (3.19) and make the change of variable t = s + r . 2 ( f, g) is dominated by It follows that the absolute value of I p,φ Z ∞ Z 0 Cp F1 (s)e2|ρ|s/ p G 1 (s + r )e2|ρ|(s+r )/ p ds 0 R Z ∞ h i · Ae−2r + A(r + x)k p (x) d x dr. 0

R∞ Since p < 2, it follows from (3.11) that 0 k p (x) d x = C p < ∞. Estimate (3.18) follows from Hölder inequality and the estimate above. LEMMA

7

One has 3 |I p,φ ( f, g)| ≤ C p ||| ∗ φ||| L p (R) || f || L p (N×R,dµ) ||g|| L p0 (N×R,dµ) .

(3.20)

Proof 3 ( f, g) is equal to The change of variable t = s + r shows that the integral I p,φ Z Z Z 2/ p 2|ρ|s 2|ρ|s/ p f (m, s)G 2 (n, s)P δ−s (m −1 n) e e ds dm dn, (3.21) N N R

where G 2 (n, s) =

Z

g(n, s + r )e2|ρ|(s+r )/ p φ(r ) dr. 0

R

One has Z R

p0

0

|G 2 (n, s)| p ds ≤ ||| ∗ φ||| L p (R)

Let G 3 (s) =

Z

Z

0

|g(n, t)| p e2|ρ|t dt.

(3.22)

R 0

|G 2 (n, s)| p dn

1/ p0

.

N

It follows by Hölder inequality and (3.11) that the absolute value of the integral (3.21) is dominated by Z F1 (s)G 3 (s)e2|ρ|s/ p ds,

R

which, again by Hölder inequality, is dominated by Z 1/ p0 Z 1/ p 0 G 3 (s) p ds F1 (s) p e2|ρ|s ds . R

Estimate (3.20) follows from (3.22).

R


111

Part (i) of the theorem now follows from (3.16), (3.18), and (3.20). For the converse, notice that it suffices to show that for any smooth compactly supported functions a, b : R → C, one has Z Z a(s)φ(t − s)b(t) ds dt ≤ C p A + ||| ∗ K p,φ ||| L p (X) ||a|| L p (R) ||b|| L p0 (R) . R R

(3.23) Let Jφ (a, b) denote the left-hand side of (3.23). We construct two functions f a , gb : 3 ( f , g ). Assume that the functions a and b N × R → C such that Jφ (a, b) = I p,φ a b are supported inside the interval [−N , N ]. Let B N be a large ball in N, for example, B N = {n(X, Y ) : |X | ≤ e N and |Y | ≤ e2N }, where X ∈ g−α , Y ∈ g−2α , and n(X, Y ) = exp(X +Y ) (see [12, Chap. II, Sec. 6]). It is well known that δs (n(X, Y )) = n(e−s X, e−2s Y ). Let ψ N : N → {0, 1} be the characteristic function of the ball B N , R and notice that N ψ N (v) dv = Ce2|ρ|N . For any s ∈ [−N , N ], let Z Z q(s) = e−2|ρ|N ψ N (m)P(v)2/ p ψ N m · δs (v) dm dv N N Z Z 2/ p −2|ρ|N 2|ρ|s ψ N (m)P δ−s (m −1 n) ψ N (n) dm dn. (3.24) =e e N N

Notice that c ≤ q(s) ≤ C p for all s ∈ [−N , N ]. Let f a (m, s) = e−2|ρ|N / p ψ N (m)a(s)e−2|ρ|s/ p q(s)−1 and

gb (n, t) = e−2|ρ|N / p ψ N (n)b(t)e−2|ρ|t/ p . 0

0

It follows from (3.15) and (3.24) that 3 Jφ (a, b) = I p,φ ( f a , gb );

therefore 1 2 |Jφ (a, b)| ≤ |I p,φ ( f a , gb )| + |I p,φ ( f a , gb )| + |I p,φ ( f a , gb )|.

(3.25)

Finally, notice that ( || f a || L p (N×R,dµ) ≈ ||a|| L p (R) , ||gb || L p0 (N×R,dµ) ≈ ||b|| L p0 (R) , and (3.23) follows from (3.16), (3.18), and (3.25). 4. L p -Fourier multipliers The Fourier transform on the symmetric space X associates to any smooth compactly supported function f on X a function f˜ : a∗C × K/M → C, where a∗C is the complex

112


dual of a. By definition, one has e f (λ, b) =

Z

f (z)e(−iλ+ρ)A(z,b) d x, X

where A(z, b) is an a-valued analogue of the usual scalar product on Euclidean spaces (see [12, Chap. III]). For any g ∈ G and k ∈ K, one has, by definition, A(gK, kM) = −H (g −1 k). The Fourier transform extends to an isometry of L 2 (X) onto L 2 (a∗+ × K/M, |c(λ)|−2 dλ db), where, if G has real rank one, a∗+ = {c · α : c ∈ R+ } and c is the Harish-Chandra function (Plancherel theorem). Let a∗ be the real dual of a. By Plancherel theorem, any bounded even multiplier m : a∗ → C defines a bounded operator Tm on L 2 (X) given by Tg m f (λ, b) = m(λ) f˜(λ, b). Assume that p ∈ (1, 2) ∪ (2, ∞) is a fixed exponent, and let ρ p = |2/ p−1|ρ. For any λ ∈ a∗C , let, by definition, |λ| = |λ(H0 )|, and let T p = a∗ + i(−ρ p , ρ p ) = {λ ∈ a∗C : |=(λ(H0 ))| < |ρ p |}. Assume that the multiplier m : a∗ → C extends to an even holomorphic function in the interior of the tube T p and satisfies the differential inequalities ∂j (4.1) j m(λ) ≤ A j |λ + iρ p |− j + |λ − iρ p |− j ∂λ for any j = 0, 1, . . . and λ ∈ T p . 8 If p ∈ (1, 2) ∪ (2, ∞) and if m satisfies the differential inequalities (4.1), then the operator Tm extends to a bounded operator on L p (X). THEOREM

An operator Tm defined by a multiplier m that satisfies (4.1) can be thought of as a singular integral operator on the symmetric space X. In this section the constants may depend on finitely many of the constants A j in (4.1). Proof One can clearly assume that p ∈ (1, 2). In order to ensure the convergence of the integrals throughout this section, we also assume that the multiplier m(λ) is premulti2 plied with a factor of the form e−δλ(H0 ) , where 0 < δ ≤ 1. Our estimates are uniform in δ; once one proves suitable uniform estimates, standard limiting arguments allow one to pass to the general theorem. The multiplier m is holomorphic inside the tube T p ; therefore we can assume that for any ξ ∈ (−ρ p , ρ p ) the function λ → m(λ + iξ ) is a Schwartz function on a∗ . To simplify the notation, we identify the complex plane C with a∗C using the map λ → λ · α. Notice that this map also gives an identification of R with a∗ . By the Fourier inversion formula, the kernel K of the operator Tm is given by Z K k1 (exp(r H0 ))k2 = C m(λ)8λ exp(r H0 ) |c(λ)|−2 dλ R


113

for any k1 , k2 ∈ K and r ≥ 0. The functions 8λ are the elementary spherical functions on X, and c(λ) is the Harish-Chandra function. Let χ : [0, ∞) → [0, 1] be a smooth cutoff function supported in the interval [1, ∞) and equal to 1 on the interval [2, ∞). It is shown in both [20, Secs. 4, 5] and [1, Sec. 4] that the kernel (1 − χ (r ))K (k1 (exp(r H0 ))k2 ) (i.e., the local part of our kernel) defines a bounded operator on L p (X). It remains therefore to prove a similar statement for the kernel χ (r )K (k1 (exp(r H0 ))k2 ). It is well known (see, e.g., [14, App. A]) that the function λ → c(−λ)−1 is holomorphic inside the region =λ ≥ 0 and satisfies the estimates ∂k d−k k c(−λ)−1 ≤ Ck 1 + |λ| ∂λ

(4.2)

for all integers k = 0, 1, . . . and for all λ with the property that 0 ≤ =λ ≤ |ρ|. The number d in (4.2) is equal to (dim X − 1)/2 = (m 1 + m 2 )/2. It is also shown in [14, App. A] that if r ≥ 1/2, then the spherical functions 8λ can be written in the form 8λ exp(r H0 ) = e−|ρ|r eiλr c(λ)(1 + e−2r a(λ, r )) + e−iλr c(−λ)(1 + e−2r a(−λ, r )) . (4.3) The function a in (4.3) satisfies the inequalities ∂ j ∂l − j a(λ, r ) j l ≤ C j (1 + |λ|) ∂λ ∂r

(4.4)

for all integers j = 0, 1, . . . and l ∈ {0, 1}, and for all r ≥ 1/2 and λ with the property 0 ≤ =λ ≤ |ρ|. One also has |c(λ)|2 = c(λ)c(−λ) for any λ ∈ R. Since the symbol m is even, it follows from Corollary 2 and (4.3) that it suffices to prove that the function Z |ρ p |r φ0 (r ) = χ(r )e eiλr m(λ) 1 + e−2r a(λ, r ) c(−λ)−1 dλ R

satisfies inequalities (3.5) and (3.6). An easy argument based on (4.4) shows that the error term introduced by the factor a(λ, r ) in the definition above and its derivative are dominated by Ce−r . Therefore it remains to prove that the function Z |ρ p |r φ(r ) = χ(r )e eiλr m 1 (λ) dλ (4.5) R

satisfies inequalities (3.5) and (3.6), where m 1 (λ) = m(λ)c(−λ)−1 . It follows from (4.1) and (4.2) that the function m 1 is holomorphic in the interior of the region 0 ≤ =λ < |ρ p | and satisfies the inequalities ∂j − j d j m 1 (λ) ≤ C 0j λ − i|ρ p | 1 + |λ| ∂λ

(4.6)

114


for any j = 0, 1, . . . and λ in the region 0 ≤ =λ < |ρ p |. To prove (3.5), one moves the contour of integration in (4.5) to the line R + i|ρ p |(1 − r −1 ) (notice that there is nothing to prove if r ≤ 1), and the function φ in (4.5) becomes Z (4.7) φ(r ) = Cχ(r ) eiλr m 1 λ + i|ρ p |(1 − r −1 ) dλ. R

To prove the first inequality in (3.5), we integrate by parts N times and notice that if r ≥ 1, then Z d r −N 1 + |λ| |λ − i|ρ p |r −1 |−N dλ ≤ C/r R

if N > d + 1. The first inequality in (3.5) now follows from (4.6). The proof of the second inequality in (3.5) is similar, the only difference being that differentiation with respect to r brings down an extra factor of λ in the integral in (4.7) (or a factor of r −2 ) that weakens the singularity at λ = 0, and a similar integration by parts allows one to obtain the desired estimate. Thus it remains to prove that the function φ in (4.5) satisfies the cancellation condition (3.6). Let ψ be a smooth function supported in the interval [−1, 1] and equal to 1 in the interval [−1/2, 1/2]. Since the function φ already satisfies the first inequality in (3.5), it suffices to prove that for any ε ≤ 1 one has Z ∞ φ(r )ψ(εr ) dr ≤ C. 0

We move the integration in (4.5) to the line R + i|ρ p |(1 − ε), and we have to prove that Z Z eiλr χ (r )ψ(εr )e|ρ p |εr dr m 1 λ + i|ρ p |(1 − ε) dλ ≤ C (4.8) R

R

for any ε ∈ (0, 1]. Let ψ1 (r ) = ψ(r )e|ρ p |r , and notice that ∂j ψ (r ) j 1 ≤C ∂r

(4.9)

for any integer j ∈ [0, d + 2], where d = (dim X − 1)/2. Let F denote the Euclidean R Fourier transform on R given by F (a)(ξ ) = R a(x)e−i x·ξ d x for Schwartz functions a : R → C. Let L be the distribution on R with the property that F (L) = χ . Then the inner integral in (4.8) can be written as ξ − λ 1 L ξ → (F ψ1 ) . ε ε It follows from (4.9) that the function F ψ1 is sufficiently rapidly decreasing at infinity; thus (4.8) is established once we prove that for any x ∈ R, L ξ → m 1 (ξ + x + i|ρ p |(1 − ε)) ≤ C 1 + |x| d (4.10)


115

with a constant C independent of x and ε. Notice that F ((−iξ )L) = χ 0 . Let η be the inverse Fourier transform of the smooth compactly supported function χ 0 . Thus (−iξ )L = η(ξ ), and one has Z L(a) = R

2

η(ξ )(a(ξ ) − a(0)e−ξ ) dξ + c1 a(0) −iξ 2

for any Schwartz function a : R → C, where c1 = L(ξ → e−ξ ). Therefore the term on the left-hand side of (4.10) is equal to Z

2

η(ξ )[m 1 (ξ + x + i|ρ p |(1 − ε)) − m 1 (x + i|ρ p |(1 − ε))e−ξ ] dξ −iξ R + c1 m 1 x + i|ρ p |(1 − ε) .

One first moves the contour of integration in the integral above to the line −i|ρ p |(1 − ε) + R (this is possible since η is the inverse Fourier transform of a smooth compactly supported function; thus η is holomorphic in C), and (4.10) follows from (4.6) and the observation that the function ξ → η(ξ − i|ρ p |(1 − ε)) is rapidly decreasing at ∞. This completes the proof of the cancellation condition (3.6), and the theorem follows from Corollary 2. 5. Imaginary powers of Laplacians Natural examples of multipliers satisfying the differential inequalities (4.1) are provided by exponential powers of modified Laplacians on X. Let 1 be the LaplaceBeltrami operator on X. One has gf (λ, b) = − λ(H0 )2 + |ρ|2 e 1 f (λ, b) for any smooth compactly supported function f : X → C and any λ ∈ a∗C . For any p ∈ [1, ∞] and u ∈ R, let −iu m p,u (λ) = λ(H0 )2 + |ρ p |2 . Notice that m p,u is bounded and holomorphic in the interior of the tube T p and has singularities at the points λ = iρ p and λ = −iρ p (if p 6 = 2). Let L p,u be the operator defined by the Fourier multiplier m p,u . Notice that L p,u = [−1−(|ρ|2 −|ρ p |2 )I ]−iu . THEOREM 9 If p ∈ (1, 2) ∪ (2, ∞) and u ∈ R, then there exist constants c p and C p such that

c p eπ|u|/2 (1 + |u|)|1/ p−1/2| ≤ |||L p,u ||| L p (X)→L p (X) ≤ C p eπ |u|/2 (1 + |u|)|1/ p−1/2| . (5.1)

116


The standard notation |1/ p−1/2| |||L p,u ||| L p (X)→L p (X) ≈ p eπ|u|/2 1 + |u| is used in this section to denote the double inequality (5.1). L p → L q boundedness properties of operators of the form (z I − 1)s for suitable values of z and s have been studied extensively in various settings (see [2, Sec. 4] for a detailed discussion of the problem and appropriate references). The upper estimate in (5.1) is a slight improvement over the estimate in [9, Cor. 4.2]. It is not hard to see, however, that this improvement can also be obtained as a consequence of Lemma 10 at the end of this paper and the method used in [9]. (This was observed by Andreas Seeger.) On the other hand, the lower estimate in (5.1) is new and cannot be obtained by the argument in [5]. It is shown in [4] that if q ∈ (1, ∞) is such that |2/q − 1| < |2/ p − 1|, then the operator L p,u is bounded on L q (X) and |||L p,u ||| L q (X)→L q (X) ≈ p,q e|u| arcsin(|2/q−1|/|2/ p−1|) .

(5.2)

Notice that the expression on the right-hand side of (5.2) agrees with the best possible lower bound given by the Clerc-Stein condition |||L p,u ||| L q (X)→L q (X) ≥ sup |m p,u (λ)| ≈ e|u| arcsin(|2/q−1|/|2/ p−1|) . λ∈Tq

However, the L p -analysis of the operator L p,u is more delicate (mainly because size estimates on the kernel of the operator are not sufficient), and the L p -norm estimate in (5.1) does not agree with the lower bound predicted by the Clerc-Stein condition. Indeed, one can easily check that sup |m p,u (λ)| = eπ |u|/2 ,

λ∈T p

which is not proportional to the bound on the right-hand side of (5.1) if |u| is large. Proof of Theorem 9 Assume as in Section 4 that p ∈ (1, 2) and that the multiplier m p,u is premultiplied 2 with a factor of the form e−δλ(H0 ) in order to ensure the convergence of all the integrals throughout this section. Also, identify C with aC via the map λ → λ · α. The kernel K of the operator L p,u is given by the Fourier inversion formula Z K k1 (exp(r H0 ))k2 = C m p,u (λ)8λ exp(r H0 ) |c(λ)|−2 dλ. R

The basic idea of the proof is to apply Theorem 1 and control the L p -norm of the operator L p,u using the norm of a convolution operator on L p (R). Notice first that


117

one can assume that |u| is large, say, |u| ≥ C p for some suitable constant C p . (The theorem for |u| small follows from Th. 8 and the observation that, by the Clerc-Stein condition, |||L p,u ||| L p (X)→L p (X) ≥ 1 for any u ∈ R.) We divide the proof of the theorem into three steps. Step 1: Reduction of the problem. We assume from now on that u is positive. As in Section 4, let χ : [0, ∞) → [0, 1] be a smooth cutoff function supported in the interval [1, ∞) and equal to 1 in the interval [2, ∞). It follows from [20, Secs. 4 and 5] that the kernel (1 − χ (r ))K (k1 (exp(r H0 ))k2 ) (i.e., the local part of our kernel) defines a bounded operator on L p (X) with norm at most C p (1 + |u|) N , where N is a large fixed integer that depends only on the group G. It remains therefore to prove that the kernel χ (r )K k1 (exp(r H0 ))k2 defines a bounded operator on L p (X) with norm proportional to eπ |u|/2 (1 + |u|)|1/ p−1/2| . Let c2 be such that ec2 = sup (z 2 + |ρ p |2 )−i , where the supremum is taken over z ∈ T p \ [B(i|ρ p |, 1/2) ∪ B(−i|ρ p |, 1/2)]. Notice that c2 < π/2. Expand the function 8λ (exp(r H0 )) as in (4.3), and notice that the part of the kernel χ (r )K (k1 (exp(r H0 ))k2 ) corresponding to the error term e−2r a(λ, r ) induces a bounded operator on L p (X) with norm at most C p ec2 u (1 + |u|) N . Thus it remains to prove that if Z K 1 k1 (exp(r H0 ))k2 = χ(r )e−|ρ|r eiλr m p,u (λ)c(−λ)−1 dλ (5.3) R

for any k1 , k2 ∈ K and r > 0, then ||| ∗ K 1 ||| L p (X) ≈ p eπ |u|/2 1 + |u|

|1/ p−1/2|

.

We move the contour of integration in (5.3) to the line i|ρ p | + R and remove the large frequencies using a smooth cutoff function. One can check easily that the error term (corresponding to large frequencies) induces a bounded operator on L p (X) with norm at most C p ec2 u (1 + |u|) N . Thus it remains to prove that the kernel Z −iu K 2 k1 (exp(r H0 ))k2 = χ(r )e−2|ρ|r/ p eiλr λ2 + 2i|ρ p |λ η(λ) dλ (5.4) R

has the property that ||| ∗ K 2 ||| L p (X) ≈ p eπ |u|/2 (1 + |u|)|1/ p−1/2| .

(5.5)

The function η in (5.4) is smooth, supported in the interval [−1, 1], and η(0) = c(−i|ρ p |)−1 6= 0. The estimate (5.5) is a consequence of Theorem 1. Let Z −iu φ p,u (r ) = χ(r ) eiλr λ2 + 2i|ρ p |λ η(λ) dλ. (5.6) R

118


Recall that we assume that u is positive (i.e., u ≥ C p ), and let Y p : R → [0, π] be the function defined implicitly by tan(Y (λ)) = λ/(2|ρ p |). An elementary computation shows that Z 4 2 2 φ p,u (r ) = eπ u/2 χ(r ) ei[λr −u log(λ +4|ρ p | λ )/2] η(λ)e−uY p (λ) dλ. (5.7) R

We first prove a rough estimate on |φ 0p,u (r )|. Notice that Y (λ) > π/2 if λ < 0 and Y p (λ) ≥ cλ if λ ∈ [0, 1]. It follows from (5.7) and the assumption that u is positive that |φ 0p,u (r )| ≤ Ceπ |u|/2 /|u| if r ≤ 2 and |φ 0p,u (r )| ≤ Ceπ|u|/2 /|u|2 if 2 ≤ r ≤ 10|u|2 . In order to estimate |φ 0p,u (r )| when r ≥ 10|u|2 , notice that the derivative of the phase function in (5.7) is equal to r−

u 4|ρ p |2 + 2λ2 . λ 4|ρ p |2 + λ2

An integration-by-parts argument shows that |φ 0p,u (r )| ≤ Ceπ |u|/2 |u|2 /r 2 if r ≥ 10|u|2 . To summarize, one has  π|u|/2 /|u|  if r ≤ 2,  Ce 0 π|u|/2 2 |φ p,u (r )| ≤ Ce /|u| if 2 ≤ r ≤ 10|u|2 ,   Ceπ|u|/2 |u|2 /r 2 if r ≥ 10|u|2 . Therefore

∞

Z 0

|φ 0p,u (r )| dr ≤ Ceπ |u|/2 ,

and it follows from Theorem 1 that (5.5) and the theorem follow once we prove that |1/ p−1/2| ||| ∗ φ p,u ||| L p (R) ≈ p eπ |u|/2 1 + |u| . Notice also that the contribution of the integral over λ < 0 in formula (5.7) defining φ p,u is negligible (since the absolute value of the derivative of the phase function is greater than or equal to r and e−uY p (λ) ≤ e−π |u|/2 if λ < 0). Thus one can replace integrals (5.6) and (5.7) with the corresponding integrals on R+ . A similar argument shows that one can also remove the factor χ(r ) in (5.6) and (5.7) at the expense of a negligible error. Finally, it remains to prove that the operator S p,u defined by the Fourier multiplier q p,u : R → C, ( 0 if ξ ≤ 0, q p,u (ξ ) = −iu log(ξ 4 +4|ρ |2 ξ 2 )/2 −uY (ξ ) (5.8) p p e e η(ξ ) if ξ > 0, extends to a bounded operator on L p (R) and |||S p,u ||| L p (R)→L p (R) ≈ p 1 + |u|

|1/ p−1/2|

.

(5.9)


119

Step 2: The lower estimate in (5.9). For any ξ > 0, let β p,u (ξ ) = e−iu log(ξ then

2 /u 2 +4|ρ

2 p | )/2

e−uY p (ξ/u) η

ξ ; u

q p,u (ξ ) = e−iu log ξ β p,u (uξ ).

(5.10)

Notice that |β p,u (ξ )| ≥ c for all ξ ∈ [0, 4] (recall that u is a large positive number) and that there exists a number c3 > 0 such that ∂j (5.11) j β p,u (ξ ) ≤ C j e−c3 ξ ∂ξ for all integers j = 0, 1, . . . and real numbers ξ > 0. Let γ be a smooth cutoff function supported in the interval [1/2, 4] and equal to 1 in the interval [1, 2], and let au be the function whose Euclidean Fourier transform is the function ξ → γ (uξ ). Then Z ∞ |S p,u au (r )| = C eiξr q p,u (ξ )γ (uξ ) dξ Z0 C ∞ i[ξr/u−u log ξ ] = e β p,u (ξ )γ (ξ ) dξ u 0 and the method of stationary phase shows that |S p,u au (r )| ≥ cu −3/2 for any r ∈ [u 2 /2, u 2 ]. Therefore ||S p,u f u || L p (R) ≥ cu 2/ p−3/2 . On the other hand, ||au || L p (R) ≈ u 1/ p−1 , and the lower estimate in (5.9) follows. Step 3: The upper estimate in (5.9). This is based on the following lemma. LEMMA 10 If q ∈ (1, ∞) and u ∈ R, then the operator Su defined by the Fourier multiplier lu : R → C, ( 0 if ξ ≤ 0, lu (ξ ) = −iu log ξ e if ξ > 0,

extends to a bounded operator on L q (R) and |1/q−1/2| |||Su ||| L q (R)→L q (R) ≤ Cq 1 + |u| . Assuming this lemma for a moment, notice that the operator S p,u is the composition of the operator Su and the operator defined by the Fourier multiplier ξ → β p,u (uξ ). By (5.11), this second operator is bounded on L q (R) for any q ∈ [1, ∞] and its L q norm is dominated by an absolute constant. Thus the upper estimate in (5.9) follows from Lemma 10.

120


Proof of Lemma 10 Variants of this lemma are certainly known; however, for lack of a precise reference, we give a short proof of Lemma 10. Notice that we can assume that q ∈ (1, 2] and u ≥ 0. By Plancherel theorem, |||Su ||| L 2 (R)→L 2 (R) ≤ C.

(5.12)

Therefore it suffices to prove that the operator Su is bounded from the Hardy space H 1 (R) to L 1 (R) and that 1/2 |||Su ||| H 1 (R)→L 1 (R) ≤ C 1 + |u| . (5.13) It is well known that singular integrals are bounded from H 1 (R) to L 1 (R) (see [10]); thus we only have to prove (5.13) when u is large. The operator Su is translation and scale invariant. Therefore it suffices to prove that if a : R → C is a measurable function supported in the interval [−1, 1] with the properties |a(x)| ≤ 1/2 and Z a(x) d x = 0, R

then ||Su a|| L 1 (R) ≤ C 1 + |u|

1/2

.

In the terminology of [22, Chap. III], the function a is a standard atom on the real line. It follows from (5.12) that Z 2u 1/2 |Su a(x)| d x ≤ Cu 1/2 ||Su a|| L 2 (R) ≤ C 1 + |u| . −2u

The kernel k of the operator Su is a smooth function away from the origin of R, and the method of stationary phase shows that |k 0 (y)| ≤ Cu 3/2 |y|−2 for any y ∈ R \ {0}. Therefore if |x| ≥ 2u ≥ 2, then Z |Su a(x)| =

1 −1

Z a(y)k(x − y) dy =

1 −1

a(y) k(x − y) − k(x) dy

≤ Cu 3/2 |x|−2 , which shows that

Z |Su a(x)| d x ≤ C 1 + |u| |x|≥2u

Estimate (5.13) and the lemma follow.

1/2

.


121

Acknowledgments. I would like to thank Andreas Seeger for suggesting the problem during a short visit to Madison, Wisconsin. I would also like to thank Jean-Philippe Anker for providing a copy of the preprint [4] and Elias M. Stein for his time, interest, and helpful discussions on the subject. References [1]

[2] [3] [4] [5] [6] [7] [8]

[9]

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J.-P. ANKER, L p Fourier multipliers on Riemannian symmetric spaces of the

noncompact type, Ann. of Math. (2) 132 (1990), 597 – 628. MR 92e:43006 102, 113 , Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257 – 297. MR 93b:43007 116 J.-P. ANKER and N. LOHOUE´ , Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), 1303 – 1353. MR 88c:43008 102 J.-P. ANKER and A. SEEGER, Spectral multipliers of noncompact Riemannian manifolds, preprint, 1999. 116, 121 J.-L. CLERC and E. M. STEIN, L p -multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911 – 3912. MR 51:3803 102, 116 R. R. COIFMAN and G. WEISS, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1976. MR 58:2019 104 M. COWLING, The Kunze-Stein phenomenon, Ann. of Math. (2) 107 (1978), 209 – 234. MR 58:22398 , “Herz’s ‘principe de majoration’ and the Kunze-Stein phenomenon” in Harmonic Analysis and Number Theory (Montreal, 1996), CMS Conf. Proc. 21, Amer. Math. Soc., Providence, 1997, 73 – 88. MR 98k:22040 M. COWLING, S. GIULINI, and S. MEDA, L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces, I, Duke Math. J. 72 (1993), 109 – 150. MR 95b:22031 102, 116 C. FEFFERMAN and E. M. STEIN, H p spaces of several variables, Acta Math. 129 (1972), 137 – 193. MR 56:6263 120 S. GIULINI, G. MAUCERI, and S. MEDA, L p multipliers on noncompact symmetric spaces, J. Reine Angew. Math. 482 (1997), 151 – 175. MR 98g:43006 102 S. HELGASON, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr. 39, Amer. Math. Soc., Providence, 1994. MR 96h:43009 103, 107, 109, 111, 112 C. HERZ, Sur le phénomène de Kunze-Stein, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A491 – A493. MR 43:6741 102, 105 A. D. IONESCU, Fourier integral operators on noncompact symmetric spaces of real rank one, J. Funct. Anal. 174 (2000), 274 – 300. MR 2001h:43009 113 A. W. KNAPP and E. M. STEIN, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489 – 578. MR 57:536 R. A. KUNZE and E. M. STEIN, Uniformly bounded representations and harmonic analysis of the 2 × 2 real unimodular group, Amer. J. Math. 82 (1960), 1 – 62. MR 29:1287

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ALEXANDRU D. IONESCU N. LOHOUE´ , Puissances complexes de l’opérateur de Laplace-Beltrami, C. R. Acad.

Sci. Paris Sér. A-B 290 (1980), A605 – A608. MR 81h:58062 , Estimées de type Hardy pour l’opérateur 1 + λ d’un espace symétrique de type non compact, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 11 – 14. MR 90d:43008 N. LOHOUE´ and T. RYCHENER, Some function spaces on symmetric spaces related to convolution operators, J. Funct. Anal. 55 (1984), 200 – 219. MR 85d:22024 R. J. STANTON and P. A. TOMAS, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251 – 276. MR 58:23365 102, 113, 117 E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. MR 44:7280 , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993. MR 95c:42002 101, 120 ¨ J.-O. STROMBERG , Weak type L 1 estimates for maximal functions on noncompact symmetric spaces, Ann. of Math. (2) 114 (1981), 115 – 126. MR 82k:43010 106 M. E. TAYLOR, L p -estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773 – 793. MR 91d:58253 102

School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA


RANKIN-SELBERG L-FUNCTIONS IN THE LEVEL ASPECT E. KOWALSKI, P. MICHEL, and J. VANDERKAM

Abstract In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 2. Arithmetic interpretations of the results . . . . . . 3. Review of automorphic forms . . . . . . . . . . 4. Review of Rankin-Selberg convolution L-functions 5. Evaluation of the first partial moment . . . . . . . 6. Another approach for the first partial moment . . . 7. The second moment . . . . . . . . . . . . . . . 8. Quadratic forms and linear independence . . . . . 9. Mollification . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . A. Summation formulae . . . . . . . . . . . . . . . B. Shifted convolutions of modular forms . . . . . . C. Properties of Bessel functions . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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

123 129 132 135 139 141 150 160 166 175 175 186 187 188

1. Introduction In this paper we continue the program sketched in [KMV1] on the central values of Lfunctions in the level aspect, building on the method developed by Duke, Friedlander, and Iwaniec in [DFI1] and then further refined in [KMV1]. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 1, Received 9 May 2001. Revision received 9 August 2001. 2000 Mathematics Subject Classification. Primary 11F66; Secondary 11M41, 11G40. Michel’s work partially supported by National Science Foundation grant number DMS-97-2992, by the Ellentuck fund (through grants to the Institute for Advanced Study), and by Institut Universitaire de France. 123

124

KOWALSKI, MICHEL, and VANDERKAM

Let D be square-free, and fix g a modular form (not necessarily cuspidal or holomorphic) on 00 (D). Let k > 2 be even, and let q be coprime to D. We let Sk∗ (q) denote the set of primitive cuspidal newforms on 00 (q) with trivial nebentypus and weight k. In this paper we investigate the distribution of the values of the Rankin-Selberg convolution L-functions {L( f ⊗ g, s)} f ∈Sk∗ (q) for s on the critical line (<e s = 1/2) as q grows. Recall that L( f ⊗ g, s) admits an analytic continuation to all of C and a functional equation of the form q D s L ∞ ( f ⊗ g, s)L( f ⊗ g, s) = ε( f ⊗ g) L ∞ ( f ⊗ g, 1 − s)L( f ⊗ g, 1 − s), 4π where L ∞ ( f ⊗ g, s) is an explicit product of 0 factors (see Sec. 4) and ε( f ⊗ g) is called the root number. When ε( f ⊗ g) depends only on g and q, we denote it as ε(g). One important challenge in the theory of L-functions is to provide upper bounds for their values on the critical line, in particular to improve, by a positive exponent, the “convexity” bound arising from the Phragmen-Lindelöf principle (see [IS2] for a motivating introduction to these questions). For our first result, we do so in the q-aspect (see [DFI1], [DFI2], [CI] for some other examples). 1.1 Let g be a primitive cuspidal holomorphic newform of integral weight, or a nonexceptional weight-zero Maass form, on 01 (D) with D square-free. Then for all > 0, all integers j > 0, and all f ∈ Sk∗ (q) with (q, D) = 1, B 1 ( j) f ⊗ g, + it ,k, j,g 1 + |t| q +1/2−1/80 , L 2 where the exponent B is absolute. THEOREM

As one can check from the proof, the constant involved in ,k, j,g depends polynomially on the parameters of g (the level, the weight, or the eigenvalue). Remark. In Theorem 1.1, a Maass form is called “nonexceptional” if its eigenvalue under −1 is greater than 1/4. According to a conjecture of A. Selberg, exceptional Maass forms do not exist. For g exceptional, a weaker but still convexity-breaking bound can be obtained, but we have preferred to limit ourselves to the simplest (and presumably only) case. Theorem 1.1 can be seen as the generalization to the cuspidal case of a famous convexity-breaking result of Duke, Friedlander, and Iwaniec [DFI1]. 1.2 be a primitive character of conductor D. Then for all > 0, all integers

THEOREM

Let χ D

RANKIN-SELBERG L-FUNCTIONS IN THE LEVEL ASPECT

125

j > 0, and all f ∈ Sk∗ (q), one has 2 1 ( j) f ⊗ χ D , + it ,k, j D B (1 + |t|) B q +1/2−1/96 , L 2 where the exponent B is absolute. To see the analogy between Theorem 1.1 and this one, note that for χ D primitive there is a (nonholomorphic, weight-zero) Eisenstein series E χ D on 00 (D 2 ) with nebentypus 2 such that L( f ⊗ E , s) = L( f ⊗ χ , s)2 . The slightly better exponent 1/80 of χD χD D Theorem 1.1 also applies in Theorem 1.2; it is the result of exploiting some extra averaging in the large sieve inequality of Section 7.1. Remark. In fact, our result is a little bit more general: one can also break convexity for f ’s with nontrivial nebentypus as long as the conductor of the nebentypus is relatively small with respect to q (we refer the reader to Th. 7.2 for the general statement). Moreover, by using slightly more sophisticated arguments, the same method yields the convexity-breaking bound without any assumption on D (see [Mi]). As a corollary, we obtain the first unconditional improvement of the “trivial bound” for the problem of distinguishing modular forms by their first Fourier coefficients. This analogue of the smallest quadratic nonresidue problem was suggested to us by P. Sarnak. COROLLARY 1.3 Let g be a primitive cusp form as in Theorem 1.1 with D > 1, k > 2 an even integer, and ε > 0. There exists a constant C = C(g, k, ε) depending only on g, k, and ε such that for any primitive holomorphic form f ∈ Sk∗ (q), there exists n 6 Cq 1−1/40+ε such that λ f (n) 6 = λg (n). (1.1)

Here the λ f (n), λg (n) are the Fourier coefficients of f and g. The “obvious” bound is n 6 Cq 1+ε for any ε > 0. Under the generalized Riemann hypothesis for Rankin-Selberg L-functions, (1.1) would be true for n = C(log q D)2 with an absolute C. Using modularity, this applies, in particular, to elliptic curves (see [DK] for a related result “on average”). Let us mention another possible application of Theorem 1.1. Sarnak has recently obtained a convexity-breaking bound for Rankin-Selberg convolution L-functions in the weight aspect, using somewhat different techniques (see [Sa]). His main motivation was the quantum unique ergodicity problem: for f ∈ Sk∗ (q), let µ f be the

126


probability measure on X 0 (q) given by µf =

3 | f (z)|2 k d x dy y , π ( f, f ) y2

where ( f, f ) is the Petersson inner product. 1.4 ([RuSa]) Let q > 1. Let { f j } j>1 be any sequence of primitive holomorphic forms on X 0 (q) with increasing weights k j . As j → +∞, the sequence of probability measure µ f j , j > 1, converges weakly to the Poincaré measure on X 0 (q), µ = 1/ vol(X 0 (q)) (d x dy/y 2 ). CONJECTURE

The main application of the convexity-breaking bound of [Sa] is the proof of this conjecture when the primitive forms are of CM type (in the sense of [Ri]). The proof uses a formula of T. Watson [W], relating the central value of the triple product Lfunction of 3-modular forms to the square of the integral of the product of the 3-forms. This formula shows that proving a nontrivial estimate for the Weyl sums corresponding to this equidistribution problem is tantamount to proving a convexity-breaking bound (in the weight aspect) for the central value of the triple product L-function of f j ⊗ f j ⊗ g, where g is a weight-zero Maass form (or even an Eisenstein series). For f j a CM-form, this reduces to a convexity-breaking bound for a Rankin-Selberg L-function. A possible analog of Conjecture 1.4 in the level aspect is the following: for any q > 1, let πq : X 0 (q) → X 0 (1) be the canonical projection. 1.5 For k > 2 even and fixed, let { f j } j>1 be any sequence of primitive holomorphic forms of weight k with increasing levels q j . As j → +∞, the sequence of probability measure πq j ,∗ (µ f j ), j > 1, converges weakly to the Poincaré measure on X 0 (1), µ = (3/π )(d x dy/y 2 ). CONJECTURE

Although it has not yet been fully established in this context, it is likely that Watson’s formula continues to hold, so that, combined with Theorem 1.1, it should give a proof of Conjecture 1.5 when the f j are CM-forms. Theorem 1.1 follows from the amplification method invented by Friedlander and Iwaniec, applied to bounds of the form Mg,g

1 2

+ µ; ` =

2 X B q 1 1 L f ⊗ g, + µ λ (`) 1 + |t| f ,g |Sk∗ (q)| 2 `1/2 ∗ f ∈Sk (q)


127

for µ = δ + it, δ, t ∈ R, |δ| 6 1/ log q, and 0 < ` < q α for some small α > 0. We prove these bounds in the process of obtaining an explicit asymptotic expansion of this second moment. While we prove the bounds in the general case, we calculate the asymptotics only for the more restrictive case when q is prime and k < 12, so that the space of cusp forms is spanned by the newforms.∗ An asymptotic formula for Mg,g appears in Theorem 7.3, which is a generalization of the main result of [KMV1], where an asymptotic formula is established for the second moment Mg,g (1/2 + µ; `) for g the level-one Eisenstein series 1 X ∂ E 0 z, := E(z, s)|s=1/2 = y 1/2 log y + 4y 1/2 τ (n) cos(2π nx)K 0 (2π ny). 2 ∂s n>1

(1.2) Here E(z, s) is the Eisenstein series for the full modular group (see [I1, Chap. 3, p. 68]), and τ (n) is the divisor function. In the latter case, the second moment is the fourth moment of L( f, 1/2 + µ). However, there are several differences: since g is cuspidal, a very complicated term from our previous work (so-called “off-offdiagonal” in [KMV1]), vanishes here; on the other hand, we have to take care of the perturbations created by the (possibly nontrivial) level and nebentypus of g using the theory of Atkin-Lehner-Li operators, showing a nice matching between the diagonal and off-diagonal terms. An immediate application of Theorem 7.3 is to take the specific values µ = 0, ` = 1 in these asymptotics, and to assume, for example, that χ D is trivial and g is holomorphic. We have the following result. 1.6 For q prime, k < 12, χq and χ D trivial, PROPOSITION

X 1 2 1 L f ⊗ g, = P(log q) + Og,k,ε (q −1/12+ε ) |Sk∗ (q)| 2 ∗ f ∈Sk (q)

for all ε > 0, where P(x) is a cubic polynomial, depending on g. The lead Q ing coefficient of P is 1/(3ζ (2)2 ) L(sym2 g, 1) p|D (1 − p −1 )2 /(1 + p −1 ), where L(sym2 g, s) is the symmetric square L-function of g. ∗ This

hypothesis is rather technical and comes from the fact that we are using, rather crudely, Petersson’s trace formula to average over newforms. Recently, Iwaniec provided a very convenient variant of Petersson’s formula in the square-free level case, in order to average over the set of primitive newforms (rather than over a full orthogonal basis), and we suspect that, using this, one can prove the results of this paper valid for any square-free level with no small prime divisor (see [IS1] for a striking application of this formula).

128


The asymptotics for small `, combined with those for the first twisted moment 1 1 X 1 Mg + µ; ` := ∗ L f ⊗ g, + µ λ f (`) 2 |Sk (q)| 2 f

at µ = 0, allow us to apply the older mollification technique to infer various nonvanishing results for the critical values L( f ⊗ g, 1/2). 1.7 There exists a positive constant c such that, given g as in Theorem 1.1, k < 12, and any sufficiently large prime q, • if either g has real Fourier coefficients and ε(g) = 1 or g has nonreal Fourier coefficients, then THEOREM

|{ f ∈ Sk∗ (q), L( f ⊗ g, 1/2) 6 = 0}| > c + og (1), |Sk∗ (q)| •

(1.3)

if g has real Fourier coefficients and ε(g) = −1, then |{ f ∈ Sk∗ (q), L 0 ( f ⊗ g, 1/2) 6 = 0}| > c + og (1). |Sk∗ (q)|

(1.4)

This second case is needed because L( f ⊗g, 1/2) is identically zero when ε(g) = −1. The dependence on D is much weaker here than in Theorem 1.2 since we use a zerofree region for L(g ⊗ g, s). Finally, the precise evaluation of the second moment for µ 6= 0 allows one, following the methods of [KM1] and [HM], to study the order of vanishing of L( f ⊗ g, s) at the critical point s = 1/2, which we denote by r ( f ⊗ g), in order to infer that there exists an absolute constant A > 0 such that, for all q prime, X exp Ar( f ⊗ g) g |Sk∗ (q)|. (1.5) f ∈Sk∗ (q)

Since the methods are essentially identical to those of the cited papers, we do not pursue this proof in detail here. Our paper is organized as follows. First, we discuss some arithmetic interpretations of these results. In the next section we recall some basic facts about various types of modular functions. Next, we recall the definition of the Rankin-Selberg convolution L-function and its functional equation. In Sections 5 and 6 we compute the first moment in two ways, the second of which introduces tools necessary to attack the second moment. We evaluate the second moment in Section 7, while proving the convexity-breaking bound (Theorem 1.1) along the way. In Section 8 we use our estimates of the first moment to deduce various linear independence lemmas in the spirit


129

of Theorem 2.1. We include the mollification in Section 9. In Appendix A we review the Atkin-Lehner theory of newforms and deduce from it several Poisson-type summation formulae that are our basic tools. Then in Appendix B we prove a bound on shifted convolutions in a manner based on [DFI1]. Finally, in Appendix C we review some basic identities and bounds for Bessel functions. In several places we use the following notational convenience: given L(s) = Q Q (D) (s) := L (s) an Euler product, we write L D (s) := p|D L p (s) and L Qp p ( p,D)=1 L p (s). 2. Arithmetic interpretations of the results This project was motivated by the fact that in many situations the central value L( f ⊗ g, 1/2) is known (or, more often, conjectured) to have deep arithmetical interpretations. One of these is found in the work of D. Rohrlich [Ro], based on a remark of Kazhdan: let A be an abelian variety defined over Q, and let G Q = Gal(Q/Q) be the absolute Galois group. A central object of study is the Galois module defined by the algebraic points on A, A(Q) ⊗Z C (G Q acting trivially on the second factor). By the Mordell-Weil theorem, this Galois representation decomposes as an algebraic direct sum of finite-dimensional complex irreducible representations of G Q , each of them occurring with finite multiplicity. Given ρ : Gal(Q/Q) → GLn (C) an irreducible continuous complex Galois representation (automatically the image of ρ is finite, and ρ factorizes through a finite quotient of G Q := Gal(Q/Q)), the remark of Kazhdan gives a conjectural formula for the multiplicity of ρ in A(Q) ⊗Z C. Under various standard conjectures (including the Birch–Swinnerton-Dyer conjecture for A ker ρ over the subfields of Q ), one has the following formula: multiplicity of ρ in A(Q) ⊗Z C = ords=1 L(AQ ⊗ ρ, s), where A in L(AQ ⊗ ρ, s) refers to the Galois representation on an `-adic Tate module of AQ (see [Ro] for the definition of the L-function). In our applications we take A = Jacnew (X 0 (q)) = J0new (q) the new part of the Jacobian of the modular curve. Up to a finite number of Euler factors, Y 1 L J0new (q)Q ⊗ ρ, s + = L( f ⊗ πρ , s), 2 ∗ f ∈S2 (q)

where πg is the conjectural GLn automorphic representation associated to ρ by the Langlands correspondence and L( f ⊗ πρ , s) is the Rankin-Selberg convolution of f (more accurately, π f , the GL2 automorphic representation canonically associated with f ) and πρ . Hence, conjecturally, X multiplicity of ρ in J0new (q)(Q) ⊗Z C = ords=1/2 L( f ⊗ πρ , s), f ∈S2∗ (q)

130


so it seems worthwhile to study the distribution of the “analytic ranks” r ( f ⊗ πρ ) := ords=1/2 L( f ⊗ πρ , s) when f varies over the family S2∗ (q). When ρ is 1-dimensional, πρ corresponds to a Dirichlet character. When this character is trivial, the multiplicity is simply the rank of J0new (q)(Q). This was investigated first in [Br] and [Mu] and then more thoroughly in [IS1], [KM1], [KM2], [KMV1], and [V1] when q is prime (so that J0 (q) = J0new (q)); the latter analytical works, combined with the partial progress of B. Gross and D. Zagier, V. Kolyvagin, and Kolyvagin and D. Logachëv towards the Birch–Swinnerton-Dyer conjecture, imply both conditional and unconditional results regarding the rank of J0 (q)(Q) for q prime. We address the question of nontrivial characters in the paper [MV]. In this paper we focus on the case when n = 2. Here the very existence of πρ is a deep question, which has now been solved in many cases: if ρ is of dihedral, tetrahedral, or octahedral type, it follows from work of E. Hecke, H. Maass, R. Langlands, and J. Tunnell, while for certain infinite families of icosahedral type it follows from recent work of K. Buzzard, M. Dickinson, N. Shepherd-Barron, and R. Taylor. In cases where the existence is known (now using classical terminology), one may associate to an irreducible ρ a cuspidal modular form gρ that is either a holomorphic form of weight one (if det ρ is odd) or a Maass form with eigenvalue 1/4 (if det ρ is even). In particular, assuming the Birch–Swinnerton-Dyer conjecture, Theorem 1.7 has some bearing on the multiplicity of ρ in J0 (q)(Q). In [Ro], the main concern is to produce nontrivial explicit examples of vanishing of L( f ⊗ gρ , s) at s = 1/2; here our result is in the other direction, giving a large collection of f ’s for which L( f ⊗ gρ , s) does not vanish at this point (or at least vanishes to the minimal possible order). Moreover, inequality (1.5) suggests that f ’s with a high order of vanishing are very rare. Concerning the conjecture of B. Birch and P. Swinnerton-Dyer, one can say much more when ρ is dihedral. We recall briefly the theory of Heegner points and the Gross√ Zagier formula in [GZ]. Let K = Q( −D) be an imaginary quadratic field with ring of integers O K , let HK be the Hilbert class field, and let χ˜ be a character of G Gal(HK /K ). Let ρχ˜ = IndG QK denote the 2-dimensional representation induced by χ. ˜ It is irreducible if and only if χ˜ is nonquadratic. The associated modular form gρχ˜ is the theta series on 00 (D) of weight one and nebentypus χ D (the Kronecker symbol of K ) given by X gχ˜ (z) = δχ˜ =1 L(χ D , 1) + χ˜ (a) e N K /Q (a)z , 06 =a⊂O K

where we have identified χ˜ with a character of the ideal class group of O K . If χ˜ is nonquadratic, gχ˜ is cuspidal; otherwise, gχ˜ is an Eisenstein series. When every


131

prime factor of q splits in K , one finds in X 0 (q)(HK ) the set of Heegner points corresponding to pairs of elliptic curves, linked by a cyclic isogeny of degree q, with complex multiplication by O K . Letting e K ∈ J0 (q)(HK ) denote the image of one of these points in the Jacobian, the Gross-Zagier formula (for D odd at least) interprets the central value of the derivative L 0 ( f ⊗ gχ˜ , 1/2) in terms of the Neron-Tate height of the (χ˜ , f )-eigencomponent of e K . In particular, for q prime we obtain from (1.4) a lower bound for the dimension of the χ-isotypical ˜ component of the Gal(Q/K )module J0 (q)(HK ); let J0 (q)(HK )χ˜ ⊂ J0 (q)(HK ) ⊗Z Z[χ]; ˜ then rankZ J0 (q)(HK )χ˜ > c + o K (1) dim J0 (q). (2.1) This can also be shown for χ˜ real by more elementary methods (it amounts to proving a double nonvanishing result for two real characters) by combining a slight variant of the methods of [IS1], [KM2], and [KMV2]; in particular, we have rankZ J0 (q)(HK ) > c + o K (1) |HK | dim J0 (q). Note also that Theorem 1.1 provides a nontrivial upper bound for the Neron-Tate height of the ( f, χ)-eigencomponent ˜ e K ,χ˜ , f in the q-aspect. Concerning the original question of estimating the multiplicity of ρχ˜ , observe that if χ˜ is not real and e K ,χ, ˜ f 6 = 0, then the vector space spanned by {e K ,χ˜ , f , σ (e K ,χ, ˜ f )} (σ the complex conjugation) is 2-dimensional and realizes ρχ˜ ; in fact, σ (e K ,χ, ˜ f) = e0 , where e0K = σ (e K ), so the two vectors e K ,χ˜ , f and σ (e K ,χ, ˜ f ) cannot be K ,χ, ˜ f

colinear. In conclusion, we have for χ˜ nonreal, multiplicity of ρχ˜ in J0 (q)(Q) ⊗Z C > c + o K (1) dim J0 (q). Following the methods of [V2], the techniques used to prove Theorem 1.7 can also be used to give the following linear independence result, which, along with some variants given in Section 8, may have interesting arithmetic applications. THEOREM 2.1 Let K be an imaginary quadratic field of odd discriminant −D, let HK be the Hilbertclass field of K , and let χ˜ be a character of Gal(HK /K ). Let q be a prime number that splits in K , and let e K be a Heegner divisor in J0 (q)(HK ). For ε > 0 fixed and L ε (q/D 5 )1/2− an integer, the images of the χ˜ -eigencomponent of e K by the first Hecke operators T1 e K ,χ˜ , T2 e K ,χ˜ , . . . , TL e K ,χ˜

are linearly independent in J0 (q)(HK ) ⊗Z C. If q is inert in K , or if K is real and q splits, one has very similar constructions of Hecke modules generated by Heegner points (or cycles) endowed with a proper height

132


pairing, whose values at the Heegner eigencomponents are expressed in terms of the central values L( f ⊗ gχ˜ , 1/2); we refer the interested reader to [G], [GKZ], [D], and [BD]. In particular, when K is imaginary, q is inert, and χ˜ is a character of order less than or equal to 4, then (1.3), combined with the results of M. Bertolini and H. Darmon [BD], proves the existence, for q a sufficiently large prime, of a quotient Jχ˜ defined over Q of dimension comparable to dim J0 (q) such that the χ-eigencomponent ˜ of the Gal(Q/K )-module Jχ˜ (Q) ⊗Z C is zero-dimensional. 3. Review of automorphic forms In this section we review the types of automorphic forms to be considered in the rest of the paper. 3.1. Holomorphic cusp forms For k and q two integers, k > 2, and χq a Dirichlet character of modulus q and conductor q, ˆ let Sk (q, χq ) denote the complex vector space of weight k holomorphic cusp forms with level q and nebentypus χq . These are the bounded holomorphic functions on the upper half-plane which satisfy the automorphy relation f (γ z) a b ∀γ = ∈ 00 (q), f |γ (z) := = χq (d) f (z). (3.1) c d (cz + d)k We represent these elements by their Fourier series X X f (z) = fˆ(m)e(mz) = ψ f (m)m (k−1)/2 e(mz). n>1

n>1

This space is equipped with Petersson’s inner product Z ( f, g)k = f (z)g(z)y k−2 d x dy. X 0 (q)

The Hecke operators Tn , with (n, q) = 1, are normal with respect to the inner product; more precisely, T ∗ = χq (n)Tn for (n, q) = 1, where T ∗ denotes the adjoint. One can thus find an orthogonal basis of Sk (q, χq ), Bk (q), formed of eigenvectors of all the {Tn , (n, q) = 1}. For f a Hecke eigenvector, let λ f (n)n (k−1)/2 denote the eigenvalue of Tn (sometimes one speaks of λ f (n) of the normalized eigenvalue). We have the adjointness relation λ f (n) = χq (n)λ f (n),

ψ f (n) = χq (n)ψ f (n) for (n, q) = 1

(3.2)


133

and for (n, q) = 1 the recursion formulae m n X , ψ f (m)λ f (n) = χq (d)ψ f d d d|(m,n) m n X ψ f (mn) = µ(d)χq (d)ψ f λf . d d

(3.3) (3.4)

d|(m,n)

The space of newforms Sk (q, χq )new is the orthogonal complement of the (old) subspace generated by the forms f (dz) with f ∈ Sk (q 0 , χq 0 ), dq 0 |q q 0 6= q, and χq 0 inducing χq . This space is stable under all the Tn , and in [ALi] it is shown that the Tn can be simultaneously diagonalized. For f such a newform, equalities (3.3) and (3.4) hold for all m, n, and in particular ψ f (1) 6 = 0. We say that such an f is primitive if ψ f (1) = 1; in that case, for all n, ψ f (n) = λ f (n). We let Sk∗ (q, χq ) denote the set of primitive new forms; it forms an orthogonal basis of Sk (q, χq )new . 3.1.1. Petersson’s trace formula Let Bk (q) be an orthogonal basis of Sk (q, χq ). Petersson’s formula states that 0(k − 1) (4π)k−1

ψ f (m)ψ f (n)

X

( f, f )

f ∈Bk (q)

with 1(m, n) := 2πi −k

= δm,n + 1(m, n)

(3.5)

4π √mn X S(m, n; c) Jk−1 c c

c≡0(q) c>0

and S(m, n; c) the (twisted) Kloostermann sum X

S(m, n; c) =

χq (x)e

x(c),(x,c)=1

mx + nx . c

Note that (3.5) is independent of the choice of basis; in what follows, we assume that Bk (q) is a Hecke eigenbasis and that it contains Sk∗ (q, χq ). For Bk (q) such a basis, we use the following notation: Xh f ∈Bk (q)

α f :=

0(k − 1) (4π )k−1

X f ∈Bk (q)

αf , ( f, f )

and we refer to it as the harmonic average. Remark. Sometimes we make the stronger hypothesis that there are no old forms: Bk (q) = Sk∗ (q, χq ). This is the case when χq is primitive or if k < 12 and χq is trivial.

134


3.2. Maass forms ˆ and Let D be a positive integer, let χ D be a character of modulus D and conductor D, let λ be a positive real number. Let Mλ (D, χ D ) denote the (finite-dimensional) space of weight-zero Maass forms of level D, nebentypus χ D , and eigenvalue λ. In other words, Mλ (D, χ D ) consists of functions g, satisfying relation (3.1) for k = 0, which satisfy (1 + λ)g = 0 for the hyperbolic Laplacian 1 with λ = 1/4 + r 2 and either r ∈ R+ or ir ∈ [−1/4, 1/4].∗ Let Sλ (D, χ D ) ⊂ Mλ (D, χ D ) denote the subspace of Maass cusp forms. Any g ∈ Mλ (D, χ D ) has a Fourier expansion near infinity of the form X g(z) = ψg (n)e(nx)2|y|1/2 K ir 2π|ny| , n∈Z

where z = x + i y, K s is the K Bessel function, and ψg (0) = 0 for g ∈ Sλ (D, χ D ). The Hecke operators act on the Hilbert space Sλ (D, χ D ) (equipped with the Petersson inner product), and the theory of old/new forms is identical to that in the holomorphic case. We again call a new form f primitive if ψ f (1) = 1; in that case, ψ f (n) = λ f (n) for all n, where λ f (n) is the eigenvalue of Tn associated to f and equalities (3.3) and (3.4) are valid (replacing q by D) for all m and n. Let Sλ (D, χ D )∗ denote the set of primitive new forms. There is another operator acting on Mλ (D, χ D ), namely, the reflection operator R f (z) = f (−z). Since it commutes with the Tn , a primitive cusp form g satisfies Rg = εg g with εg = ±1. We call g even or odd according to the value of εg . 3.3. Bounds on Fourier coefficients Given g a primitive form of one of the types presented above, we often need upper bounds for the normalized Fourier coefficients λg (n). For p prime, let αg,1 ( p), αg,2 ( p) be the complex roots of the quadratic polynomial X 2 − λg ( p)X + χ D ( p); when g is holomorphic (by work of M. Eichler, G. Shimura, Y. Ihara, and P. Deligne), the Ramanujan-Petersson bounds are valid; namely, |αg,1 ( p)|, |αg,2 ( p)| 6 1,

so that ∀n > 1,

|λg (n)| 6 τ (n).

(3.6)

When g is a Maass form, this bound is not known in general. (Although it holds for the forms of type gρ given in the introduction.) Nevertheless, the following bound of Serre (see [Sh]) is sufficient for our purposes: |αg,1 ( p)|, |αg,2 ( p)| 6 p 1/5 , ∗ The

so that ∀n > 1,

|λg (n)| 6 τ (n)n 1/5 .

(3.7)

latter comes from the “Selberg bound” for exceptional eigenvalues on congruence subgroups. We use it only in Appendix A.4; everywhere else we assume that r ∈ R.


135

More often we use the results of Rankin-Selberg theory, which gives the RamanujanPetersson bound “on average”: X ∀X > 0, |λg (n)|2 g X. (3.8) n6X

Finally, we need to introduce the following function: σg (n) := this function is almost multiplicative; by (3.3) and (3.4),

P

d|n

|λg (d)|. First

(mn)−ε σg (mn) σg (m)σg (n) (mn)ε σg (mn)

(3.9)

for all ε > 0, the implied constant depending only on ε, and from (3.8) we have X ∀X > 0, σg (n)2 X 1+ε (3.10) n6X

for all ε > 0, the implied constant depending on ε, and g. 4. Review of Rankin-Selberg convolution L-functions Let χ D and χq be Dirichlet characters modulo q and D, respectively, and let χ := χq χ D denote the character modulo q D. Let f ∈ Sk (q, χq )∗ and g ∈ S∗ (D, χ D )∗ be normalized newforms, g either a holomorphic form of weight k 0 or a Maass form with eigenvalue λ = 1/4 + r 2 > 1/4 (so that r ∈ R). The Rankin-Selberg convolution L-function is 2 Y 2 X λ f (n)λg (n) Y Y α f,i ( p)αg, j ( p) −1 L( f ⊗ g, s) := L(χ , 2s) = 1 − , ns ps p i=1 j=1

n>1

(4.1) where for each prime p, α f,1 ( p), α f,2 ( p) and αg,1 ( p), αg,2 ( p) are the roots of the quadratic equations X 2 − λ f ( p)X + χq ( p) = 0,

X 2 − λg ( p)X + χ D ( p) = 0.

R. Rankin, Selberg, and others proved that L( f ⊗g, s) admits an analytic continuation over the whole complex plane except when f = g, in which case there are simple poles at s = 0, 1. Moreover, this L-function admits a functional equation linking s to 1 − s. When (q, D) = 1, we set (see [Li2, Th. 2.2 and Exam. 2]∗ ) 3( f ⊗ g, s) := ∗ This

q D s 4π 2

0g (s)L( f ⊗ g, s)

gives the holomorphic case, but the proof also works when g is a Maass form.

(4.2)

136


with |k − k 0 | k + k0 0g (s) = 0 s + 0 s+ −1 for g holomorphic, 2 2 k + 2ir − 1 k − 2ir − 1 0g (s) = 0 s + 0 s+ for g a Maass form. 2 2 We then have 3( f ⊗ g, s) = ε( f ⊗ g)3( f ⊗ g, 1 − s)

(4.3)

with ε( f ⊗ g) =

( χ D (−q)χq (D)η f (q)2 ηg (D)2 χ D (q)χq (−D)η f (q)2 ηg (D)2

if g is holomorphic and k 0 > k, else.

(4.4)

Here η f (q), ηg (D) are the pseudo-eigenvalues of f, g for the Atkin-Lehner-Li operators Wq , W D , and g is the primitive form proportional to W D g; that is, W D g = ηg (D)g (see App. A.1). A particularly important case arises when η f (q)2 = 1 (e.g., if χq is real), so we define ( χ D (q)ηg (D)2 if g is holomorphic and k 0 > k, ε(g) = (4.5) χ D (−q)ηg (D)2 else. 4.1. The twisted moments In what follows we use µ to denote a complex number of the form µ = δ + it with |δ| 6 1/ log q. Let g and g 0 be primitive cusp forms with weights k g , k g0 or eigenvalues λg , λg0 , square-free levels D and D 0 , and nebentypus χ D and χ D 0 , respectively, which are not exceptional. Following the methods of [KMV1], we wish to compute the twisted moments Mg (µ; `) :=

Xh

1 L f ⊗ g, + µ λ f (`) 2

(4.6)

1 1 L f ⊗ g, + µ L f ⊗ g 0 , + µ0 λ f (`). 2 2

(4.7)

f ∈Sk∗ (q,χq )

and Mg,g0 (µ, µ0 ; `) :=

Xh f ∈Sk∗ (q,χq )

We use the functional equation (4.3) to represent L( f ⊗ g, 1/2 + µ) as a rapidly converging series. To this end, we set G g,µ (s) :=

(4π 2 )µ ξ(1/2 + s − µ) 5 Pg (s) . 0g (1/2 + µ) ξ(1/2) Pg (µ)


137

Here ξ(s) = s(1 − s)π −s/2 0(s/2)ζ (s) is the completed ζ -function of Riemann∗ , and Pg (s) is an even polynomial with real coefficients, depending only on k, k g , such that Pg (s)0g (1/2 + s) is holomorphic in the region −A < <e s for some A > 1/2. The functional equation of ξ(s) implies that G g,µ (−s) =

(4π 2 )µ 0g (1/2 − µ) G g,−µ (s). (4π 2 )−µ 0g (1/2 + µ)

(4.8)

With this in mind, we define Hg,µ (s) := (4π 2 )−s 0g

1 2

+ s G g,µ (s).

Note that Hg,µ (µ) = 1. Using contour shifts and the functional equations of L( f ⊗ g, s) and (4.8), one shows that n X λ (n)λ (n) 1 f g (q D)µ L f ⊗ g, + µ = V g,µ 2 qD n 1/2 n>1

+ εµ ( f ⊗ g)

n X λ f (n)λg (n) V g,−µ qD n 1/2

(4.9)

n>1

with Z 1 ds Hg,µ (s)L(χ, 1 + 2s)y −s , 2πi (3) s−µ Z 1 ds Vg,−µ (y) := Hg,−µ (s)L(χ, 1 + 2s)y −s , 2πi (3) s+µ Vg,µ (y) :=

and εµ ( f ⊗ g) =

(4.10)

(4π 2 )µ 0g (1/2 − µ) ε( f ⊗ g). (4π 2 )−µ 0g (1/2 + µ)

In the particular case where the λ f (n), λg (n) are real, ε(g) = −1, and µ = 0, the functional equation then automatically forces L( f ⊗g, 1/2) to vanish, so any analysis must focus on the first derivative, which is given in much the same way by X λ f (n)λg (n) n 1 L 0 f ⊗ g, =2 Wg 2 qD n 1/2 n>1

with Wg (y) := ∗ The

1 2πi

Z (3)

Hg (s)L(χ, 1 + 2s)y −s

ds . s2

(4.11)

arithmetic nature of this function is not needed for the proof; it is here merely to force polynomial growth in the t-variable in the forthcoming computations.

138


In what follows, for notational simplicity, we suppress dependence on µ and µ0 . However, to be consistent, we make the following convention explicit. CONVENTION 4.1 An expression Expg,µ depending on g and µ is usually written Expg ; accordingly, by Expg0 we mean implicitly Expg0 ,µ0 , and by Exp g (resp., Exp g0 ) we always mean Exp g,−µ (resp., Exp g0 ,−µ0 ).

In this paper these notations never conflict. By a contour shift to the right, the definition of ξ , and Stirling’s formula, B Vg (y) A 1 + |t| y −A (4.12) for all A > 0, and by shifting the contour to the left to <e s = −1/ log(q D), we pass a pole at s = µ and get B Vg (y) g 1 + |t| τ (q D)(log q D + | log y|), (4.13) while by shifting the contour to <e s = −1/4, we obtain, using Burgess’s bound on L(χ, s), Vg (y) = (ress=0 + ress=µ )Hg (s)L(χ , 1 + 2s) ˆ 3/16 y 1/4 + O (1 + |t|) B (q D)ε (qˆ D)

y −s s−µ (4.14)

for all ε > 0, the implied constant depending onε and g. We compute the moments (4.6) and (4.7). Closely related to these are the more general, “spectrally complete” moments defined by extending the averaging over the whole Hecke eigenbasis Bk (q), Mg (`) :=

X λg (n) n

Mg,g0 (`) =

n 1/2

Vg

n Xh ψ f (n)ψ f (`), qD

(4.15)

f ∈Bk (q)

X λg (m)λg0 (n) m n Xh Vg Vg0 ψ f (m)ψ f (n)λ f (`), qD q D0 m 1/2 n 1/2

m,n>1

f ∈Bk (q)

(4.16) where we use Convention 4.1 but also take as a definition that the χ 0 appearing in Vg0 = Vg0 ,µ0 is χ 0 = χ q χ D 0 . Note that, to have λ f (`) well-defined for all f ∈ Bk (q), we must assume either that Bk (q) = Sk∗ (q, χq ) or that (`, q) = 1. For simplicity we give the expression of (4.6) and (4.7) only when χq is trivial and Bk (q) = Sk∗ (q, χq ); then ε( f ⊗ g) = ε(g) is independent of f (see (4.5)). From


(4.9) we have

(q D)µ Mg (`) = Mg (`) + εµ (g)Mg (`)

139

(4.17)

with

(4π 2 )µ 0g (1/2 − µ) ε(g). (4π 2 )−µ 0g (1/2 + µ) Similarly, (4.7) is a sum of four terms: εµ (g) =

(4.18)

µ0

q µ+µ D µ D 0 Mg,g0 (`) 0

= Mg,g0 (`) + εµ (g)Mg,g0 (`) + εµ0 (g 0 )Mg,g0 (`) + εµ (g)εµ0 (g 0 )Mg,g0 (`). (4.19) 5. Evaluation of the first partial moment We now evaluate the first partial moment Mg (`) in two ways. • The first approach to evaluating this sum, given in the present section, is largely axiomatic and uses very little information about the λg ’s. This very robust method is based on large sieve inequalities for Kloosterman sums developed by J.-M. Deshouillers and Iwaniec [DI]. In particular, we use the form given in [DFI1, Prop. 1]. • The second approach, given in Section 6, uses the modularity properties of the λg and rests on the summation formulae of Appendix A.5. While this approach is less general, it is more powerful, and we use its techniques to compute the second moments as well. 5.1. The first approach: The large sieve We compute the linear form L g (E x ) :=

X

x` Mg (`)

`6L

for xE = (x1 , x2 , . . . , x L ) ∈ C L . In what follows, L is smaller than a fixed power of q D. 5.1.1. Treatment of L g (µ; xE) Applying (3.5), we obtain L g (E x ) = L gD (E x ) + L gN D (E x) with λg (`) ` Vg , qD `1/2 `6L X X λg (n) n L gN D (E x) = x` Vg 1(n, `). qD n 1/2 n L gD (E x) =

X

`6L

x`

140


√ Note that, by Weil’s bound for Kloosterman sums, 1(`, n) `nq −3/2 . Using the rapid decay of Vg (y) for y > 1, we may assume that n 6 (q D)1+ for any small positive , since the contribution of the remaining terms is a O,A,g ((1 + |t|) B ||E x ||1 (q D)−A ) for all A. Now we can use the large sieve for Kloosterman sums (see [DFI1, Prop. 1 and the remark following it]). PROPOSITION 5.1 Let k > 2 be an integer. For η a smooth function supported in [C, 2C] such that η(i) i C −i for all i > 0, set 4π √`n X S(n, `; c) −k 1η (n, `) := 2πi Jk−1 η(c). c c c≡0(q) c>0

Then for any sequences of complex numbers x` , yn , √ L N k−3/2 X X L 1/2 N 1/2 x` yn 1η (`, n) ,k C 1+ 1+ ||x||2 ||y||2 C q q `6L n6N

with any > 0. Moreover, the exponent k − 3/2 can be replaced by 1/2. Proofs of this proposition appear only in the literature for χq trivial and k even, but one can show, following the methods of [DFI1] and [DeI] (using N. Proskurin’s generalization of the Kuznetzov trace formula for integral weight), that it holds in our more general case as well. Thus for all > 0, B DL 1/4 L 1/2 1/2 L gN D (E x ) 1 + |t| q 1+ D q q ,g 1 + |t|

B

q

L 1/4 q

D 3/4 ||E x ||2

X n6(q D)1+

|λg (n)|2 1/2 ||E x ||2 n (5.1)

so long as L 6 q (here we have used (3.8)). Next we apply formula (4.14) to evaluate L D (E x ). The resulting error term is 1/4 B 3/16 L ˆ O,g 1 + |t| (q D) (qˆ D) ||E x ||2 . qD Thus we obtain the following result.


141

PROPOSITION 5.2 For L 6 q and xE ∈ C L , X x` Hg (s)L(χ, 1 + 2s) ` −s L g (E x) = λ (`) res g s=µ s−µ qD `1/2 `6L L 1/4 B + O,g 1 + |t| (q D) (qˆ 3/16 + D 3/4 )||E x ||2 . q

If χ = χ D χq is nontrivial and µ = 0, then ress=0

Hg (s)L(χ D , 1 + 2s) ` −s = L(χ D , 1), s qD

while if χ = χ D χq is the trivial character modulo q D and µ = 0, then q D ϕ(D) Hg (s)L(χ D , 1 + 2s) ` −s 1 ϕ(D) ress=0 = log + 9(q D) + Cq D s qD 2 D ` D P with 9(D) := p|D log p/( p − 1), and Cq D = Og (1). The evaluation of the original first moment Mg (`) then follows quickly by restricting the sums to one term. 6. Another approach for the first partial moment In this section we give another approach, which stems from the techniques of Iwaniec and Sarnak [IS1] and is based on the summation formulae of Appendix A, thus relying on the automorphic nature of the λg (n). We assume that g is a modular form of square-free level. We again compute Mg (`), providing a slight improvement on Proposition 5.2 (although this has no significant contribution to applications) and also allowing us to attack the second partial moment. Applying (3.5), we obtain Mg (`) = MgD (`) + MgN D (`) with λg (`) ` MgD (`) = 1/2 Vg , (6.1) qD ` X λg (n) n MgN D (`) = Vg 1(n, `) qD n 1/2 n 4π √ n X 1 X 1 = 2πi −k J `n . c λ (n)S(n, `; c) V g g k−1 qD c c2 n n 1/2 c≡0(q),c>0

(6.2) We partition the n-sum using a smooth function η(x) that is 0 for x 6 1/2, 1 for x > 1, and partitions further into smooth functions by X η(x) = η M (x) M>1

142


with η M compactly supported in [M/2, 2M] such that x i η M (x)(i) i 1 for any i > 0. P We also require that M6X 1 log X . We set F(x) :=

1

x

V 1/2 g

x n X 1 X η(x) = η (x) := FM (x) V M g qD qD n 1/2 M>1

M>1

so that for all i, A > 0 (using (4.13) and (4.12)), xi

A B ∂i −1/2 1+i q D F (x) 1 + |t| τ (q)M (log q) . M g,A,i ∂i x x

We also define TM (c) := c

X

λg (n)S(n, `; c)FM (n)Jk−1

n

so that X

MgN D (µ; `) = 2πi −k

4π √ `n c

X 1 TM (c). c2

c≡0(q),c>0 M

For C > it is enough to bound the sum on c > C through the use of Weil’s bound on Kloosterman sums, getting (by (3.8)) r √ B ` B ` O,g 1 + |t| q = O,g 1 + |t| q . C q q2

Thus we may assume that c 6 C. We may also assume that M√6 q 1+ since the contribution of the M’s such that M > q 1+ is an Og ((1 + |t|) B `M −A ) for some large A > 0. We open the Kloosterman sum and apply the formulas of Section A.5 to the nsum (which is possible since D is square-free) to get ηg (D2 ) X λg D2 (n)G χq D1 (` − n D2 ; c) TM (c) = χ D2 (−c) √ D2 n Z ∞ 4π √ 4π √nx `x Jg √ × FM (x)Jk−1 dx c c D2 0 ηg (D2 ) X + χ D2 (c) √ λg D2 (n)G χq D1 (` + n D2 ; c) D2 n Z ∞ 4π √ 4π √nx × F(x) M Jk−1 `x K g √ d x, c c D2 0 where G χq D1 (a; c) =

X x(c) (x,c)=1

χq D1 (x)e

ax , c

(6.3)


143

D1 = (c, D), and D2 = D/D1 . We split this sum into two parts: the term corresponding to n = `D2 , which we denote ηg (D2 ) TMO D (c) = χ D2 (−c) √ λg D2 (`D2 )G χq D1 (0; c) D2 Z ∞ 4π √`x 4π √`x × FM (x)Jk−1 Jg d x, c c 0

(6.4)

6=

and the remaining terms for which n 6= `D2 , which we denote TM (c). In Section 6.1 we show that (under the Ramanujan-Petersson conjecture for the Fourier coefficients 6= of g) the TM (c)-terms are negligibly small: p √ X X 1 6= B qˆ ` T (c) = O,g 1 + |t| q . q c2 M 1+ M6q

c6Cq|c

6.1. The off-diagonal remainder term 6= The term TM (c) splits naturally as the sum of two terms corresponding to ` − D 2 n and ` + D 2 n. We consider the case of ` − D 2 n (the case of ` + D 2 n being treated in the same way). The full sum is then X 1 ηg (D2 ) X λg D2 (n)G χq D1 (` − n D2 ; c) χ D2 (−c) √ 2 c D2 n6= D2 `

c6C q|c

∞

Z × 0

4π √ 4π √nx d x. FM (x)Jk−1 `x Jg √ c c D2 0

We exploit the oscillations of Jg (x), which is either 2πi k Jk 0 −1 (x) (with k 0 > 1 an integer), −2π Y0 (x), or −π(J2ir (x) − J−2ir (x))/ sin(πir ) (with r ∈ R∗ by our assumptions on g). LEMMA 6.1 Let h(x) be a smooth function supported in [M, 2M] which satisfies |x i h (i) (x)| i a i 1 + | log x|

for some a > 1 and all i > 0, x > 0. For ν complex and j > 0, ∞

Z 0

Jν (x)h(x) d x ν, j

a j (1 + | log M|) M <e ν+ j+1 . j−1 M (1 + M)<e ν+ j+1/2

144


Proof We integrate by parts, using (C.1) and (C.4): Z ∞ Z ∞ ν+1 i0 h M ν+1 x Jν (x)h(x) d x = − h(x) dx J (x) ν+1 x M ν+1 0 0 Z ∞ Jν+1 (x)h 1 (x) d x = M −1 0

with h 1 (x) := −(x/M)ν+1 M[(M/x)ν+1 h(x)]0 , so that (i)

|x i g1 (x)| i,ν a i+1 1 + | log x| for all i > 0, x > 0. Iterating, we have Z ∞ Z −j Jν (x)h(x) d x = M 0

0

∞

Jν+ j (x)h j (x) d x

with h j a smooth function supported on [M, 2M] with (i)

|x i h j (x)| i, j,ν a i+ j 1 + | log x|

for all i > 0, x > 0. The lemma then follows from (C.4) applied to Jν+ j . Remark. The lemma also holds for Y0 , up to an extra factor of (1 + | log M|). We apply this to the integral r Z ∞ Z 4π √ 4π √nx c D2 ∞ dx = 2 FM (x)Jk−1 `x Jν h(y)Jν (y) d x √ 4π n 0 c c D2 0 with c h(y) = 4π

r

r r `D D2 c D2 2 2 y FM y Jk−1 y . n 4π n n

√ The function h satisfies with M 0 = (4π/c) Mn/D2 √ the conditions of Lemma 6.1 and a = log q(1 + `M/c). In particular, if n > q (c2 /M + `), the integral is very small. For n 6 q (c2 /M + `) we use (C.4), k > 2, and <e ν > 0 to bound the integral by √ B M ` O,g q 1 + |t| . c To finish bounding this term with minimal effort, we assume that the RamanujanPetersson conjecture holds for g, so that for n > 1, D2 |D, > 0 we have |λg D2 (n)| n . Using the trivial bound p |G χq D1 (` ± D 2 n; c)| 6 qˆ D1 (n ± D2 `, c),


we find that these terms are then bounded (using g, q 1 + |t|

B p √ X qˆ `

M

M6q 1+

g, q 1 + |t|

B

P

X 1 c3

c6C q|c

145

M6X

1 6 log X ) by X

(n − D2 `, c)

n6q (c2 /M+`)

p q` ˆ ` 1+ . q q

Remark. The Ramanujan-Petersson conjecture is not vital here, and it could be avoided with more work, but since this section is more to demonstrate techniques, we used the shorter proof. The main point here is Section 6.2 where the off-diagonal term is evaluated quite precisely (without any hypothesis). It only contributes a remainder term (as we already know by the results of Section 5), but the identity we obtain is essential for the computation of the second moment. 6.2. The off-diagonal main term Now we evaluate TMO D (c) (the n = `D2 term of (6.3)), which also contributes as a remainder term to the first moment. However, since terms very much like it contribute to the second moment, we evaluate it in a little more detail. Convergence issues are less of a worry here, so we can sum TMO D (c) over all M (including those with M > q 1+ , which we know to be negligible) to get T O D (c), defined like TMO D (c) with FM replaced by F. If χq D1 is nontrivial, G χq D1 (0; c) = 0, while if χq D1 is trivial, G χ D1 (0; c) = ϕ(c). So (6.4) vanishes identically if χq is nontrivial, in which case we are done. Thus we may assume that χq D1 is trivial (and thus the λ f (n) are real). Since (c, D2 ) = 1, ˆ From (A.2) c must be coprime to the primitive conductor of χ D , which we call D. and the first part of Proposition A.1, we have g D2 = g D = g. Thus we may replace ˆ 2 , and (c, D2 ) = 1. Summing over the new c with cq D1 , where D1 D2 = D, D|D c-variable, we obtain the main term of MgN D (`), which we call the off-diagonal term: OD Mg,C (`) = 2π

×

i −k χ D (−q) q2

X c6C/q D1

X D1 D2 =D ˆ (D1 , D)=1

χ Dˆ (D1 ) ηg (D2 ) λg (`D2 ) √ D12 D2

ϕ(cq D1 ) χ D2 (c) c2

Z 0

∞

4π √`x 4π √`x F(x)Jk−1 Jg d x. cq D1 cq D1

We can now replace η(x) by 1 in the formula for F(x) at a cost of at most Og, (q −1 ), by Lemma C.2. √ We can also remove the constraint c 6 C/q D1 with an error term of Og, (q `/C) (using Lemma C.2 and k > 2). Setting y =

146


√ 4π `x/(cq D1 ), the main term of MgN D (`) becomes MgO D (`) =

i −k λg (`) √ χ D (−q) q ` 1 × 2πi

Z

X D1 D2 =D ˆ (D1 , D)=1

χ Dˆ (D1 ) ηg (D2 ) λg (D2 ) √ D1 D2

(4π)2 D ` s ds 2 Hg (s)L(χ, 1 + 2s)Z (s)H (s) D1 q s−µ (1)

(6.5)

with X ϕ(cq D1 )χ D (c) 2 Z (s) := c1+2s

H (s) :=

and

∞

Z

Jk−1 (y)Jg (y)y −2s dy. 0

c>1

Since ϕ and χ D2 are multiplicative, Z (s) = ϕ(q D1 )L D2 (χ Dˆ , 2s)−1 where L n (χ, s)−1 :=

Y p|n

1−

L(χ Dˆ , 2s) , L(χ, 1 + 2s) χ( p) . ps

Shifting the contour in (6.5) to <e s = 1/2 + and using the standard bounds (see Lemma C.2) for the Bessel functions, we have √ B `σg (`) OD , (6.6) Mg (`) = O,g 1 + |t| q q while if we shift to <e s = , we have MgO D (`) = O,g

B σg (`) 1 + |t| q 1/2 . `

(6.7)

This completes our bound for the nondiagonal term. 6.3. A more precise computation of the off-diagonal term We now calculate the off-diagonal term more precisely. For this we use the precise expression for H (s). In the holomorphic case, from [EMO2, Chap. 6, Sec. 8, (33)], H (s) =

2πi kg 2−2s 0(2s)0((k + k g )/2 − 1/2 − s) . 0((k + k g )/2 − 1/2 + s)0(|k − k g |/2 + 1/2 + s)0(−|k − k g |/2 + 1/2 + s) (6.8)


147

Using (A.15) and (A.16), we have √ H (s) = i kg π

0g (1/2 − s) 0(s)0(s + 1/2) , π cos(π(s − |k − k g |/2)) 0g (1/2 + s)

so the off-diagonal term is λg (`) ϕ(q) MgO D (`) = √ χ D (−q) q ` Z × (3)

(4π)s 0g

1 2

X D1 D2 =D ˆ (D1 , D)=1

ϕ(D1 ) ηg (D2 ) 1 λg (D2 ) √ D1 2πi D2

− s G g (s)

i kg −k 0(s)0(s

+ 1/2) Dˆ s L(χ Dˆ , 2s)L D2 / Dˆ (χ Dˆ , 2s)−1 π cos(π(s − |k − k g |/2)) π D ` s ds 2 × . (6.9) Dˆ D1 q s − µ ×

√ π

The functional equation for L(χ Dˆ , s) is (see [D]) Dˆ 1/2−s 1 a Dˆ s a 0 s+ L(χ Dˆ , 2s) = εχ Dˆ 0 + − s L(χ Dˆ , 1 − 2s), π 2 π 2 2 G(χ ) ˆ D εχ Dˆ = (−i)a , (6.10) Dˆ 1/2 where a = 0 if χ Dˆ is even and a = 1 if it is odd. We consider these cases separately. 6.3.1. χ even In this case χ D (hence χ Dˆ ) is even, and k g too, so using (A.15) we have i kg −k 0(s)0(s + 1/2) 0(s) = . π cos(π(s − |k − k g |/2)) 0(1/2 − s)

(6.11)

Applying (6.10), the integral in (6.9) simplifies and, after making the change of variable s ↔ −s and using (4.8), we obtain ˆ G(χ Dˆ )λg ( D) λg (`) ϕ(q) p MgO D (`) = − √ χ D (−q) q ` Dˆ X ϕ(D1 ) D λg (D2 ) × χ Dˆ (D1 ) ηg √ D1 D1 D2 D1 D2 =D/ Dˆ

(4π 2 )µ 0g (1/2 − µ) H (s) 2 −µ 0 (1/2 + µ) g g (−3) (4π ) D ` −s ds 2 × L(χ Dˆ , 1 + 2s)L D2 (χ Dˆ , −2s)−1 . s+µ Dˆ D1 q 1 × 2πi

Z

(6.12)

148


6.3.2. χ odd In this case χ D (hence χ Dˆ ) is odd, and k g too, so using (A.15) we have i kg −k 0(s + 1/2)0(s) 0(s + 1/2)0(s) = i kg −k+|k−kg |−1 π cos(π(s − |k − k g |/2)) π sin(πs) 0(s + 1/2) = i |k−kg |−1 . 0(1 − s) Again using (6.10), the integral in (6.9) simplifies and we find that MgO D (`) equals the right-hand side of (6.12) multiplied by −i kg −k+|k−kg | = ±1. 6.4. When g is a Maass form Here χ Dˆ is even and k g = 0. Using [EMO2, Chap. 6, Sec. 8, (33)], we obtain (using 0 < <e s 6 3/4 < 1/2 + (k − 1)/2 − |<e ir |) H (s) =

−π2−2s h B(2s, 1/2 − s + (k − 1 + 2ir )/2) sin(πir ) 0(1/2 + s + (k − 1 − 2ir )/2)0(1/2 + s − (k − 1 − 2ir )/2) i B(2s, 1/2 − s + (k − 1 − 2ir )/2) − , 0(1/2 + s + (k − 1 + 2ir )/2)0(1/2 + s − (k − 1 + 2ir )/2)

where B(x, y) = 0(x)0(y)/ 0(x + y). Taking a common denominator, applying (A.15) to the numerator, and transforming some cosines in the denominator into products of 0-functions through (A.15), we get k 0g (1/2 − s) H (s) = 21−2s 0(2s) cos π s − 2 0g (1/2 + s) √ 0g (1/2 − s) 0(s) = ik π . 0(1/2 − s) 0g (1/2 + s) This leads to (6.12) after applying the procedure given after (6.11).


149

6.5. End of the computations We use the equalities below, which follow from Proposition A.1. Recall that we have ˆ where Dˆ is the conductor of χ D : the decomposition D = D1 D2 D, ˆ = λg ( D) ˆ −1 , ˆ = 1 so λg ( D) |λg ( D)| p p p −1 |λg (D2 ) D2 | = 1 so λg (D2 ) D2 = λg (D2 ) D2 , ˆ G(χ Dˆ )λg ( D) ˆ p = ηg ( D), 1 = χ Dˆ (D1 )ηg (D1 )2 , ˆ D D ˆ ηg = χ Dˆ (D2 )ηg (D2 )ηg ( D), D1 λg (D2 ) µ(D2 )ηg (D2 ) ˆ 2 = ηg (D)2 . = , χ Dˆ (D1 D2 )ηg (D1 )ηg (D2 )ηg ( D) √ D2 D2 We plug these formulae into (6.12); after some straightforward computations, we obtain, using (4.5) and (4.18), Z λg (`) ϕ(q) 1 OD M (`) = −εµ (g) √ Hg (s)L(χ Dˆ , 1 + 2s) q 2πi (−3) ` X ` −s ds ϕ(D1 ) µ(D2 )χ Dˆ (D2 ) Y 2s 1 − χ ( p) p × . ˆ D ˆ s+µ D11−s D21+s Dq D1 D2 =D/ Dˆ

p|D2

The arithmetic function inside the integral equals Y p − 1 χ ˆ ( p) D s Y χ Dˆ ( p) 2s D 1 − − 1 − χ ( p) p = , Dˆ p 1−s p 1+s p 1+2s Dˆ

p|D/ Dˆ

p|D/ Dˆ

so we have M

OD

Z ` −s ds λg (`) ϕ(q) 1 (`) = −εµ (g) √ Hg (s)L(χ D , 1 + 2s) q 2πi (−3) qD s+µ ` λg (`) ϕ(q) 1 ` −s = ress=0,−µ εµ (g) √ Hg (s)L(χ D , 1 + 2s) q s+µ Dq ` Z −s λg (`) ϕ(q) 1 ` ds Hg (s)L(χ D , 1 + 2s) . − εµ (g) √ q 2πi (3) Dq s+µ ` (6.13)

Note that there may be another pole at s = 0 if χ D is trivial. Note also that the second term of (6.13) is very similar to the diagonal term Z ` −s ds 1 Hg (s)L (q) (χ D , 1 + 2s) . (6.14) −εµ (g)MgD (`) = −εµ (g) 2πi (3) Dq s+µ

150


More precisely, since L (q) (χ D , 1 + 2s) −

ϕ(q) L(χ D , 1 + 2s) q Y χ D (q) Y 1 = 1 − 1+2s − L(χ D , 1 + 2s), 1− q q p|q

p|q

the difference between the second term of (6.13) and (6.14), after shifting the contour in the integrals √ for the V−’s to <e s = > 0, is at most O (1 + B − |t|) q (|λg (`)|/(q `)) , where q is the smallest prime divisor of q. 6.2 Let qˆ be the conductor of χq , and let q − be the smallest prime factor of q. If qˆ 6= 1, then MgO D (`) = 0. Otherwise, we have PROPOSITION

` −s 1 λg (`) ϕ(q) ress=0,−µ Hg (s)L(χ D , 1 + 2s) MgO D (`) = εµ (g) √ q Dq s+µ ` σg (`) B − εµ (g)MgD (`) + O,g 1 + |t| q √ . q− ` Moreover, assuming that the λg (n) satisfies the Ramanujan-Petersson bound, p B q` ` ˆ D OD 1+ . Mg (`) = Mg (`) + Mg (`) + O,g 1 + |t| q q q 7. The second moment In this section we compute the twisted second partial moment Mg,g0 (µ, µ0 ; `) defined by (4.16). Recall that we have to assume either that ` is coprime with q or that Sk∗ (q, χq ) = Bk (q) so that λ f (`) is well defined for all f ∈ Bk (q). As with the first moment, it is useful to consider for xE = (x1 , . . . , x` , . . . , x L ) ∈ C L the linear form X x) = x` Mg,g0 (`). (7.1) L g,g0 (E `6L (`,q)=1

From our hypothesis on `, we may apply the Hecke recursion formula (3.3) for f , followed by (3.4) for g 0 (since g 0 is primitive, (3.4) holds without restriction on m, n), and we obtain X χq (d) X µ(a)χ D 0 (a) Mg,g0 (`) = λg0 (b) d 1/2 a 1/2 de=`

ab=d

X λg (m)λg0 (n) m adn Xh × Vg Vg0 ψ f (m)ψ f (aen). (7.2) qD q D0 (mn)1/2 m,n>1

f


151

D (`) + M N D (`) We apply Petersson’s formula (3.5) to obtain Mg,g0 (`) := Mg,g 0 g,g 0 with X χq (d) X µ(a)χ D 0 (a) D λg0 (b) Mg,g 0 (`) = d 1/2 a 1/2 ab=d de=` X λg (aen)λg0 (n) aen adn Vg Vg0 , × qD q D0 (ae)1/2 n n X 1 X µ(a)χ D 0 (a) ND −k Mg,g λg0 (b) √ 0 (`) = 2πi d 1/2 a ab=d de=` X 1 X λg (m)λg0 (n) × √ c m,n mn c≡0(q) 4π √aemn m adn 0 Vg S(m, aen; c)Jk−1 . × Vg qD q D0 c

Applying (3.4) in the reverse direction, we have (by (4.13), (3.9), and (3.10)) 1 X X λg (en)λg0 (dn) en dn D Vg Vg0 Mg,g 0 (`) = n q D q D0 `1/2 de=` n σg (`) = O,g,g0 (1 + |t| + |t 0 |) B q 1/2 . `

(7.3)

To save notation, we now assume that |t| > |t 0 |. 7.1. The nondiagonal main term We next evaluate M N D (`) following the methods used in [DFI1] and [KMV1]. We evaluate the sum precisely for restricted ranges of the variables m, n, c, and the remaining ranges are bounded using the large sieve inequality of Section 5.1. As in Section 6, we define m dan 1 FM,N (m, n) := Vg0 η M (m)η N (n) V g qD q D0 (mn)1/2 and F(m, n) :=

m dan X 1 0 V V = FM,N (m, n). g g 0 qD qD (mn)1/2 M,N >1

We also define TM,N (c) := c

X

λg (m)λg0 (n)S(m, aen; c)FM,N (m, n)Jk−1

m,n

TM,N :=

X 1 TM,N (c), c2 q|c

4π √aemn c

, (7.4)

152


so that X

M N D (`) = 2πi −k

de=`

1 d 1/2

X µ(a)χ D 0 (a) X 0 (b) λ TM,N . g a 1/2

ab=d

(7.5)

M,N

Note that the derivatives of FM,N satisfy the bounds xi y j

A q D 0 A0 B ∂i ∂ j −1/2 i+ j q D F (x, y) 1 + |t| (M N ) (log q) (7.6) M,N ∂i x ∂ j y x day

for all i, j, A, A0 > 0. By taking i = j = 0 and either A or A0 large in (7.6), we have X B −A ND Mg,g q 0 ,M,N ε,A,g,g 0 1 + |t| M+N q 1+ε

for any ε > 0 and any A > 0. Thus we may assume that M 6 (q D)1+ε ,

N 6 (q D 0 )1+ε .

(7.7)

It also proves convenient to remove large values of c through the large sieve inequality (see Proposition 5.1) together with (3.8), which implies that, for M, N q 1+ε , X q|c,c>C

B `3/4 (D D 0 )1/2 (M N )1/4 1 TM,N (c) ε 1 + |t| q ε . 2 c C 1/2

(7.8)

Note also that one can do slightly better when averaging over `. Namely, using also (3.7) to bound λg0 (b), the corresponding term for L g,g0 (E x ) (see (7.1)) is bounded by X `6L (`,q)=1

x`

X de=`

1 d 1/2

X µ(a)χ D 0 (a) X λg0 (b) 1/2 a

ab=d

q|c,c>C

1 TM,N (c) c2

B L 3/4 (D D 0 )1/2 (M N )1/4 ε 1 + |t| q ε ||E x ||2 . (7.9) C 1/2 Accordingly, in Sections 7.2 and 7.3 we attach a smooth compact function ηC (c) vanishing for c > 2C and equal to one for c 6 C. 7.2. Applying the summation formula We apply the summation formula in Section A.5 on the m-variable with the effect of splitting − + OD TM,N (c) = TM,N (c) + TM,N (c) + TM,N (c) (7.10)


153

with (see (6.3)) ηg (D2 ) OD TM,N (c) = δχq =εq |χ Dˆ (c)|ϕ(c)χ D2 (−c) √ D2 X − × λg D2 (aen D2 )λg0 (n)G (aen, n), n ± TM,N (c)

ηg (D2 ) = χ D2 (±c) √ D2 m X , n , (7.11) × λg D2 (m)λg0 (n)G χq D1 (aen ± m D2 ; c)G ± D2 m,n Z

m6 =±aen D2 4π √zx ∞

4π √aex y

FM,N (x, y) d x, c c Z ∞ √ 4π √aex y 4π zx G + (z, y) := Kg Jk−1 FM,N (x, y) d x. c c 0 G − (z, y) :=

Jg

Jk−1

0

(7.12)

± 7.2.1. Treatment of TM,N (c) ± We rewrite TM,N (c) as

ηg (D2 ) X ± G χq D1 (±h D2 ; c)Th± (c) TM,N (c) := χ D2 (±c) √ D 2 h6=0 with Th± (c) =

X m,n m±aeD2 n=h

λg D2 (m)λg0 (n)G ±

m ,n . D2

Following [DFI1], we set P := 1 + c−1 (aeM N )1/2 , Y = aeN , Z := c2 P 2 M −1 , and f (z, y) = G ± (m/D2 , y/ae). By integrating by parts (cf. [DFI1, p. 229]), one has for all A, i, j > 0, z −A y −A (ae)1/2 M i+ j−3/2 z i y j f (i j) f (z, y) A,i, j 1 + P . 1+ Z Y c We evaluate expressions like Th± (c) in Proposition B.1, which gives p + − TM,N (c) + TM,N (c) = O,g,g0 (1 + |t|) B qˆ Z (ae P)3/4 N 1/4 M 1/2 c p 3/4 11/4 1/4 −1/2 2+ = O,g,g0 (1 + |t|) B q(ae) ˆ P N M c . Remark. Note that at this point there is a major simplification compared with the treatment given in [DFI1] or [KMV1] (and also the forthcoming [MV]). In these + − papers, g is an Eisenstein series, with the effect that TM,N (c) + TM,N (c) contains a main term (coming from the constant coefficient of g) thus contributing to the second moment.

154


Setting qˆ = q β and summing over c, we bound the contribution of these terms by p N 1/4 C p 17/8 M 7/8 N 13/8 q` ˆ 3/4 1/2 + q` ˆ q M q 11/4 B ε 3/4 −(1−2β)/12 1 + |t| q ` q + `17/8 q −(1−2β)/4

ε,g,g0 1 + |t| ε

B

qε

(7.13)

on choosing C = M 1/2 q 2/3(1−β/2) . The first error term agrees with the one in (7.8). If we now consider the corresponding terms for the average L g,g0 (E x ), then their contribution is bounded by p p N 1/4 C M 7/8 N 13/8 ||E x ||1 qˆ L 3/4 1/2 ||E x ||1 + qˆ L 17/8 11/4 q M q

ε 1 + |t|

B

qε

ε 1 + |t|

B

q ε L 3/4 q −(1−2β)/12 ||E x ||2 ||E x ||1 2/3

1/3

+ L 17/8 q −(1−2β)/4 ||E x ||1

(7.14)

on choosing C = M 1/2 q 2/3(1−β/2) (||E x ||2 /||E x ||1 )2/3 so that the first error term equals that of (7.9). Note that, to beat the convexity bound by a positive power with our current methods, the conductor of χq has to be smaller than q 1/2−δ for some positive δ. We suspect that this can be improved through more careful analysis of the cancellation between various Gauss sums, but for the purposes of this paper such a restriction is acceptable. O D (c) are This finishes our estimate if χq is nontrivial, since in that case the TM,N zero. From (7.14) we obtain for nontrivial χq the equality D B ε 3/4 −(1−2β)/12 Mg,g0 (`) = Mg,g q + `17/8 q −(1−2β)/4 ) , 0 (`) + Oε,g,g 0 (1 + |t|) q (` D (`) is defined in (7.3), while from (7.1) we have where Mg,g 0

L g,g0 (E x) =

X

D x` Mg,g 0 (`)

`6L (`,q)=1

+ Oε,g,g0 (1 + |t|) B q ε (L 3/4 q −(1−2β)/12 ||E x ||2 ||E x ||1 + L 17/8 q −(1−2β)/4 ||E x ||1 ) . 2/3

1/3

(7.15)

7.3. Expanding the c-sum Assume now that χq is the trivial character. We concentrate on OD TM,N :=

X q|c ˆ (c, D)=1

1 OD T (c), c2 M,N

where the sum is performed over c 6 C. We first add back the terms from large c at an admissible cost. To be specific, the bounds Jg (x), K g (x) log(x), Jk−1 (x) x


155

O D (c) (1 + |t|) B q `1/2 M N , so imply that TM,N

X q|c,c>C

B `1/2 B `1/2 M N 1 OD 0 1 + |t| 0 1 + |t| q 1/6 . T (c) q ,g,g ,g,g M,N Cq c2 q

Next we consider T O D :=

X

OD TM,N ,

M,N

where the sum is over the indices such that M 6 (q D)1+ , N 6 (q D 0 )1+ . Again, we may reinsert large values of M and N at a cost which is O A ((1 + |t|) B q −A ) for all A > 0. Thus, up to an admissible term, T O D can be rewritten like (7.11) but with FM,N (x, y) replaced by η(x)η(y)F(x, y) in (7.12). We can replace η(y) by 1 since it leaves the n-sum unchanged. We can also replace η(x) by 1 at an admissible cost, namely, the size of the term obtained by replacing η(x) by 1 − η(x), which is O ((1 + |t|) B q `1/2 (D 0 /q)). Thus we have, up to admissible error, 2πi −k T O D =

X λg0 (n) adn MgO D (aen), √ Vg0 0 q D n n

where MgO D (`) is the nondiagonal term defined in (6.5). From (7.5), the equality immediately above, (7.13), and the Hecke recursion formulae (3.3) and (3.4), we obtain that the nondiagonal term is given by OD B ε 3/4 −(1−2β)/12 M N D (`) = Mg,g q + `17/8 q −(1−2β)/4 ) , 0 (`) + Oε,g,g 0 (1 + |t|) q (` where OD Mg,g 0 (`) =

X de=`

1 X λg0 (dn) dn O D Vg0 Mg (en). √ q D0 d 1/2 n n

(7.16)

Applying the bound (6.7) gives OD B σg (`) , Mg,g √ 0 (`) = O,g,g 0 (1 + |t|) q ` and adding this to (7.3), we obtain, in view of (7.13), (7.14), and (7.15), the following proposition. 7.1 Let D be a square-free integer, and denote by qˆ = q β the conductor of χq . For L > 1 an integer and xE = (x1 , . . . x L ) ∈ C L , let L g,g0 (E x ) be the linear form defined in (7.1). PROPOSITION

156


We have B L g,g0 (E x ) q ε 1 + |t| + |t 0 | X σg (`) 2/3 1/3 x ||2 ||E x ||1 × |x` | 1/2 + L 3/4 q −(1−2β)/12 ||E ` `6L + L 17/8 q −(1−2β)/4 ||E x ||1

(7.17)

for all ε > 0, the implied constants depending only on ε, k, g, g 0 . In particular, for ` 6 q (1−2β)/15 coprime with q, we have Mg,g0 (`) 1 + |t| + |t 0 |

B

σg (`) qε √ . `

Before continuing our evaluation of the second moment under somewhat restrictive conditions, we derive a generalization of the main result of [DFI1]. 7.4. Breaking convexity for Rankin-Selberg convolutions We now use Proposition 7.1 to bound L( f ⊗ g, s) and its derivatives along the critical line. THEOREM 7.2 Let g be cuspidal with square-free level D, and let qˆ = q β be the conductor of χq . Then for all f ∈ Sk∗ (q, χq ), t ∈ R, and j > 0,

B 1 ( j) f ⊗ g, + it ε 1 + |t| q 1/2−(1−2β)/80+ε L 2 for all ε > 0, the exponent B being absolute and the implied constant depending on k, g, ε. Proof We use the amplification technique of [DFI1]. Consider a sequence of real numbers xE := (x` )`6L indexed by the integers coprime with q. For µ = δ + it with |δ| 6 1/ log q, we consider the quadratic form Q (E x ) :=

Xh f ∈Sk∗ (q,χq )

X 2 2 1 x` λ f (`) L f ⊗ g, + µ . 2 `6L (`,q)=1

We want an upper bound for this quadratic form. Using the functional equation, we


157

have X λ (m)λ (m) m 2 2 1 g f Vg L f ⊗ g, + µ 1/2 2 qD m m X λ (m)λ (m) m 2 g f . + Vg 0 1/2 q D m m Since we need only an upper bound, we can extend the sum to all f ∈ Bk (q): 2 Xh X x` λ f (`)

Q (E x)

f ∈Bk (q)

`6L (`,q)=1

n X λ (m)ψ (m) m 2 X λ (m)ψ (m) m 2 o g f g f × Vg + Vg . 0 0 1/2 1/2 q D q D m m m m Using (`, q) = 1, along with (3.3), we have X 2 X X x` λ f (`) = `6L (`,q)=1

Now for ` 6 L 2 we define X ` =

xd`1 χ q (`2 )xd`2 λ f (`1 `2 ).

`1 ,`2 6L/d (d`1 `2 ,q)=1

d

P P d

`1 `2 =`

x d`1 χ q (`2 )xd`2 so that

Q (E x ) L g,g ( XE ) + L g,g ( XE ).

Hence, from Proposition 7.1 applied to (g, µ), (g 0 , µ0 ) = (g, µ), and from the estimates X 2 X X X σg (`) |X ` | 1/2 L ε |x` |2 , |X ` | L ε |x` | , ` 2 2 `6L

`6L

X `6L 2

|X ` |2 L ε

`6L

`6L

X

|x` |2

2

,

`6L

we obtain for each f ∈ Sk∗ (q, χq ), X 2 2 1 x` λ f (`) L f ⊗ g, + µ Q (E x ) (1 + |t|) B ( f, f )−1 2 `6L (`,q)=1

× q ε ||E x ||22 + L 3/2 q −(1−2β)/12 ||E x ||2 ||E x ||1 4/3

2/3

+ L 17/4 q −(1−2β)/4 ||E x ||21 .

We finish the proof by choosing (similarly to [DFI1, p. 236]) the following for (x` ): we take   if ` = p 2 , p a prime 6 L 1/2 , −χ q ( p)  x` =

χ q ( p)λ f ( p) if ` = p a prime 6 L 1/2 ,   0 else.

158


The x` satisfy X 2 x` λ f (`) q −ε L

and

||E x ||1 + ||E x ||22 L 1/2 .

`6L (`,q)=1

Using this for L = q 1/20 and the upper bound ( f, f ) k q log3 q, we obtain the theorem in the case when j = 0 and for the variable s = 1/2 + µ in a (1/ log q)neigborhood of the critical line. Cauchy’s formula then provides the bound for 1/2+µ on the critical line for higher derivatives. 7.4.1. Proof of Corollary 1.3 Let ϕ be a smooth test function on [0, +∞[, equal to 1 for 0 6 x 6 1/2 and equal to 0 for x > 1. Let f ∈ Sk∗ (q) as in the statement. Let X = X ( f ) be the largest integer such that λ f (n) = λg (n) for all n 6 X . If X < D, we are finished. Otherwise, we have n X X X X g |λg (n)|2 = λ f (n)λg (n) 6 λ f (n)λg (n)ϕ , X n6X/2

n6X/2

n6X

where the first inequality follows by the fact that the Rankin-Selberg convolution L(g ⊗ g, s) has a simple pole at s = 1, and the others by the nonnegativity of ϕ and of λ f (n)λg (n) for n 6 X . The last sum is estimated by the Mellin transform and a contour shift to <e (s) = 1/2 + δ for any δ > 0; applying the Phragmen-Lindelöf principle with the bound of Theorem 1.1, one finds that n X λ f (n)λg (n)ϕ ϕ,δ X 1/2+δ q (1/2−δ)(1−2γ ) , X n6X

where γ = 1/80 − ε is the exponent improving on the convexity bound for the Rankin-Selberg convolution. It follows that X g,δ q 1−2γ +δ for any δ > 0, which is Corollary 1.3. 7.5. Asymptotic evaluation To prove Theorem 1.7, we need the asymptotics of the second moment, not just an upper bound. Our techniques accomplish this only when Sk∗ (q) = Bk (q), so we add the assumptions that χq is trivial, q is prime, and k < 12. At this point essentially all of the work has been done. It is a matter of combining all the preceding estimates.


159

First by (7.16), along with Proposition 6.2, we have X ϕ(q) OD Mg,g res e−s 0 (`) = εµ (g) s=0,−µ q`1/2 de=`

×

X λg (en)λg0 (dn)

L(χ D , 1 + 2s)Hg (s)Vg0

dn (Dq)s q D0 s + µ

n 1+s X X λg0 (dn) dn − εµ (g) d −1/2 Vg0 MgD (en) 0 1/2 q D n n de=` B + Oε,g,g0 (1 + |t|) q ε−1 , n

(7.18)

and from (7.3) (applied to the pairs (g1 , µ1 ) = (g, −µ), (g2 , µ2 ) = (g 0 , µ0 )) and (6.1) we know that the second term in (7.18) equals D B ε−1 −εµ (g)Mg,g . 0 (`) + Oε,g,g 0 (1 + |t|) q Combining (4.19) with the equality D OD Mg,g0 (`) = Mg,g 0 (`) + M g,g 0 (`)

+ Oε,g,g0 (1 + |t| + |t 0 |) B q ε (`3/4 q −1/12 + `17/8 q −1/4 ) , we see that the various M D -terms cancel each other as well as the possible extra residue at s = 0 for µ 6= 0 (since in that case χ D is trivial and g = g); hence we obtain the following result. THEOREM 7.3 Let g and g 0 be primitive (nonexceptional) cusp forms of square-free level D, D 0 and nebentypus χ D , χ D 0 , respectively. Assume that q is prime, coprime with D D 0 , that χq is the trivial character, and that Sk∗ (q) = Bk (q). Let µ := δ + it, µ0 = δ 0 + it 0 with |δ|, |δ 0 | 6 1/ log q and t, t 0 ∈ R. For any ` < q and any ε > 0, 0 main main (q D)µ (q D 0 )µ Mg,g0 (`) = Mg,g 0 (`) + εµ (g)εµ0 (g )M g,g 0 (`) 0

main main + εµ0 (g 0 )Mg,g 0 (`) + εµ (g)M g,g 0 (`)

+ Oε (1 + |t| + |t 0 |) B q ε (`3/4 q −1/12 + `17/8 q −1/4 ) , where the implied constant depends on ε, k, g, g 0 and where εµ (g) is defined in (4.18), and with Z ϕ(q) 1 dt t main Mg,g0 (`) := 1/2 ress=µ Jg,g0 (s, t)q s+t D s D 0 , 2πi (3) (s − µ)(t − µ0 ) q` Jg,g0 (s, t) := Hg (s)L(χ D , 1 + 2s)Hg0 (t)L (q) (χ D 0 , 1 + 2t)L(g ⊗ g 0 ; `; s, t), L(g ⊗ g 0 ; `; s, t) = νg,g0 (`; s, t)

L(g ⊗ g 0 , 1 + s + t) , L(χ D χ D 0 , 2 + 2s + 2t)

160


where νg,g0 (`; s, t) is the multiplicative function of ` defined by νg,g0 (`; s, t) :=

X de=`

∞ ∞ 1 Y X λg ( p k+v p (d) )λg0 ( p k+v p (e) ) X λg ( p k )λg0 ( p k ) −1 , d s et p k(1+s+t) p k(1+s+t) p|de

k=0

k=0

and the three other main terms are defined following Convention 4.1. Proof of Proposition 1.6 We suppose that g has trivial nebentypus (so g is self-dual) and is holomorphic. Applying Theorem 7.3 for µ = µ0 = 0 and ` = 1, we find that, up to remainder terms of order q −1/12+ε , the second moment equals Z 1 φ(q) ress=0 Hg (s)Hg (t) Mg,g (1) = 4 q 2πi (3) L(g ⊗ g, 1 + s + t) dt × (q D)s+t ζ (D) (1 + 2s)ζ (D) (1 + 2t) (D) . ζ (2 + 2s + 2t) st We can shift the t-contour to <e t = −1/2 without hitting any poles other than at t = 0, so it remains to calculate the asymptotics at that point since the remaining contour contributes as an O(q −1/2+ε ). The residue at s = t = 0 clearly gives a polynomial in log q. The leading term comes from replacing ζ (D) (1 + 2z) by (φ(D)/D)(2z)−1 and L(g ⊗ g, 1 + s + t) by Rg (s + t)−1 , where ζ (s) L(sym2 g, s) Rg = ress=1 L(g ⊗ g, s) = ress=1 ζ D (s) ζ (2s) Q −1 p|D (1 − p ) = L(sym2 g, 1). ζ (2) Since ress,t=0 q s+t /(s 2 t 2 (s + t)) = (log q)3 /3, we can conclude the proof of Proposition 1.6. 8. Quadratic forms and linear independence In this section we restrict to the case when χq is trivial, q is prime, and k < 12, so Bk (q) = Sk∗ (q) and ε( f ⊗ g) = ε(g). We restrict to the case when µ = 0 and g has real coefficients (thus ε(g) = ±1 and χ D is real), and we assume that the coefficients λg (n) satisfy the Ramanujan-Petersson bound (3.6). We use the results of Sections 5 and 6 to analyze the behavior of certain quadratic forms related to the values L( f ⊗ g, 1/2). This in turn allows us to prove a series of linear independence results similar to those of [V2].


161

8.1. When ε(g) = 1 We compute the quadratic form Xh

Q (E x ) :=

f ∈Sk∗ (q)

2 1 X x` λ f (`) . L f ⊗ g, 2 `6L

We show that, under certain restrictions, this is positive definite. 8.1 For any fixed 1/2 > > 0, for L 6 (q/D 5 )1/2− , and for q sufficiently large (depending on ε, g), the quadratic form Q (E x ) is positive definite. THEOREM

Proof For the moment we assume that χ is nontrivial. We have ` ` X 1 2 λ f (`1 )λ f (`2 ) = εq (d)λ f , d2 d|(`1 ,`2 )

so that X

Q (E x) =

εq (d)

Let

=

L 2,

xd`1 xd`2 M1 (`1 `2 ).

`1 ,`2

d

L0

X

and define, for ` 6

L 0,

x` :=

X

εq (d)

d

X

xd`1 xd`2 ,

`1 `2 =`

so that Q (E x ) = L ( XE ),

0 XE ∈ R L .

This is now a linear form in XE , so we can use the results of Section 5.1 to show that Q (E x ) = Q M (E x ) + Err(E x)

with

L 0 1/4 L 0 1/2 3/4 Err(E x ) q D || XE ||2 , + q q 1/2 D 1/2 the implied constant depending on ε, g, and X X xd` xd` 1 2 Q M (E x ) = 2L(χ D , 1) εq (d) λ (` ` ). 1/2 1/2 g 1 2 d `1 ,`2 `1 `2

By the Hecke recursion, λg (`1 `2 ) =

X b|(`1 ,`2 )

χ D (b)µ(b)λg

` ` 1 2 λg , b b

(8.1)

162


so Q M (E x ) = 2L(χ D , 1)

X X λg (`) 2 ν(k) xk` 1/2 := 2L(χ D , 1) ν(k)yk2 `

X

`

k

k

with ν(k) =

X

εq (d)µ(b)

bd=k

ε2 (b) Y 1 = 1− if k < q (since q is prime), b p p|k

and yk =

X

xk`

`

λg (`) . `1/2

In other words, we have diagonalized Q M (E x ), which is now clearly positive definite since L(χ D , 1)ν(k) is positive for all k. It remains to bound Err(E x ) in terms of Q M (E x) to prove that Q (E x ) is positive definite. We have X

|x` |2 6

`6L 2

X X X `

2 2 X X |xd`1 ||xd`2 | L |xd`1 ||xd`2 |

`1 `2 =` d

L

XX

`1 ,`2

|xd1 `1 |2

X

`1 ,`2 d1

|xd2 `2 |2 L

d

X

|x` |2

2

.

`

d2

By Möbius inversion, x` =

X

yk`

k

(λ−1 g )(k) k 1/2

,

where λ−1 g is the convolution inverse of λg : (λ−1 g )(k) =

X

µ(m)λg (m)χ D (n)µ2 (n).

mn 2 =k

Thus X

|x` |2 6

X X `

`6L

6

(λ−1 g )(k) 2

k6L/l

X X `

|yk` |

k6L/`

|yk` |2

k 1/2 X |(λ−1 )(k)|2 g k6L/`

k

q

X k

by (3.8). Combining this with (8.1), we have X L 2 1/4 L Err(E x ) q 1/2 1/2 + D 3/4 |yk |2 . q q D k

|yk |2


163

Returning to Q M (E x ), since χ D is real, we have L(χ D , 1) D −1/2 (of course, we could use L(χ D , 1) > c()D − , but we prefer the results to be effective) so that X Q D (E x ) q D −1/2 |yk |2 , k

which finishes the proof of Theorem 8.1 if χ D is nontrivial. If χ D is trivial, the analysis of the quadratic form is essentially that of [V2] and we do not repeat the details here. 8.2. When ε(g) = −1 Now we consider the case when ε(g) = −1 and χ D is nontrivial. The functional equation then implies that L( f ⊗ g, 1/2) = 0, so we must consider the first derivative to get interesting results. The corresponding quadratic form is then Q 0 (E x ) :=

Xh f ∈Sk∗ (q)

2 1 X L 0 f ⊗ g, x` λ f (`) . 2 `6L

THEOREM 8.2 For any fixed 1/2 > > 0, for L 6 (q/D 5 )1/2− , and for q large enough (depending on ε, g), the quadratic form Q 0 (E x ) is positive definite.

Proof We can again use the functional equation to get X λ f (n)λg (n) n 1 L 0 f ⊗ g, =2 Wg 2 qD n 1/2

(8.2)

n>1

with

1 Wg (y) := 2πi

Z (3)

Hg (s)L(χ, 1 + 2s)y −s

ds . s2

Note that, by the usual argument, for all A > 0 we have Wg (y) A y −A and

y −s + O τ (D) Dˆ 1/2 y 1/2 . 2 s Following the methods of Section 8.1, we can again write Wg (y) = ress=0 Hg (s)L(χ, 1 + 2s)

M

Q 0 (E x ) = Q 0 (E x ) + Err(E x)

(8.3)

164


with Err(E x ) satisfying the same bound as before and M Q 0 (E x ) = L(χ, 1) log q D + 2L 0 (χ, 1) + Hg0 (0)L(χ, 1) R (E x ) − L(χ , 1)S (E x ), where R (E x ), S (E x ) are as above. Thus X q D 0M 0 Q (E x ) > L(χ, 1) log 1 + o(1) + 2L (χ, 1) ν(k)yk2 . L2 k

From the Hadamard factorization theorem in [D] for L(χ, s), we have L0 1 (χ, 1) > − log D + O(1) L 2 so that M

Q 0 (E x ) > L(χ, 1) log

q X 1 + o(1) ν(k)yk2 , L2 k

and we finish as before. Remark. There are stronger variants of Theorems 8.1 and 8.2 which can be obtained along the same lines. It is possible to replace L( f ⊗g, 1/2) by L( f, 1/2)L( f ⊗g, 1/2) to get the same result at the cost of sharpening the condition L (q/D 5 )1/2− to L (q/D 5 )1/4− . Using Theorem 7.3, it is also possible to replace L( f ⊗ g, 1/2) by L( f ⊗ g, 1/2)L( f ⊗ g 0 , 1/2) with the stronger condition that L ,g,g0 q 1/30− . 8.3. Linear independence results We now use Theorems 8.1 and 8.2 to prove the linear independence of the actions of Hecke operators in various settings. The general situation is the following: we have a complex Hermitian vector space (V, (, )) equipped with a linear action of the Hecke algebra T , generated by {T1 , . . . , T` , . . .}(`,q)=1 , which is symmetric with respect to the inner product. Suppose that there is an element eg ∈ V with f -eigencomponents eg, f satisfying the orthogonality relations 1 if ε(g) = 1, (8.4) (eg, f , eg, f 0 ) = δ f = f 0 cg L f ⊗ g, 2 1 (eg, f , eg, f 0 ) = δ f = f 0 cg L 0 f ⊗ g, if ε(g) = −1, (8.5) 2 for some positive cg > 0. In the settings we consider, the fact that (eg, f , eg, f 0 ) = 0 for f 6 = f 0 follows immediately from the symmetry of the action of the T` and the strong multiplicity one theorem. THEOREM 8.3 For any fixed 1/2 > > 0, for L 6 (q D −5 )1/2− , and for q large enough, the vectors T1 eg , T2 eg , . . . , TL eg are linearly independent.


165

Proof Suppose that there exists xE = (x1 , . . . , x L ) ∈ C L such that X x` T` eg = 0V . `

Then for each f ∈ Sk∗ (q, χq ), X

x` T` eg, f = 0V =

`

X

x` λ f (`)eg, f ,

`

and taking the inner product of this last vector with itself, and summing over f , gives Q (E x) = 0 =

2 X 1 X L f ⊗ g, x` λ f (`) . 2 f

`6L

Since λ f (`) and L( f ⊗ g, 1/2) are real numbers, Theorem 8.1 implies that xE = 0 if L is small enough. √ Theorem 8.3 may have interesting arithmetic interpretations. Let K := Q( −D) be an imaginary quadratic field of discriminant D, let χ D = (−D/∗) be the associated Kronecker symbol, and let HK be the Hilbert class field of K . As discussed in the introduction, given a character χ˜ of Gal(HK /K ), we can associate to it a theta-function gχ˜ (z) that is a modular form of weight one on 00 (D) with nebentypus χ K . The sign of the functional equation of L( f ⊗ gχ˜ , s) is ε(g) = χ K (−q) = −χ K (q). When every prime factor of q splits in K , the above axiomatic applies to eg = e K ,χ˜ , the χ˜ eigencomponent of a Heegner divisor in J0 (q)(HK ). In this case formula (8.5) is the celebrated formula of Gross and Zagier [GZ]. Theorem 8.3 thus implies Theorem 2.1. If q is inert in K (see [G], [BD]), the vector space V is defined to be M ⊗C, where M is the free finite Z-module of degree-zero divisors supported on the set of supersingular points in the fiber (of bad reduction) of X 0 (q)(Fq ). (This fiber is formed by the union of two projective curves intersecting transversally at the supersingular points.) The action of the Hecke algebra extends to this fiber, and the Hecke module M is equipped with a nondegenerate inner product that gives (, ) on V . Alternatively, M can be described as the character group of the toric part of the fiber at q of the Jacobian J0 (q), the inner product becoming the monodromy pairing. Associated to K is a finite set of Heegner divisors {e K } living on M which are acted on by Gal(HK /K ). For any e K , the χ-eigencomponent ˜ e K ,χ˜ can be taken as our eg of Theorem 8.3. In this case formula (8.4) was proven by Gross [G]. In higher weight (k > 2), there is a similar theory and there are similar formulae which involve higher-dimensional Sato-Kuga varieties instead of X 0 (q). The analogue of the Gross-Zagier formula is due to S. Zhang [Z1], [Z2]. In this setting

166


the inner product is not known to be positive (and L 0 ( f ⊗ g, 1/2) is not known to be nonnegative), but we do not use the individual positivity in our argument, so the linear independence of the image of the corresponding Heegner cycles still holds. 9. Mollification In this section we compute the mollified moments and thus prove Theorem 1.7. We assume for the moment that L( f ⊗ g, 1/2) is not identically zero (i.e., if g is real, then ε(g) = 1), so we wish to evaluate Mg :=

Xh f ∈Sk∗ (q)

and Mg,g :=

1 M( f ⊗ g) L f ⊗ g, 2

Xh 1 2 L f ⊗ g, |M( f ⊗ g)|2 2 ∗

f ∈Sk (q)

for a particular mollifier. Since the exact proportion of nonvanishing for L( f ⊗g, 1/2) (or its derivatives) is not terribly important, we take a very coarse mollifier. Moreover, since we do not want to assume the Ramanujan-Petersson conjecture in general, we are also going to sieve a certain number of “small” primes. Let C > 1 be a constant Q to be chosen later, and set P = p6C p the product of the primes less than C. For L > 1, not an integer, we define the mollifier Z X µ(`)λ f (`)λg (`) 1 L z dz P M ( f ⊗ g) = 2πi (3) ` z 3 `1/2 `6L ,(`,P)=1

=

X ` L, the integral defining x` vanishes, as is easily seen by shifting the contour to the right. Since our focus in this section is usually on poles of holomorphic functions rather than their values, we denote by ν(z 1 , z 2 , . . . , z n ) any function, holomorphic and nonvanishing in the domain <e z i > −1/10, i = 1, . . . , n, which is uniformly bounded in this domain, as are its inverse and all of its low partial derivatives. In particular, we can use this notation for Y ν(z) = L P (g ⊗ g, 1 + z) := L p (g ⊗ g, 1 + z), p|P

the product over small primes of the local factors of the Rankin-Selberg convolution L-function. By (3.6) and (3.7), the above function satisfies the required properties in every domain of the form <e z > −4/5 + for > 0. In the sequel, the value of the ν-function may change from one line to another.


167

9.1. The first mollified moment Shifting the z-contour in (9.1) to <e z = , we find by (3.8) that X `6L ,(`,P)=1

|x` |2 g, L . `

By (4.6) and Proposition 5.2 (for µ = 0), Mg = Mgmain + ε(g)Mgmain + Oε,g (L 1/4 q −1/4+ ),

where Mgmain = ress=0 = ress=0

Hg (s)L(χ D , 1 + 2s)(q D)s X x` λg (`) s `1+s ` Z X µ(`)λ2g (`) L z dz Hg (s)L(χ D , 1 + 2s)(q D)s 1 . s 2πi (3) `1+s+z z3 (`,P)=1

(9.2) The error term is admissible when L = q 1 for some fixed 1 < 1. We have the factorization X µ(`)λ2g (`) (`,P)=1

`1+s+z

= ν1 (s + z) = ν1 (s, z)

2 , 2 + 2s + 2z) L (P) (χ D L (P) (g ⊗ g, 1 + s + z)

2 , 2 + 2s + 2z) L(χ D , L(g ⊗ g, 1 + s + z)

where for P large enough, ν1 (s, z) is of the type described above. If P is chosen large enough, then |ν1 (0, 0)| is bounded from below by a positive constant depending only on P. 9.1.1. Shifting the z-contour As in [KMV2, Sec. 4], we evaluate Mgmain by calculating the s-residue and deforming the z-contour to the left of the line <e z = 0, so that the main term comes from the residue of the resulting expression at z = 0, the integral along the new contour being negligible. We use the following lemma, which can be proved by combining the modularity of the symmetric square L-function established by Gelbart and Jacquet, the modularity of GL2 × GL2 Rankin-Selberg L-functions established recently by D. Ramakrishnan [Ra] with the method of J. Hadamard and C. de la Vallée-Poussin, and the general theory of Rankin-Selberg L-functions for automorphic forms on GL3 and GL4 (see [Mo]).

168


LEMMA 9.1 Given g as above, there exists cg > 0 such that the functions L(g ⊗ g, 1 + z), L(g ⊗ g, 1 + z), and L(g ⊗ g, 1 + z) have no zeros in the domain n o −cg z, <e z > . log(|=m z| + 2)

Moreover, on the border γ := {z, <e z = −cg /log(|=m z| + 2)}, these functions, their inverses, and all their derivatives up to any order α are bounded in modulus by C g,α,δ (1 + |=m z|)δ for any δ > 0. Let γ denote the contour {z, <e z = −cg /log(|=m z| + 2)}.∗ By shifting the zcontour of (9.2) to γ , we encounter a√pole at z = 0 and a contour integral that, 0

by Lemma 9.1, is bounded by Og (e−cg since the residue is a power of log q. So the main term of (9.2) is ress,z=0

log L

). This turns out to be a negligible term

2 , 2 + 2s + 2z)L z ν1 (0, 0)Hg (s)L(χ D , 1 + 2s)(q D)s L(χ D . s L(g ⊗ g, 1 + s + z)z 3

(9.3)

What is important to us are the highest powers of log q or log L coming from this expression, so in the forthcoming computation we replace any function holomorphic and smaller than any power of log q near (0, 0) by its value at that point. Evaluating this term depends on the number of poles at the origin, which in turn depends on whether χ D is trivial or not. We thus break into two cases. 9.1.2. χ D nontrivial Suppose first that g has complex coefficients. The only poles in (9.3) come from the 1/sz 3 factor, so the main term is 2 , 2) 2 , 2) L(χ D , 1)L(χ D (q D)s L z ν1 (0, 0) L(χ D , 1)L(χ D ress,z=0 = log2 L , L(g ⊗ g, 1) 2 L(g ⊗ g, 1) sz 3 (9.4) the remaining terms being smaller by a power of log q. (Note also the importance of η1 (0, 0) being nonzero.) Putting this into Mgmain , we have

ν1 (0, 0)

X µ(`)|λg (`)|2 ` ∗ We

`1+s+z

= ν1 (s, z)

L(χ D χ D , 2 + 2s + 2z) , L(g ⊗ g, 1 + s + z)

take cg < 1/10 so that we may bound any function of the form ν1 on γ .


169

so that, setting Rg = ress=1 L(g ⊗ g, s), ν1 (0, 0)

L(χ D , 1)L(χ D χ D , 2) (q D)s L z (s + z) ress,z=0 Rg sz 3 = ν1 (0, 0)

2 , 2) L(χ D , 1)L(χ D log L , Rg

which is smaller than (9.4) by a factor of log L. Thus this term can be ignored, and the main term for the first mollified moment is given by (9.4). Similarly, if g = g, the main term of the mollified first moment is given by 2ν1 (0, 0)

2 , 2) L(χ D , 1)L(χ D (q D)s L z (s + z) ress,z=0 Rg sz 3

= 2ν1 (0, 0)

2 , 2) L(χ D , 1)L(χ D log L . (9.5) Rg

9.1.3. χ D trivial If χ D is the trivial character, then g = g and L(χ D , 1+2s) has a simple pole at s = 0. The same sort of calculations as above lead to a main term of 2ν1 (0, 0)

2 , 2) ϕ(D) L(χ D (q D)s L z (s + z) ress,z=0 2D Rg s2 z3 2 , 2) ϕ(D) L(χ D 1 = ν1 (0, 0) log q D log L + log2 L . (9.6) D Rg 2

9.2. The second mollified moment Next we turn to the second moment. We start with the square of the mollifier. By (3.3), we obtain X λ f (`) X λ f (`) |M P ( f ⊗ g)|2 = = x` , (9.7) 1/2 ` `1/2 2 2 ` 0. This is certainly sufficient, so we may ignore the resulting z 1 -contour and deal only with the residues passed during the shift. 9.3.1. The pole at z 1 = −z 2 From (9.14) this pole is simple, so to evaluate it we replace L(g ⊗ g, 1 + z 1 + z 2 ) by Rg and all other z 1 ’s by −z 2 . At this point there are no powers of L remaining, so we are left with the residues at s = t = 0, and for these the main contribution comes only from the Jg,g (s, t)(q D)s+t -portion. (Recall that z 2 is along −γ /2, which is faraway from zero.) We find that the contribution of this pole is Og (log q 3 ) Og (log q)

if χ D is trivial, if χ D is nontrivial.

(9.15) (9.16)

In each case these turn out to be smaller than the poles encountered at z 1 = z 2 = 0. Note that the powers of log q in (9.15) and (9.16) are independent of the power of z, which we have taken for the mollifier (here 3). Had we chosen that power to be 2, these terms would have been the main contribution, which is very inefficient for the mollification. 9.3.2. The pole at z 1 = 0 We are left with the contribution of the pole at z 1 = 0. We evaluate the residue at z 1 = 0; then we shift the z 2 -contour to γ . In the process we meet a pole at z 2 = 0, and the resulting z 2 -integral along γ contributes a negligible error term as above. So main (L) is given by the residue of (9.13) at s = t = z = z = 0. the main term of Mg,g 1 2 The term with the highest power of log q and log L is 2 , 2)|2 res ν2 (0, 0, 0, 0)|L(χ D s,t,z 1 ,z 2 =0

(q D)s+t L z 1 +z 2 L(χ D , 1 + 2s)L(χ D , 1 + 2t)L(g ⊗ g, 1 + z 1 + z 2 )L(g ⊗ g, 1 + s + t) st z 13 z 23 L(g ⊗ g, 1 + s + z 2 )L(g ⊗ g, 1 + t + z 1 )L(g ⊗ g, 1 + s + z 1 )L(g ⊗ g, 1 + t + z 2 )

.

Again, it makes a considerable difference whether χ D is trivial or not, and whether g = g or not. 9.3.3. χ D nontrivial and g 6 = g When χ D is nontrivial, we may replace L(χ D , 1 + 2s)L(χ D , 1 + 2t) by |L(χ D , 1)|2 since they contribute no poles to the expression. We may also replace L(g ⊗ g, 1 + s + z 1 )L(g ⊗ g, 1 + t + z 2 ) by |L(g ⊗ g, 1)|2 , so it is enough to compute ress,t,z 1 ,z 2 =0

(q D)s+t L z 1 +z 2 (s + z 2 )(t + z 1 ) . (s + t)(z 1 + z 2 ) st z 13 z 23


173

The pole at s = 0 is simple, so this equals rest,z 1 ,z 2 =0

log4 L log3 L (q D)t L z 1 +z 2 (t + z 1 ) = + log q D . (z 1 + z 2 ) 8 3 t 2 z 13 z 22

Note that from (9.16) the z 1 + z 2 = 0 pole does not contribute as a main term. main (L) differs. In that case, η This is the first time that the evaluation of Mg,g g,g is replaced by ηg,g , and Jg,g is transformed into Jg,g , so arguing as before, we have a main term of 2 , 2)|2 res ν2 (0, 0, 0, 0)|L(χ D s,t,z 1 ,z 2 =0

(q D)s+t L z 1 +z 2 L(χ D , 1 + 2s)L(χ D , 1 + 2t)L(g ⊗ g, 1 + s + t)L(g ⊗ g, 1 + z 1 + z 2 ) st z 13 z 23 L(g ⊗ g, 1 + s + z 1 )L(g ⊗ g, 1 + t + z 1 )L(g ⊗ g, 1 + s + z 2 )L(g ⊗ g, 1 + t + z 2 )

.

This time L(g ⊗ g, 1 + s + t) has no pole at s + t = 0, so the residue is smaller by a main (L) do not contribute as main factor log L; thus this term and that coming from Mg,g terms. main (L), the main term for the second mollified Adding in the contribution from Mg,g moment is 2ν2 (0, 0, 0, 0)

2 , 2)|2 |L(χ , 1)|2 |L(χ D log4 L log3 L D + log q D . 8 3 |L(g ⊗ g, 1)|2

(9.17)

Comparing this with the square of the modulus of (9.4), we see that the powers of log L and log q match, so the fraction of L( f ⊗ g, 1/2) which does not vanish is at least Xh |M1 (g)|2 |ν1 (0, 0)|2 1> 1 1 + og (1) . M2 (g) ν2 (0, 0, 0, 0) ∗ f ∈Sk (q) L( f ⊗g,1/2)6 =0

In particular, the positive constant 1(|ν1 (0, 0)|2 /ν2 (0, 0, 0, 0)) is bounded from below independent of g. 9.3.4. χ D nontrivial and g = g main (L)-terms are equal, so the main term is In this case all four Mg,g 2 , 2)|2 res 4ν2 (0, 0, 0, 0)|L(χ D s,t,z 1 ,z 2 =0

(q D)s+t L z 1 +z 2 |L(χ D , 1)|2 L(g ⊗ g, 1 + z 1 + z 2 )L(g ⊗ g, 1 + s + t) , 3 3 st z 1 z 2 L(g ⊗ g, 1 + s + z 1 )L(g ⊗ g, 1 + s + z 2 )L(g ⊗ g, 1 + t + z 1 )L(g ⊗ g, 1 + t + z 2 )

174


which reduces to 4ν2 (0, 0, 0, 0)

2 , 2)|2 |L(χ , 1)|2 |L(χ D D Rg2

× ress,t,z 1 ,z 2 =0 = 4ν2 (0, 0, 0, 0)

(q D)s+t L z 1 +z 2 (s + z 1 )(s + z 2 )(t + z 1 )(t + z 2 ) (s + t)(z 1 + z 2 ) st z 13 z 23

2 , 2)|2 |L(χ , 1)|2 |L(χ D D (log2 L + log L log q D). Rg2

(9.18)

Again, from (9.16), the z 1 + z 2 = 0 pole does not contribute as a main term, and the ratio of the square of the modulus of (9.5) to the modulus of (9.18) is bounded from below by an absolute positive constant. 9.3.5. χ D trivial In this case, g = g and we are reduced to ϕ(D) 2 L(χ 2 , 2)2 D 4ν2 (0, 0, 0, 0) 2D Rg2 (q D)s+t L z 1 +z 2 (s + z 1 )(s + z 2 )(t + z 1 )(t + z 2 ) (s + t)(z 1 + z 2 ) s 2 t 2 z 13 z 23 ϕ(D) 2 L(χ 2 , 2)2 D = ν2 (0, 0, 0, 0) D Rg2 log4 L log L(log q D)3 × + log3 L log q D + log2 L log2 q D + , 4 3 × ress,t,z 1 ,z 2 =0

(9.19)

and again this matches the square of (9.6) in magnitude. 9.4. Nonvanishing of the first derivative Finally, if g = g and ε(g) = −1, we have L( f ⊗ g, 1/2) = 0 identically, so we turn to computing the first and second mollified moments of the first derivative: Xh f ∈Sk∗ (q)

L ( f ⊗ g)M( f ⊗ g) 0

and

Xh

L 0 ( f ⊗ g)2 M( f ⊗ g)2 .

f ∈Sk∗ (q)

The case is completely analogous to the one above. Starting from (4.11), all that we do is replace ds/s and dt/t by ds/s 2 and dt/t 2 , which affects the orders of magnitude of the first and second moments in the same way (increasing the order of magnitude of the former by one factor of log q, and increasing the order of the latter by two factors of log q, while changing the constants slightly).


175

9.5. Removing the harmonic weight So far we have proved (1.3) with |{ f ∈ Sk∗ (q), L( f ⊗ g, 1/2) 6 = 0}| 0(k − 1) replaced by ∗ |Sk (q)| (4π )k−1

X f ∈Sk∗ (q) L( f ⊗g,1/2)6=0

1 ( f, f )

and (1.4) with |{ f ∈ Sk∗ (q), L 0 ( f ⊗ g, 1/2) 6 = 0}| 0(k − 1) replaced by ∗ |Sk (q)| (4π )k−1

X f ∈Sk∗ (q) 0 L ( f ⊗g,1/2)6 =0

1 . ( f, f )

In [KMV1, Sec. 6], a procedure is described for passing from harmonic weights to natural weights. This same procedure applies in our case, the key point being the convexity-breaking bound of Theorem 1.1. (In [KMV1] the necessary convexitybreaking bound was provided by Th. 1.2.) From this, one can conclude the proof of Theorem 1.7.

Appendices A. Summation formulae In this appendix we derive formulae for expressions of the type an X λg (n)e F(n), c n where g is one of the forms discussed in Section 3, with nebentypus χ D , a and c relatively prime, and F a smooth function, decaying rapidly at infinity, which vanishes in a neighborhood of the origin. Such formulae do not seem to be in the literature for nontrivial levels (although they are certainly known to several people). We mix the methods of M. Jutila [J] and Duke and Iwaniec [DI], starting first by establishing the analytic continuation and P functional equation of the Dirichlet series L(g, a/c, s) = n>1 λg (n)e(n(a/c))n −s , using the automorphic properties of g. The summation formulae then follow through Mellin inversion. Recall that the action (of weight k) of GL+ 2 (R) on the space of functions on the upper half is detk/2 γ f |γ (z) := f (γ · z), j (γ , z)k

176


where γ = ac db , j (γ , z) = cz + d, and γ · z = (az + b)/(cz + d). The form g satisfies the automorphy equation g|γ (z) = χ D (γ )g(z), where χ D (γ ) = χ D (d). A.1. A review of Atkin-Lehner theory The theory of holomorphic newforms in the classical setting is primarily due to A. Atkin and J. Lehner [AL] in the case of trivial nebentypus, and it was extended by W. Li in [Li1] and [ALi] to the general case. We review these results in this section, noting that they apply equally well to Maass forms. In what follows, let S∗ (D, χ D ) denote either a space of holomorphic forms (in which case ∗ may be replaced by a k) or Maass forms. An important role in this theory is played by the Atkin-Lehner operators, which are defined as follows. Given a factorization D = D1 D2 with (D1 , D2 ) = 1, χ D factors uniquely into χ D = χ D1 χ D2 , where χ Di is a Dirichlet character of modulus Di . For any y ≡ 1(D1 ), x ≡ 1(D2 ), and integers z, w such that D12 xw − Dyz = D1 , the matrix x D1 y z D w D1 defines a linear map W D1 : S∗ (D, χ D1 χ D2 ) =⇒ S∗ (D, χ D1 χ D2 ) which is independent of the choice of w, x, y, and z. In addition, if W D0 1 comes from another matrix of the same form but with no assumption on congruence classes x 0 (mod D2 ) and y 0 (mod D1 ), then W D0 1 = χ D1 (y 0 )χ D2 (x 0 )W D1 .

(A.1)

For (D1 , D2 ) = 1 with D1 D2 |D such that (D1 D2 , D/D1 D2 ) = 1, we have W D2 ◦ W D1 = χ D2 (D1 )W D1 D2 ,

W D1 ◦ W D1 = χ D1 (−1)χ D/D1 (D1 )I

(A.2)

with I the identity operator. Note that if D1 = D, W D is the Atkin-Lehner involution represented by the matrix 0 1 WD = −D 0 These operators act in a convenient way on the space of newforms. To be more precise, we enclose in Propositions A.1 and A.2 the properties of these operators that we use in the sequel.


177

PROPOSITION A.1 ([Li1], [ALi]) If D = D1 D2 with (D1 , D2 ) = 1, the operator W D1 sends newforms to newforms. • For g ∈ S∗ (D, χ D1 χ D2 )∗ ,

g|W D1 =: ηg (D1 )g D1

•

with g D1 ∈ S∗ (D, χ D1 χ D2 )∗ and |ηg (D1 )| = 1. The constant ηg (D1 ) is called the pseudo-eigenvalue of the operator W D1 . If χ D1 is trivial, then W D1 is an endomorphism of S∗ (D, χ) and ηg (D1 ) is a true eigenvalue: g D1 = g. If λg (D1 ) 6 = 0, G(χ D1 ) (A.3) ηg (D1 ) = √ , λg (D1 ) D1 where G(χ D1 ) is the Gauss sum associated with the character χ D1 . Moreover, for D1 D2 |D such that (D1 , D2 ) = 1 = (D1 D2 , D/D1 D2 ) and λg (D1 D2 ) 6= 0, we have ηg (D1 D2 ) = χ D1 (D2 )χ D2 (D1 )ηg (D1 )ηg (D2 ).

•

•

The Fourier coefficients of g D1 are ( χ D1 (n)λg (n) if (n, D1 ) = 1, λg D1 (n) = χ D2 (n)λg (n) if n|D1∞ . If D1 = D, then g|W D = ηg (D)g with g(z) =

P

n>1

(A.4)

g(n)e(nz), ˆ and

λg (n) = χ D (n)λg (n) if (n, D) = 1.

(A.5)

In [ALi] formula (A.3) is given only for D1 a prime power, but it extends to composite D1 by (A.2) and by the multiplicative relation for the Gauss sums: G(χ D1 χ D2 ) = χ D1 (D2 )G(χ D1 )χ D2 (D1 )G(χ D2 ). PROPOSITION A.2 ˆ where Dˆ is the conductor of χ D . For g ∈ S∗ (D, χ D1 χ D2 )∗ and p a Let D = D0 D, prime factor of D, one has • if p|D0 , ( χ Dˆ ( p) p −1 if ( p, D/ p) = 1, 2 λg ( p) = (A.6) 0 if p 2 |D, •

ˆ ˆ |λg ( p)| = 1. if p| D/(D 0 , D),

178


A.2. A factorization lemma LEMMA A.3 Let D > 1 be an integer, and let a, c be two coprime integers. We suppose that D1 := (c, D) is coprime with D2 := D/D1 ; then there exists a matrix of the form ab c d ∈ SL2 (Z), such that the following identity holds (as operators acting on weight k modular forms, not as matrices): 0 a b a b0 1 0 γ = = 0 ◦ W D2 ◦ c d c D d0 0 D2 0 0 for some γ1 = ca0 D db 0 ∈ 00 (D). The integer d 0 satisfies the congruences d 0 ≡ a(D1 ),

d 0 ≡ −c(D2 ).

(A.7)

Proof Since (a, c) = 1 and, (c, D) is coprime with D2 , we may select a matrix moreover, ab y of the form γ = c d with D2 |d. Now pick any matrix W D2 = xzDD2 w D rep2 resenting the Atkin-Lehner operator (in particular, y ≡ 1(D2 ), x ≡ 1(D1 )); then form 1 0 aw D2 − zbD1 −ay + xb −1 γ1 := D2 Id γ · · W = ∈ 00 (D) D2 cw D2 − zd D1 −cy + xd 0 D2−1 by our choice of d. Moreover, d 0 = −cy + xd ≡ d ≡ a(D1 ) and d 0 = −cy + xd ≡ −c(D2 ); hence the lemma follows since scalar matrices act as the identity. A.3. The summation formula: Holomorphic cusp forms Given a, c coprime integers such that D1 = (c, D) is coprime with D2 = D/D1 , in this section and Sections A.4 and A.5 we return to showing that a a X L g, , s := λg (n)e n n −s c c n>1

admits an analytic continuation over C with a functional equation that we describe below. We then obtain the desired summation formula by Mellin inversion. We start with g a holomorphic cusp form of weight k. Take γ ∈ SL2 (Z ) as in Lemma A.3. For any z with <e z > 0, we have, in view of Lemma A.3, z −k/2 . g(γ z) = (cz + d)k g|γ (z) = (cz + d)k D2 χ(γ1 )ηg (D2 )g D2 D2 From (A.7) we have χ(γ1 ) = χ D1 (d 0 )χ D2 (d 0 ) = χ D1 (a)χ D2 (−c),


179

so that z −k/2 g(γ z) = χ D1 (a)χ D2 (−c)ηg (D2 )D2 (cz + d)k g D2 D2 z −k/2 k := κ D2 (cz + d) g D2 . D2

(A.8)

For t ∈ R∗+ , set z t := −d/c + i/(ct), so that cz t + d = i/t, γ z t = a/c + it/c. By Lemma A.3, D2 divides d, so d d D2 dc d D2 a D2 e −n =e −n −n =e −n =e −n , cD2 c D2 c c which in turn implies that g(γ z t ) =

X n>1

a 2πnt λg (n)e n n (k−1)/2 exp − c c

2πn κ X a D2 n (k−1)/2 = ik √ λg D2 (n)e − n exp − t −k . c D cD t D 2 n>1 2 2 (A.9) Since t → g(γ z t ) has exponential decay at zero and infinity, the Mellin transform R∞ s+(k−1)/2 dt/t is analytic on C. For <e s sufficiently large, by the first 0 g(γ z t )t equality in (A.9) we have Z ∞ c s+(k−1)/2 k − 1 a dt g(γ z t )t s+(k−1)/2 = 0 s+ L g, , s , t 2π 2 c 0 which provides the analytic continuation of 0(s + (k − 1)/2)L(g, a/c, s) over C. Multiplying both sides of (A.9) by t s+(k−1)/2−1 with −<e s sufficiently large, then integrating over t, we have c √ D s k − 1 a 2 0 s+ L g, , s 2π 2 c √ c D 1−s k − 1 D2 a 2 , 1 − s , (A.10) = ikκ 0 1−s+ L g D2 , − c 2π 2 the functional equation. Note, in particular, that since the left-hand side is holomorphic for s > (k + 1)/2, the poles of the 0-function on the right-hand side must be canceled by zeros of the L-function. Now let F : R+ → R be smooth, vanish near the origin, and decay (along with ˆ all its derivatives) rapidly at infinity. Let F(s) denote its Mellin transform. Equation

180


(A.10) then gives Z Z a c√ D 1−2s 1 1 2 k ˆ ˆ F(s)L g, , s ds = i κ F(s) 2πi (1) c 2πi (1) 2π 0(1 − s + (k − 1)/2) D2 a × L g D2 , − , 1 − s ds. 0(s + (k − 1)/2) c By Mellin inversion (and a contour shift slightly to the right), the left-hand side equals a X λg (n)e n F(n). c n>1

To evaluate the right-hand side, we shift the contour to <e s = −1 (crossing no poles, by the above argument) and change variables, replacing 2(1 − s) with s. The sum on n is then absolutely convergent, and we can write our sum as √ nD a 1 1 Z 2πi k κ X s 0(s/2 + (k − 1)/2) 2π n −s 2 λ D2 (n)e − Fˆ 1 − ds. √ √ c 2 2πi (4) 2 0(1 − s/2 + (k − 1)/2) c D2 c D2 n>1

Next we open the Mellin transform, with the goal of writing this expression as the integral of a single function against F. There is a slight problem of absolute convergence when k = 1. To avoid it, we deform the line <e s = 4 to the contour joining the ˆ points −1/4+i∞, −1/4+i, 3/4+i, 3/4−i, −1/4−i, −1/4−i∞, replace F(1−s/2) with the integral of F against x −s/2 , and shift the s-contour back to <e s = 3/4. (Note that the shift is justified since we perform a shift in the domain <e s < 1 for which the residual horizontal integral goes to zero as |t| → +∞.) Finally, we use the identity (see [EMO2]) Z x −s 0(s/2 + (k − 1)/2) 1 1 ds for 0 < σ < 1 Jk−1 (x) = 2 2πi (σ ) 2 0(1 − s/2 + (k − 1)/2) to obtain c

a ηg (D2 ) X a D2 λg (n)e n F(n) = χ D1 (a)χ D2 (−c) √ λg D2 (n)e − n c c D 2 n>1 n>1 √ Z ∞ h 4π nx i × F(x) 2πi k Jk−1 √ d x. c D2 0 X

A.4. The summation formula: Maass cusp forms Suppose now that g is a primitive Maass form with nebentypus χ D , with a Fourier development at infinity of the form X g(z) = λg (n)cεg (nx)2|y|1/2 K ir (2π|y|), n>1


181

where c+ (x) = 2 cos(2π x),

c− (x) = 2i sin(2π x).

Since these forms satisfy the same sorts of modular relations as do the holomorphic forms, Atkin-Lehner theory applies to them as well. Note also that since R( f |γ ) = (R( f ))|γ˜ , where γ˜ is the matrix γ with the antidiagonal multiplied by −1, we can use (A.1) to show that, for D2 |D, W D2 g has parity χ D2 (−1)εg . For (a, c) = 1, we first evaluate the sums a X λg (n)c± n F(n), c n>1

of which our desired sum is a simple linear combination. As before, we start by establishing the analytic continuation and functional equation of a a X L + g, , s := λg (n)cεg n n −s , c c n>1 a X a λg (n)c−εg n n −s . L − g, , s := c c n>1

Once again, relation (A.8) is valid (with k = 1), where g D2 is some primitive Maass form of nebentypus χ D1 χ D2 . Using the same z t as before, we get X n>1

a rt t λg (n)c n 2 K ir 2πn c c c εg

=κ

X n>1

λg D2 (n)c

εg D

2

s D2 a 1 1 2 K ir 2π n . (A.11) −n c cD2 t cD2 t

For <e s sufficiently large, we use the identity (see [EMO2, Chap. 6, Sec. 8, (26)]) ∞

Z 0

r

t t dt K ir 2πn t s c c t 1 c s+1/2 s−3/2 s + 1/2 + ir s + 1/2 − ir =√ 2 0 0 2 2 c 2πn

to show that Z ∞ dt g(γ z t )t s t 0 s + 1/2 + ir s + 1/2 − ir a 1 1 = cs π −s−1/2 0 0 L + g, , + s . 2 2 2 c 2

182


The exponential decay of g(γ z t ) at zero and infinity implies the analytic continuation of s + 1/2 + ir s + 1/2 − ir a 0 L + g, , s . 0 2 2 c Arguing as before, we infer the functional equations a D2 a 3+ g, , s = κ3+ g D2 , − ,1 − s , c c a c√ D s s + ir s − ir a 2 3+ g, , s = 0 0 L + g, , s . c π 2 2 c

(A.12)

To get L − , we apply the differential operator (1/(2πi))(∂/∂ x) on both sides of (A.8), getting, after replacement of z by z t and cancellation of some term (since γ ∈ SL2 (Z)), −t

2

X n>1

λg (n)nc

−εg

a rt t K ir 2πn n 2 c c c

s 1 κ X D2 a 1 −εg D 2 2 K ir 2πn . = λg D2 (n)nc −n c cD2 t cD2 t D2 n>1

The same argument as above leads to the functional equations a D2 a ,1 − s , 3− g, , s = −κ3− g D2 , − c c a c√ D s 1 + s + ir 1 + s − ir a 2 3− g, , s = 0 0 L − g, , s . c π 2 2 c

(A.13)

Taking a linear combination, we have a a a 2L g, , s = L + g, , s + L − g, , s c c c c√ D 1−2s h D2 a 2 ,s =κ C + (s) − C − (s) L g D2 , − c π D2 a i ,s + εg D2 C + (s) + C − (s) L g D2 , c with 0((1 − s + ir )/2)0((1 − s − ir )/2) , 0((s + ir )/2)0((s − ir )/2) 0((2 − s + ir )/2)0((2 − s − ir )/2) . C − (s) = 0((1 + s + ir )/2)0((1 + s − ir )/2) C + (s) =


183

This in turn leads to the identity a X 2 λg (n)e n F(n) c n>1

nD a 1 1 2πκ X 2 λ D2 (n)e − = √ c 2 2πi c D2 n>1 Z s + s × Fˆ 1 − C 1− − C− 1 − 2 2 (4) X 2πκ n D2 a 1 1 λ D2 (n)e + ε g D2 √ c 2 2πi c D2 n>1 Z s + s Fˆ 1 − C 1− + C− 1 − × 2 2 (4)

√ s 2π n −s ds √ 2 c D2

√ s 2π n −s ds. √ 2 c D2 (A.14)

A.4.1. Computation of C + (s) ± C − (s) We compute C + (s) ± C − (s) using the basic recursion properties of 0 (the functional equation and the duplication formula): 1 1 π π 0(s)0(1 − s) = , 0 +s 0 −s = , (A.15) sin(πs) 2 2 cos(πs) 1 = 21−2s π 1/2 0(2s). (A.16) 0(s)0 s + 2 Some straightforward trigonometric calculations give cos(πir ) 0(1 − s + ir )0(1 − s − ir ), 2−2s π 22s−1 h 0(1 − s − ir ) 0(1 − s + ir ) i C + (s) − C − (s) = − . sin(πir ) 0(s − ir ) 0(s + ir ) C + (s) + C − (s) =

(A.17) (A.18)

At this point, we repeat the arguments of Appendix A.3, arriving at integrals of the form Z s s 4π √n −s 1 s Fˆ 1 − 0 + ir 0 − ir ds √ 2πi (3/4) 2 2 2 c D2 or 1 2πi

Z

√ s h 0(s/2 − ir ) 0(s/2 + ir ) i 4π n −s ˆ ds. F 1− − √ 2 0(1 − s/2 − ir ) 0(1 − s/2 + ir ) c D2 (3/4)

ˆ We again replace F(1−s/2) with the integral of F against x −s/2 and exchange orders of integration. In order to maintain convergence, and to avoid the poles at s = ±2ir ,

184


instead of staying at <e s = 3/4 we use the s-contour linking the points −1/4 + i∞, −1/4 + 2i|r |, 3/4 + 2i|r |, 3/4 − 2i|r |, −1/4 − 2i|r |, −1/4 − i∞. In particular, if ir ∈ R∗ (the case of exceptional eigenvalues), we still have from Selberg’s bound that 3/4 > 1/2 > |2ir |. From this and the identities (see [EMO2]) Z s x −s s 1 1 + ir 0 − ir ds, −<e 2ir < σ, K 2ir (x) = 0 2 2πi (σ ) 2 2 2 Z −s 1 1 x 0(s/2 + ir ) J2ir (x) = ds, −<e 2ir < σ < 1, 2 2πi (σ ) 2 0(1 − s/2 + ir ) we have c

a ηg (D2 ) X a D2 λg D2 (n)e − n λg (n)e n F(n) = χ D1 (a)χ D2 (−c) √ c c D 2 n>1 n>1 Z ∞ 4π √nx × F(x)Jg √ dx c D2 0 aD ηg (D2 ) X 2 λg D2 (n)e n + εg D2 χ D1 (a)χ D2 (−c) √ c D 2 n>1 Z ∞ 4π √nx × F(x)K g √ d x, c D2 0 X

where εg D2 = χ D2 (−1)εg and Jg (x) = −

π J2ir (x) − J−2ir (x) , sin(πir )

K g (x) = 4 cos(πir )K 2ir (x).

When r = 0, the case of greatest interest, the functions reduce to −2π Y0 and 4K 0 , respectively. A.5. The summation formula recapitulated We put everything into one formula for convenient reference. A.4 Let D be a positive integer, let χ D be a character of modulus D, and let g be one of the forms presented in Section 3. For (a, c) = 1, set D1 = (c, D), D2 = D/D1 , and assume that (D1 , D2 ) = 1, so that χ D = χ D1 χ D2 is the unique factorization of χ D into characters of modulus D1 and D2 . For F ∈ C ∞ (R∗+ ) a smooth function THEOREM


185

vanishing in a neighborhood of zero and rapidly decreasing, c

X n>1

a λg (n)e n F(n) c

Z 4π r nx ηg (D2 ) X a D2 ∞ dx λg D2 (n)e − n F(x)Jg = χ D1 (a)χ D2 (−c) √ D2 c c D 2 n>1 0 aD Z ∞ 4π r nx ηg (D2 ) X 2 λg D2 (n)e n F(x)K g d x. + χ D1 (a)χ D2 (c) √ c c D2 D 2 n>1 0 In this formula ηg (D2 ) is the pseudo-eigenvalue of the Atkin-Lehner operator W D2 ; if λg (D2 ) 6 = 0, it equals value

•

ηg (D2 ) = •

G(χ D2 ) √ ; λg (D2 ) D 2

if g is holomorphic of weight k g , then Jg (x) = 2πi kg Jkg −1 (x),

•

if g is a Maass form with (1+λ)g = 0, then let r satisfy λ = (1/2+ir )(1/2− ir ), and let εg be the eigenvalue of g under the reflection operator: Jg (x) =

•

K g (x) = 0;

−π J2ir (x) − J−2ir (x) , sin(πir )

K g (x) = εg 4 cosh(πr )K 2ir (x);

if r = 0, Jg (x) = −2π Y0 (x),

K g (x) = εg 4K 0 (x).

Finally, for completeness, we recall the summation formula of [J]: √ Z +∞ an X x log c τ (n)e F(n) = 2 + γ F(x) d x c c 0 n>1 Z an 4π √nx +∞ X + τ (n) − 2πe − Y0 c c 0 n>1 an 4π √nx + 4e K0 F(x) d x. c c This corresponds to the summation formula for the Eisenstein series E 0 (z, 1/2) given in (1.2).

186


B. Shifted convolutions of modular forms In this section we establish the bound cited in Section 7.2.1; we follow [DFI2] almost exactly. We work in a more general setting: consider two primitive forms g, g 0 of 0 which are not square-free level D and D 0 with respective nebentypes χ D and χ D 0 ∗+ exceptional. We are given a smooth test function f (z, y) on R × R∗+ satisfying z y i+ j z i y j f (i j) (z, y) i, j 1 + 1+ P Z Y for all i, j > 0 with some P, Z , Y > 1. For (a, b) = 1 we wish to evaluate the sum X D ±f (a, b; h) := λg (m)λg0 (n) f (am, bn). (B.1) am±bn=h

B.1 Suppose (a, b) = 1 and h 6= 0, and suppose that f satisfies the above conditions. Then D ±f (a, b; h) = O,g,g0 P 5/4 (Z + Y )1/4 (Y Z )1/4+ . (B.2) PROPOSITION

This is a direct generalization of [DFI2, Th. 1], which corresponds to the case when g = g 0 = E 0 (z, 1/2). We borrow the notation of [DFI2, (22), p. 214], and we set U = R 2 = min(Y, Z )P −1 . Following that proof, we have X D ±f (a, b; h) := D ±f (a, b, r ; h) 16r 0 and k > 0, we have LEMMA

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Kowalski Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000, USA; [email protected] Michel Mathématiques, Université Montpellier II, Case Courier 051, 34095 Montpellier CEDEX 05, France; [email protected]; current: Institute for Advanced Study, Princeton, New Jersey 08540, USA VanderKam Center for Communications Research, Princeton, New Jersey 08540, USA; [email protected]


CLASSES OF DEGENERACY LOCI FOR QUIVERS: THE THOM POLYNOMIAL POINT OF VIEW ´ FEHER ´ ´ and RICHARD ´ ´ LASZL O RIMANYI

Abstract The Chern classes for degeneracy loci of quivers are natural generalizations of the Thom-Porteous-Giambelli formula. Suppose that E, F are vector bundles over a manifold M and that s : E → F is a vector bundle homomorphism. The question is, which cohomology class is defined by the set 6k (s) ⊂ M consisting of points m where the linear map s(m) has corank k? The answer, due to I. Porteous, is a determinant in terms of Chern classes of the bundles E, F. We can generalize the question by giving more bundles over M and bundle maps among them. The situation can be conveniently coded by an oriented graph, called a quiver, assigning vertices for bundles and arrows for maps. We give a new method for calculating Chern class formulae for degeneracy loci of quivers. We show that for representation-finite quivers this is a special case of the problem of calculating Thom polynomials for group actions. This allows us to apply a method for calculating Thom polynomials developed by the authors. The method— reducing the calculations to solving a system of linear equations—is quite different from the method of A. Buch and W. Fulton developed for calculating Chern class formulae for degeneracy loci of An -quivers, and it is more general (can be applied to An -, Dn -, E 6 -, E 7 -, and E 8 -quivers). We provide sample calculations for A3 - and D4 -quivers. 1. Introduction The goal of this paper is to demonstrate the usefulness of the theory of Thom polynomials for group actions developed by M. Kazarian in [Kaz1] and [Kaz2] by calculating some formulas of degeneracy loci. Our calculations are based on our method, the restriction equations (see [FR1]), and a beautiful chapter of algebra, the representation theory of quivers. The paper is intended to be self-contained, except for some DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 2, Received 14 November 2000. Revision received 21 June 2001. 2000 Mathematics Subject Classification. Primary 14N10, 16G20, 58K30. Fehér’s work supported by Hungarian Higher Educational Research and Development (FKFP) grant number 0055/2001 and Hungarian National Science Foundation (OTKA) grant number D29234. Rimányi’s work supported by OTKA grant number T029759 and FKFP grant number 0055/2001. 193

´ and RIMANYI ´ FEHER

194

technical details on the existence of the Poincaré dual and standard facts from the representation theory of quivers. Certain types of Thom polynomials were studied under different names. The name Thom polynomial comes from singularity theory, where R. Thom proposed the following question: Given a smooth map f : M → N , what is the cohomology class [η( f )] ∈ H ∗ (M) defined via Poincaré duality by the closure of η( f ) ⊂ M—the points of M where f has a singularity of type η? As Thom observed, this class can be expressed as a polynomial of characteristic classes of the vector bundles T M and f ∗ T N . In homotopy theory an extensively studied question is whether a fiber bundle admits a section. One obstruction is the so-called first obstruction, which is a cohomology class of the base of the fiber bundle measuring the nonexistence of a section. Thom polynomials for group actions are first obstructions. In algebraic topology these questions can be translated to questions on equivariant cohomology theory. In algebraic geometry Thom polynomials are called classes or formulas of degeneracy loci. The earliest example, which was also the first example in singularity theory, is the following. Example 1.1 Let E and F be complex vector bundles of complex dimension n and p over the manifold M, and let s : E → F be a vector bundle homomorphism, that is, a section of Hom(E, F). Let 6k (s) denote the set of m ∈ M such that the linear map s(m) has corank k. We are looking for an expression for the cohomology class [6k (s)]—the Poincaré dual of the closure of 6k (s). The corresponding situation in singularity theory is that if we have a smooth map f : M → N , then we look for the set of points in M where d f , the Jacobian of f , has corank k. It is a special case of Example 1.1, for real vector bundles, the bundles E = T M, F = f ∗ T N , and for the section d f . The homotopy theory approach to the same problem would be to look at the subspace 6 0. We can change the integral to the integral over the whole circle h 21 + ρ 2 6 1. (Of course, we must write 1/2 in front of the integral.) Passing to the polar coordinates h 1 = r cos φ, ρ = r sin φ, we obtain 1

Z

2π

Z dr

2π 0

0

1 (1+r eiφ )λ/2 (1+r e−iφ )λ/2 (1−r 2 )2n−3r 3 · − {e2iφ −2+e−2iφ } dφ. 4

Removing the curly brackets, we obtain a sum of three integrals. The first and third integrals coincide. We obtain Jn (λ) = π

1

Z

2π

Z dr

0

−π

(1 + r eiφ )λ/2 (1 + r e−iφ )λ/2 (1 − r 2 )2n−3r 3 dφ

0 1

Z

Z

2π

dr 0

(1 + r eiφ )λ/2 (1 + r e−iφ )λ/2 (1 − r 2 )2n−3r 3 e2iφ dφ.

0

Repeating calculations (1.13) and (1.14) for each integral, we get ∞ h X (−λ/2)k+2 (−λ/2)k (2n − 1)! Jn (λ) = π 2 − k!(2n + k)! k=0

∞ X

(−λ/2)k (−λ/2)k (2n − 3)! i k!(2n + k − 1)! k=0 h (−λ/2)(−λ/2 + 1) λ λ F − + 2, − ; 2n + 1; 1 = π2 − 2n(2n − 1)(2n − 2) 2 2 λ 1 λ + F − , − ; 2n; 1 (2n − 1)(2n − 2) 2 2 λ i (−λ/2)2 λ + F − + 1, − + 1; 2n + 1; 1 . 2n(2n − 1)(2n − 2) 2 2 +

(k + 1)

Applying Gauss’s formula (1.15), we obtain (−λ/2)(−λ/2 + 1) (2n)!0(2n + λ − 1) 2 π − · 2n(2n − 1)(2n − 2) 0(2n + λ/2 − 1)0(2n + λ/2 + 1) (2n − 1)!0(2n + λ) 1 · + (2n − 1)(2n − 2) 0 2 (2n + λ/2) (−λ/2)2 (2n)!0(2n + λ − 1) + · 2n(2n − 1)(2n − 2) 0 2 (2n + λ/2) (2n − 3)!0(2n + λ − 1) = π2 · 0(2n + λ/2 − 1)0(2n + λ/2) (−λ/2)(−λ/2 + 1) 2n + λ − 1 (−λ/2)2 × − + . 2n + λ/2 2n + λ/2 − 1 2n + λ/2 − 1

252

YURII A. NERETIN

Finally, we obtain (2n − 3)!0(2n + λ + 1) 0(2n + λ/2)0(2n + λ/2 + 1) (2n − 1)!0(2n + λ + 1) π2 · . = (2n − 1)(2n − 2) 0(2n + λ/2)0(2n + λ/2 + 1)

Jn (λ) = π 2

Now we have to repeat all calculations for the case when n = 1. In the case when K = C, we have an integral over the circle | p| = 1; in the case when K = H, we obtain an integral over the sphere h 21 + h 22 + h 23 + h 24 = 1. We omit these calculations. 1.10. Another representation of the densities For definiteness, consider the case when K = R. Proposition 1.5 on multiplicativity shows that the density (1.6) can be represented in the form n Y

det(1 + [g]n−k+1 )λk −λk−1 · dσn (g) =

k=1

n λ j 0(n/2) Y n− j 1 + [ϒ (g)] · dσn (g). 1 π n/2 j=1

(1.17)

2. Examples: Calculation of some matrix integrals 2.1. Some integrals over classical groups Theorem 1.6 immediately yields the following. 2.1 Let λ0 = µ0 = 0. Then COROLLARY

Z

n Y

det(1 + [g]n−k+1 )λk −λk−1 dσn (g)

SO(n) k=1

=

n Y k=1

Z

n Y

2λk

0(k − 1)0(λk + (k − 1)/2) , 0((k − 1)/2)0(λk + k − 1)

(2.1) µk −µk−1

det(1 + [g]n−k+1 )λk −λk−1 det(1 + [g]n−k+1 )

dσn (g)

U(n) k=1

=

n Y 0(k)0(k + λk + µk ) , 0(k + λk )0(k + µk )

k=1

Z

n Y Sp(n) k=1

det(1 + [g]n−k+1 )λk −λk−1 dσn (g)

(2.2)

HUA INTEGRALS ON CLASSICAL GROUPS

=

n Y k=1

253

0(2k)0(2k + λk + 1) . 0(2k + λk /2)0(2k + λk /2 + 1)

(2.3)

The integrals are absolutely covergent under the following conditions: (a) in the real case: −(k − 1)/2, (b) in the complex case: −k, (c) in the quaternionic case: −(2k + 1), for all k. Remark. Let g ∈ SO(n). Then det(1 + [g]n−1 ) =

det(1 + g) . det(1 + [ϒ n−1 (g)]1 )

But ϒ n−1 (g) ∈ SO(1), and hence it equals 1. Thus, det(1 + [g]n−1 ) =

1 det(1 + g). 2

For this reason, integral (2.1) depends on λ1 in a nonessential way. Remark. The absolute convergence of integrals (2.1) – (2.3) follows from the absolute convergence of integrals (1.11), (1.12), and (1.16). Absolute convergence of the former integrals is a simple exercise. Formulas (1.5) and (1.17) imply a more general (and more artificial) statement. COROLLARY 2.2 Let λ0 = µ0 = 0, θ1 = 0. Then n Y

Z

det(1 + [g]n−k+1 )λk −λk−1

SO(n) k=1

n Y

1 − |[ϒ n−k (g)]1 |2

θ j /2

dσn (g)

k=2 n 0(n/2) Y λk +k+θk −2 0((k + θk − 1)/2)0(λk + (k + θk − 1)/2) = , 2 0(λk + k + θk − 1) π n/2 k=1

(2.4) Z

n Y

µk −µk−1

det(1 + [g]n−k+1 )λk −λk−1 det(1 + [g]n−k )

U(n) k=1

×

n Y k=2

1 − |[ϒ n−k (g)]1 |2

θ j

dσn (g)

254

YURII A. NERETIN

= (n − 1)!

n Y 0(k + θk − 1)0(k + θk + λk + µk ) , 0(k + θk + λk )0(k + θk + µk )

(2.5)

k=1

Z

n Y

det(1 + [g]n−k+1 )λk −λk−1

Sp(n) k=1

n Y

1 − |[ϒ n−k (g)]1 |2

2θ j

dσn (g)

k=2

= (2n − 1)!

n Y k=1

0(2(k + θk − 1))0(2(k + θk ) + λk + 1) . 0(2(k + θk ) + λk /2)0(2(k + θk ) + λk /2 + 1)

(2.6)

Proof Consider the case when K = R. By Section 1.10, the integrand in (2.4) has the form n Y

1 + [ϒ n− j (g)]1

λ j

· 1 − |[ϒ n−k (g)]1 |2

θ j /2

.

j=1

Thus, the integrand is a function in the variables x j = [ϒ n− j (g)]1 . Applying formula (1.5), we reduce the integral to the form 0(n/2) λ1 2 π n/2

Z

1

Z ···

−1

1

n Y

(1 + x j )λ j

−1 j=2

n Y

(1 − x 2j )( j+θ j −3)/2

j=2

n Y

dx j.

j=2

This integral is reduced to (1.9). In the cases when K = C, H, we obtain the same reduction to integrals (1.12) and (1.16). 2.2. Some integrals over Stiefel manifolds Recall that a Stiefel manifold Sti(m, n, K) = U◦ (n, K)/ U◦ (n − m, K) is the set of isometric embeddings Km → Kn . A projection from a group U◦ (n, K) to a homogeneous space Sti(m, n, K) is very simple: we take a unitary matrix and delete its last n − m rows. Assume that λ0 = · · · = λn−m = 0, µ0 = · · · = µn−m = 0 in integrals (2.1) – (2.3). Then the integrand depends only on the first m rows of a matrix. Therefore, we can consider the integral as an integral over the Stiefel manifold Sti(m, n, K). 2.3. Some integrals over matrix balls As in Section 1.2, we denote the matrix ball by Bm (K). Let d Z be the Lebesgue measure on Bm (K) normalized as in Lemma 1.4.


PROPOSITION

We have Z Bm (R)

255

2.3

det(1 − Z ∗ Z )(α−m−1)/2

m Y

det(1 + [Z ]m−k+1 )λk −λk−1 d Z

(2.7)

k=1 (m)

= cR

m α − m + 1 Y 0(k + α − 1)0(λk + (α + k − 1)/2) , 2 0((k + α − 1)/2)0(λk + α + k − 1)

(2.8)

k=1

Z Bm (C)

det(1 − Z ∗ Z )α−m

m Y

det(1 + [Z ]m−k+1 )λk −λk−1 µk −µk

× det(1 + [Z ]m−k+1 ) (m)

= cC (α − m + 1)

m Y k=1

Z Bm (H)

(2.9)

k=1

det(1 − Z ∗ Z )2(α−m)+1 (m)

= cH

2(α − m + 1)

dZ

0(k + α)0(k + α + λk + µk ) , 0(k + α + λk )0(k + α + µk )

m Y k=1 m Y k=1

det(1 + [Z ]m−k+1 )λk −λk−1 d Z

(2.10)

0(2(k + α))0(2(k + α) + λk + 1) , 0(2(k + α) + λk /2)0(2(k + α) + λk /2 + 1)

(m)

where the constants cK (·) are the same as in Lemma 1.4. Proof The proof is given in the case when K = R. Consider another parameter β = α + m. First, let β be an integer, β > 2m. Consider the integral (evaluated before; see Cor. 2.1) (m)

cR

β − 2m + 1 Z 2

m Y SO(β) k=1

det(1 + [g]m−k+1 )λk −λk−1 dσβ (g).

(2.11)

The integrand depends only on the matrix [g]m ∈ Bm . Hence, we can consider the integral as an integral over Bm . Using Theorem 1.3, we convert (2.11) to the form (2.7). Thus, the required statement is proved for the integer values of α > m + 1. Fix λ1 > λ2 > · · · > λm . (2.12) Then, for m + 1, the left-hand side of the integral is a bounded holomorphic function in α in the domain m + 1. Indeed, det(1 − Z ∗ Z ) 6 1 for Z ∈ Bm , and thus the integrand (for fixed λ j ) is bounded.

256

YURII A. NERETIN

It can easily be checked that the product of 0 is also bounded in the same domain. By the Carlson theorem∗ , the left part (2.7) and the right part (2.8) coincide in the whole domain m + 1. The analytic continuation allows us to omit condition (2.12). 2.4. Some integrals over spaces of anti-Hermitian matrices Let us change the variable g = −1 + 2(X + 1)−1 in integral (2.1). Obviously, X is a skew-symmetric matrix. Let us calculate the new integrand. We represent g as an (m + (n − m)) × (m + (n − m)) block matrix g = PR Q . By Proposition 1.5, we T obtain −1 det(1 + [g]m ) = det(1 + P) = det(1 + g) · det 1 + T − R(1 + P)−1 Q . (2.13) Further, 2(1 + X )

−1

P =1+g =1+ R

Q . T

Hence, the expression det(1 + g) transforms to const · det(1 + X )−1 . The Frobenius formula (1.1) gives ! ... ... −1 −1 . 2(1 + X ) = (1 + g) = 2 (2.14) ... 1 + T − R(1 + P)−1 Q (We write only the block that is interesting for us.) Therefore, by (2.13) and (2.14), the expression det(1 + [g]n−k+1 ) converts to the form const · det 1 + {X }k−1 · det(1 + X )−1 (according to the notation of Sec. 1.1). The Jacobian of the transformation g = −1 + 2(X + 1)−1 equals const · det(1 + −(n−1)/2 X) (see [12, Sec. 3.7]), and, finally, the integral transforms to the form Z Y n const · det(1 + [X ]k−1 )λk −λk−1 det(1 + X )−λn −(n−1)/2 d X, (2.15) k=2

where the integration is taken over the space of real skew-symmetric matrices. In the same way, integrals (2.2) and (2.3) transform into integrals over the space of anti-Hermitian (X = −X ∗ ) matrices over C or H. For instance, in the complex case we obtain an arbitrary integral of the form Z n Y det(1 + [X ]k )ak det(1 − [X ]k )bk d X. (2.16) X +X ∗ =0 k=1

f (z) is holomorphic and bounded for 0 and if f (n) = 0 for all n = 1, 2, . . . , then f (z) = 0 (see, for instance, [1, Th. 2.8.1]).

∗ If


257

2.5. Some integrals over spaces of dissipative matrices Let us transform integral (2.7) as in Section 2.4. It can be easily checked that det(1 − Z ∗ Z ) =

det(2(X + X ∗ )) , | det(1 + X )|2

the Jacobian is given by d Z = (1 + X )−2m d X, and integral (2.7) transforms to Z const ·

det(X + X ∗ )(α−m−1)/2 ×

m Y

det(1 + [X ]k )λk −λk−1 det(1 + X )−λm −α−m+1 d X, (2.17)

k=1

where the integration is taken over the space of real matrices T satisfying the following condition: the matrix T + T ∗ is positive definite. Integrals (2.9) and (2.10) can be transformed in a similar way. Remarks. (a) Integral (2.17) and the Cayley transforms of integrals (2.9) and (2.10) are partial cases of the matrix B-function introduced in [18]. (This B-function extends S. Gindikin’s B-function; see [10], [7].) The Cayley transform of (2.9) is also one of A. Unterberger and H. Upmeier’s integrals in [29] (see also [2]). (b) A way of separation of variables described in [18] also allows us to evaluate integrals (2.15) and (2.16), but they were not evaluated in that paper. Integral (2.16) can also be evaluated using the inverse Laplace transform in a way explained in [7] (but it was not evaluated in that book). Integral (2.1) was evaluated, and integrals (2.2) and (2.3) were announced in [19]. (c) Calculations in this paper are almost verbal (except for local difficulties in Sec. 1.9), and they provide an explanation for the existence of explicit formulas. 3. Inverse limits of orthogonal groups In order to be concrete, we consider only the case when K = R. 3.1. Inverse limit of orthogonal groups Consider a chain of the maps (defined almost everythere) ϒ1

ϒ1

ϒ1

ϒ1

· · · ← SO(k) ← SO(k + 1) ← SO(k + 2) ← · · · .

(3.1)

258

YURII A. NERETIN

Let us fix a sequence of real numbers λ1 , λ2 , . . . satisfying the condition λk > −(k − 1)/2. Consider in each space SO(k) a probability measure with the density C(λ1 , . . . , λn )−1

n Y

det(1 + [g]n−k+1 )λk −λk−1

k=1

with respect to the Haar measure. The constant C(·) is given by (2.1). By Theorem 1.6, our measures are consistent with the maps ϒ 1 . Therefore, by Kolmogorov’s theorem about inverse limits (see, for instance, [28]), we obtain a canonically defined measure on the inverse limit of the chain (3.1). (This measure depends on the sequence λ1 , λ2 , . . . .) We denote this inverse limit (equipped with the probability measure) by (Oλ1 ,λ2 ,... , νλ1 ,λ2 ,... ), and we call it the virtual orthogonal group. We also denote by ϒ ∞−k the canonical map ϒ ∞−k : Oλ1 ,λ2 ,... → SO(k). This family of measures seems too large. In Sections 3.4 – 3.6 we discuss two natural special cases. Remark. Obviously, the space Oλ1 ,λ2 ,... is not a projective limit in the category of groups. But an orthogonal group SO(n) is also the symmetric space G/K = SO(n) × SO(n)/ SO(n), where K is embedded in G as the diagonal subgroup; the maps ϒ m are quite natural as maps of symmetric spaces (see [17]). Hence, Oλ1 ,λ2 ,... can be considered as a projective limit of symmetric spaces. 3.2. Projection of (Oλ1 ,λ2 ,... , νλ1 ,λ2 ,... ) to a cube Consider the product [−1, 1]∞ of segments [−1, 1] equipped with the product of the measures 2−λ−k+2 (1 + xk )λk (1 − xk2 )(k−3)/2 d xk , B(λk + (k − 1)/2, (k − 1)/2)

(3.2)

where k = 2, 3, . . . . We define the map (Oλ1 ,λ2 ,... , νλ1 ,λ2 ,... ) → [−1, 1]∞ by the formula ω 7→ [ϒ ∞−2 (ω)]1 , [ϒ ∞−3 (ω)]1 , . . . . Obviously, the image of the measure νλ1 ,λ2 ,... is our measure on the cube. 3.3. Quasi invariance Denote by O(∞) the group of all orthogonal operators in the real Hilbert space l2 . Denote by SO(∞)fin the group of matrices g ∈ O(∞) such that


259

(a) g − 1 has only a finite number of nonzero matrix elements, (b) det(g) = 1. It is convenient to think that matrices g ∈ O(∞) are infinite upwards and to the left. We assume that the subgroup SO(k) ⊂ O(∞) corresponds to the lower right (k × k)-block of infinite matrices. Let A, B ∈ SO(k). Denote by 1n the n × n unit matrix. Consider the map ϒ n : SO(n + k) → SO(k). Obviously, 1n 0 P Q 1n 0 n n P Q ϒ = Aϒ B. 0 A R T 0 B R T This yields that for all A , B ∈ O(∞)fin the transformation S 7→ A S B

(3.3)

of the virtual orthogonal group (Oλ1 ,λ2 ,... , νλ1 ,λ2 ,... ) is well defined. 3.1 The measure νλ1 ,λ2 ,... is quasi invariant with respect to the action of the group SO(∞)fin × SO(∞)fin . The Radon-Nikodym derivative is given by the formula PROPOSITION

∞ Y 1 + [ϒ ∞− j (A S B )]1 λ j j=1

1 + [ϒ ∞− j (S)]1

.

Remark. Since A, B ∈ SO(∞)fin , only finitely many factors of this product differ from 1. Proof Let A, B ∈ SO(k). Consider the transformation 1n 0 1 S 7→ S n 0 A 0

0 B

(3.4)

of the group SO(n + k). Its Radon-Nikodym derivative is a ratio of densities, and it is equal to n+k λ j Y 1 + ϒ n+k− j 1n 0 S 1n 0 1 0 A 0 B . (3.5) 1 + [ϒ n+k− j (S)]1 j=1

By Lemma 1.1, for j > k, 1n 0 1 n+k− j ϒ S n 0 A 0

0 B

1 j−k = 0

0 1 n+k− j ϒ (S) j−k A 0

0 . B

260

YURII A. NERETIN

Thus, for j > k we have 1n n+k− j ϒ 0

1 0 S n A 0

0 B

= [ϒ n+k− j (S)]1 ; 1

Qk

and, hence, product (3.5) is reduced to j=1 . Thus, the Radon-Nikodym derivative of the transformation (3.4) depends only on ϒ n+k−k (S). Hence, the Radon-Nikodym derivatives of the maps (3.4) (where A, B ∈ SO(k) are fixed) form a compatible system of functions with respect to the chain (3.1). This implies both statements. 3.4. Hua-Pickrell measures Let λ > −1/2. Consider the probability measure on SO(n) given by the formula νλn = C(n, λ) det(1 + g)λ dg, where C(n, λ) is a constant. This corresponds to the case when λ1 = λ2 = · · · = λ in the construction of Section 3.1. Let us denote the inverse limits of the measure spaces (SO(n), νλn ) by (Oλ (∞), νλ ). We call νλ the Hua-Pickrell measure. 3.2 The measure νλ on Oλ (∞) is quasi invariant (in the case when λ = 0, it is invariant) with respect to the action of the group SO(∞)fin × SO(∞)fin . Moreover, for A, B ∈ SO(k) ⊂ SO(∞), the Radon-Nikodym derivative of the transformation S 7 → AS B is equal to

PROPOSITION

(a)

h det(1 + Aϒ ∞−k (x)B) iλ . det(1 + ϒ ∞−k (x)) (b)

The diagonal action S 7→ A−1 S A of the group SO(∞)fin extends to an invariant action of the group O(∞).

Proof (a) Let S = formation

P Q R T

∈ SO(n + k). Then the Radon-Nikodym derivative of the trans 1 S 7→ n 0

0 1 S n A 0

0 B


261

is 1n det 0

=

0 1n 0 P + 1k 0 A R 1n 0 P det + 0 1k R

Q T Q T

1n 0 λ

0 B

λ

det(1 + P)λ det(1 + AT B − A R(1 + P)−1 Q B)λ det(1 + Aϒ n (S)B)λ = . det(1 + ϒ n (S))λ det(1 + P)λ det(1 + T − R(1 + P)−1 Q)λ

The Radon-Nikodym derivative depends only on ϒ n (S) ∈ SO(k), and this implies (a). Clearly, (a) is also a corollary of Proposition 3.1. (b) This statement is very simple, but its proof uses some technique. By a criterion from [23], the representation of the diagonal group O(∞)fin in L 2 (Oλ ) is weakly continuous. Hence, it extends to the group O(∞). The group O(∞) acts by measurepreserving transformations, and, hence, the group O(∞) acts by polymorphisms (see [16, Chap. 8]). But an invertible polymorphism is a measure-preserving transformation. Remark. We see that the group of symmetries of the space (Oλ , νλ ) is larger than that for a general space (Oλ1 ,λ2 ,... , νλ1 ,λ2 ,... ). Remark. Consider the group O(∞) × O(∞) and its subgroup G that consists of pairs (g1 , g2 ) ∈ O(∞) × O(∞) such that g1 g2−1 is a Hilbert-Schmidt operator. (This is one of Olshanski’s (G, K )-pairs; see [23], [16].) It is natural to think that our action extends to a quasi-invariant action of G. Remark. Our construction for λ = 0 is the Shimomura construction in [27]. Pickrell [26] constructed a 1-parametric family of probability measures on inverse limits of Grassmannian U(2n)/ U(n) × U(n). Olshanski [25] observed that a Pickrell-type construction extends to all 10 series of classical compact symmetric spaces. (This can be observed from Hua calculations∗ in [12, Chap. 2].) In particular, it can be carried out for the classical groups U(n), SO(n), Sp(n). Our construction of the measure νλ is equivalent to this construction. (In the complex case, our construction gives an additional parameter.) ∗ In

[12, Th. 2.2.2] Hua exactly claims a projectivity of some system of measures (for noncompact symmetric spaces).

262

YURII A. NERETIN

3.5. Some integrals over the space (Oλ (∞), νλ ) Consider the space (Oλ (∞), ν λ ) equipped with Hua-Pickrell measure. The construction of Section 3.1 can be considered as a construction of a large family of functions on the space (Oλ (∞), ν λ ) with explicitly computable integrals. Let x j be the coordinates on the cube as in Section 3.2. PROPOSITION 3.3 P Consider a sequence λ1 , λ2 , · · · ∈ C such that |λk − λ| < ∞, λk > −(k − 1)/2. Let us define the function 8{λ1 , λ2 , . . . }(ω) on Oλ (∞) by the formula

8{λ1 , λ2 , . . . }(x) = 2λ1 −λ

∞ Y

det(1 + x j )λ j −λ =

j=2

1 + [ϒ ∞−k (ω)]1

λk −λ

k=1 λ j −λ j−1 j) . det(1 + ϒ ∞−k (ω))λ

Qk

j=1 det(1 + [ϒ

= lim

∞ Y

k→∞

∞−k (ω)]

(3.6)

Then the limit exists almost everywhere on (Oλ (∞), ν λ ) and Z

λ

Oλ (∞)

8{λ1 , λ2 , . . . } dν =

∞ Y

2λk −λ

k=1

0(λk + (k − 1)/2)0(λ + k − 1) . 0(λ + (k − 1)/2)0(λk + k − 1)

(3.7)

Proof Let us transform expression (3.6) to the form ∞ Y

1 + [ϒ ∞−k (ω)]1

λk −λ

.

k=1

First, let us prove existence of the functions 8. It is sufficient to prove the convergence of the product ∞ Y (1 + xk )λk −λ (3.8) k=2

[−1, 1]∞

on the cube gence of the series

equipped with measure (3.2). This is equivalent to the conver∞ X

(λk − λ) ln(1 + xk ).

(3.9)

k=2

By the Kolmogorov-Khintchin theorem on series of independent random variables (see [28]), it is sufficient to prove the absolute convergence of the series of means and


263

the convergence of the series of variances, that is, X X

Ck−1 (λk

− λ)

Ck−1 |λk − λ|2

Z

1

−1 Z 1

ln(1 + x)(1 + x)λ (1 − x 2 )(k−3)/2 d x < ∞,

(3.10)

ln2 (1 + x)(1 + x)λ (1 − x 2 )(k−3)/2 d x < ∞,

(3.11)

−1

where Z

1

Ck =

(1 + x)λ (1 − x 2 )(k−3)/2 d x.

−1

The Laplace method gives the asymptotics Ck = const · k −1/2 (1 + o(1)) and const · k −3/2 (1 + o(1)) for the integrals under sums (3.10) and (3.11). This implies the a.s. convergence of (3.9). Product (3.8) is dominated by ∞ Y

max (1 + xk )λk −λ , (1 + xk )λ−λk .

(3.12)

k=2

By Lebesgue’s theorem on dominated convergence, it is sufficient to prove that the last expression is integrable. The integral of (3.12) is ∞ Y k=2

Ck−1

Z

1

(1 + xk )|λk −λ| (1 + x)λ (1 − x 2 )(k−3)/2 d x

0

Z

0

+

(1 + xk )−|λ−λk | (1 + x)λ (1 − x 2 )(k−3)/2 d x .

−1

The Laplace method gives the asymptotics const · k −1/2 |λk − λ|(1 + o(1)) for the integrals under the product, and this implies the required statement. Remark. The author thinks that condition

P

|λk − λ| < ∞ is not necessary.

3.6. Measures on inverse limits of Stiefel manifolds Let us fix p > 0. We denote by ψk (g) the function det(1 + [g]k )λ on SO(k + p). Obviously, the function ψk (g) is invariant with respect to the action of the group SO( p) given by the formula 1 0 g 7→ g, g ∈ SO(k + p), A ∈ SO( p). 0 A

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YURII A. NERETIN

Hence, we can consider the function ψk (g) as a function on the Stiefel manifold Sti(k + p, k) (see Sec. 2.2). Denote by νkλ the probability measure on Sti(k + p, k) with the density const · ψk (g). The projections from the chain (3.1) commute with the action of SO( p). Therefore, we can consider a chain of quotient spaces Sti(k, k+ p) = SO(k + p)/ SO( p) equipped with the measures νkλ : ϒ1

ϒ1

ϒ1

ϒ1

· · · ← Sti(k, k + p) ← Sti(k + 1, k + p + 1) ← Sti(k + 2, k + p + 2) ← · · · . We denote by (Sti(∞, ∞ + p), νλ ) the inverse limit of this chain. Denote by ν λ the canonical measure on this limit. We define the group O(∞)fin × O(∞ + p)fin as the inductive limit of the groups O(k) × O(k + p) as k → ∞. 3.4 The measure ν λ is quasi invariant with respect to the action of the group O(∞)fin × O(∞ + p)fin . The action A : S 7→ AS A 1k of the diagonal group O(∞)fin on (Sti(∞, ∞+ p), ν λ ) extends to an invariant action of the group O(∞).

PROPOSITION

(a) (b)

Acknowledgments. I thank G. I. Olshanski for discussions of this subject. I also thank the referee of the paper for comments. References [1]

G. E. ANDREWS, R. ASKEY, and R. ROY, Special Functions, Encyclopedia Math.

[2]

J. ARAZY and G. ZHANG, “Invariant mean value and harmonicity in Cartan and Siegel

Appl. 71, Cambridge Univ. Press, Cambridge, 1999. MR 2000g:33001 245, 256

[3] [4] [5] [6] [7] [8]

domains” in Interaction between Functional Analysis, Harmonic Analysis, and Probability (Columbia, Mo., 1994), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 19 – 40. MR 97b:32047 257 A. BORODIN and G. OLSHANSKI, Point processes and the infinite symmetric group, Math. Res. Lett. 5 (1998), 799 – 816. MR 2000i:20020 , Distributions on partitions, point processes, and the hypergeometric kernel, Comm. Math. Phys. 211 (2000), 335 – 358. MR 2001k:33031 , Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87 – 123. CMP 1 860 761 241 , Correlation kernels arising from the infinite-dimensional unitary groups and its representations, to appear. 241 ´ J. FARAUT and A. KORANYI , Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford Univ. Press, New York, 1994. MR 98g:17031 257 M. V. FEDORYUK, Asymptotics: Integrals and Series (in Russian), Spravochn. Mat. Bibl., “Nauka,” Moscow, 1987. MR 89j:41045 245


[9]

[10] [11] [12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20] [21]

[22]

265

F. R. GANTMAKHER, The Theory of Matrices (in Russian), 4th ed., “Nauka,” Moscow,

1988; MR 90a:15001; English translation: The Theory of Matrices, Vols. 1, 2, Chelsea, New York, 1959. MR 21:6372c 241, 247 S. G. GINDIKIN, Analysis on homogeneous spaces (in Russian), Uspekhi Mat. Nauk, 19, no. 4 (1964), 3 – 92. MR 30:2167 257 S. HELGASON, Differential Geometry and Symmetric Spaces, Pure Appl. Math. 12, Academic Press, New York, 1962. MR 26:2986 243 HUA LOO KENG [L.-K. HUA], Harmonic Analysis of Functions of Several Complex Variables in Classical Domains (in Chinese), Science Press, Peking, 1958, MR 33:1483; Russian translation: Izdat. Inostr. Lit., Moscow, 1959, MR 23:A3277; English translation: Amer. Math. Soc., Providence, 1963. MR 30:2162 241, 244, 256, 261 S. V. KEROV, Subordinators and permutation actions with quasi-invariant measure (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995), 181 – 218; English translation in J. Math. Sci. (New York) 87 (1997), 4094 – 4117. MR 96k:60191 S. KEROV, G. OLSHANSKI, and A. VERSHIK, Harmonic analysis on the infinite symmetric group: A deformation of regular representation, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 773 – 778. MR 94e:20019 241 ˇ , On spectral decomposition of linear non self-adjoint operators (in M. S. LIVSIC Russian), Mat. Sbornik N.S. 34 (1954), 145 – 199, MR 16:48f; English translation in Amer. Math. Soc. Transl. Ser. 2 5 (1957), 67 – 114. 240 YU. A. NERETIN, Categories of Symmetries and Infinite-Dimensional Groups, London Math. Soc. Monogr. (N.S.) 16, Oxford Univ. Press, New York, 1996. MR 98b:22003; Russian translation: Editorial URSS, Moscow, 1998. 261 , Conformal geometry of symmetric spaces and Kre˘ın-Shmulian generalized linear fractional mappings (in Russian), Mat. Sb. 190, no. 2 (1999), 93 – 122; English translation in Sb. Math. 190 (1999), 255 – 283. MR 2000e:32035 240, 258 , Matrix analogues of B-function and Plancherel formula for Berezin kernel representations (in Russian), Mat. Sb. 191, no. 5 (2000), 67 – 100; English translation in Sb. Math. 191 (2000), 683 – 715. MR 2001k:33030 257 , On separation of spectra in harmonic analysis of Berezin kernels (in Russian), Funktsional. Anal. i Prilozhen. 34, no. 3 (2000), 49 – 62; English translation in Funct. Anal. Appl. 34 (2000), 197 – 207. MR 2001m:32012 240, 241, 257 , Plancherel formula for Berezin deformation of L 2 on Riemannian symmetric space, J. Funct. Anal. 189 (2002), 336 – 408. CMP 1 891 853 YU. A. NERETIN and G. I. OLSHANSKI˘I, Boundary values of holomorphic functions, singular unitary representations of groups O( p, q) and their limits as q → ∞ (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995), 9 – 91; English translation in J. Math. Sci. (New York) 87 (1997), 3983 – 4035. MR 97c:22018 241 N. K. NIKOLSKI˘I, Treatise on Shift Operator (in Russian), “Nauka,” Moscow, 1980, MR 82i:47013; English translation: Grundlehren Math. Wiss. 273, Springer,

266

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Berlin, 1986. MR 87i:47042 240 [23]

G. I. OLSHANSKI˘I, “Unitary representations of infinite-dimensional pairs (G, K ) and

[24] [25] [26]

D.

[27]

H.

[28]

A.

[29]

A.

the formalism of R. Howe” in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math. 7, Gordon and Breach, New York, 1990, 269 – 463. MR 92c:22043 261 , An introduction to harmonic analysis on infinite-dimensional unitary group, preprint, 2001. 241 , Inverse limits of symmetric spaces, unpublished notes. 261 PICKRELL, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), 323 – 356. MR 88d:58017 241, 261 SHIMOMURA, On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation, Publ. Res. Inst. Math. Sci. 10 (1974/75), 413 – 424. MR 51:8363 241, 261 N. SHIRYAEV, Probability (in Russian), “Nauka,” Moscow, 1980; MR 82d:60002; English translation: Probability, 2d ed., Grad. Texts in Math. 95, Springer, New York, 1996, MR 97c:60003 258, 262 UNTERBERGER and H. UPMEIER, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563 – 597. MR 96h:58170 257

Mathematical Physics Group, Institute of Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, Moscow 117259, Russia; [email protected]


COLLAPSING AND THE DIFFERENTIAL FORM LAPLACIAN: THE CASE OF A SMOOTH LIMIT SPACE JOHN LOTT

Abstract We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence. 1. Introduction A central problem in geometric analysis is to estimate the spectrum of the Laplacian on a compact Riemannian manifold M in terms of geometric invariants. In the case of the Laplacian on functions, a major result is J. Cheeger’s lower bound on the smallest positive eigenvalue in terms of an isoperimetric constant (see [11]). The problem of extending his lower bound to the case of the p-form Laplacian was posed in [11]. There has been little progress on this problem. We address the more general question of estimating the eigenvalues {λ p, j (M)}∞ j=1 of the p-form Laplacian 4 p (counted with multiplicity) in terms of geometric invariants of M. A basic fact, due to Cheeger and J. Dodziuk, is that λ p, j (M) depends continuously on the Riemannian metric g T M in the C 0 -topology (see [17]). Then an immediate consequence of the C α -compactness theorem of M. Anderson and Cheeger [1] is that for any n ∈ Z+ , r ∈ R, and D, i 0 > 0, there are uniform bounds on λ p, j (M) among connected closed n-dimensional Riemannian manifolds M with Ric(M) ≥ r , diam(M) ≤ D, and inj(M) > i 0 (cf. [10, Theorem 1.3], [14, Theorem 0.4]). In particular, there is a uniform positive lower bound on the smallest positive eigenvalue of the p-form Laplacian under these geometric assumptions. The question, then, is what happens when inj(M) → 0. For technical reasons, in this paper we assume uniform bounds on the Riemannian curvature R M . Then we wish to study how the spectrum of 4 p behaves in the collapsing limit. By collapsing DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 2, Received 20 May 2000. Revision received 20 August 2001. 2000 Mathematics Subject Classification. Primary 58J50; Secondary 35P15. Author’s research supported by National Science Foundation grant number DMS-9704633. 267

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we mean the phenomenon of a sequence of Riemannian manifolds converging in the Gromov-Hausdorff topology to a lower-dimensional space. We refer to [23, Chapters 1 and 3] for basic information about collapsing and to [12, Section I], [13], [21], and [23, Chapter 6] for information about bounded curvature collapsing. In this paper we analyze the behavior of the spectrum of 4 p under collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. The answer is in terms of a type of Laplacian on the limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence. In a subsequent paper we will extend the results to the case of singular limit space and give additional applications. From Hodge theory, dim(Ker(4 p )) = b p (M), the pth Betti number of M. Given K ≥ 0, let M (M, K ) be the set of Riemannian metrics g on M with kR M k∞ ≤ K and diam(M, g) ≤ 1. We say that M has small positive eigenvalues of the p-form Laplacian if inf λ p, j (M, g) = 0 (1.1) g∈M (M,K )

for some j > b p (M) and some K > 0. If this is the case, then we say that M has (at least) j small eigenvalues. Note that this is a statement about the (smooth) topological type of M. There are no small positive eigenvalues of the Laplacian on functions on M (see, e.g., [3]). B. Colbois and G. Courtois gave examples of manifolds with small positive eigenvalues of the p-form Laplacian for p > 0 (see [14]). Their examples were manifolds M with free isometric T k -actions, which one shrinks in the direction of the T k -orbits. In terms of the fiber bundle M → M/T k , this sort of collapsing is a case of the so-called adiabatic limit. The asymptotic behavior of the small eigenvalues of the p-form Laplacian in the adiabatic limit was related to the Leray spectral sequence of the fiber bundle in [5], [16], [18], and [26]. In another direction, K. Fukaya considered the behavior of the Laplacian on functions in the case of a sequence of manifolds that converge in the Gromov-Hausdorff metric dG H to a lower-dimensional limit space X , the collapsing assumed to be with bounded sectional curvature and bounded diameter (see [19]). He found that in order to get limits, one needs to widen the class of spaces being considered by adding a Borel measure and to consider measured metric spaces. This is the case even if X happens to be a smooth manifold. He defined a Laplacian acting on functions on the measured limit space and proved a convergence theorem for the spectrum of the Laplacian on functions, under the geometric assumption of convergence in the measured Gromov-Hausdorff topology. We consider the behavior of the spectrum of 4 p under collapse with bounded sectional curvature and bounded diameter. We find that we need a somewhat more refined structure on the limit space, namely, a superconnection as introduced by D. Quillen

COLLAPSING AND THE DIFFERENTIAL FORM LAPLACIAN

269

[27]. More precisely, we need a flat degree-1 superconnection in the sense of [8]. L Suppose that B is a smooth connected closed manifold and that E = mj=0 E j is a Z-graded real vector bundle on B. The degree-1 superconnections A0 that we need are of the form A0 = A0[0] + A0[1] + A0[2] , (1.2) where A0[0] ∈ C ∞ (B; Hom(E ∗ , E ∗+1 )), • A0[1] is a grading-preserving connection ∇ E on E, and • A0[2] ∈ 2 (B; Hom(E ∗ , E ∗−1 )). The superconnection extends by Leibniz’s rule to an operator A0 on the E-valued differential forms (B; E). The flatness condition (A0 )2 = 0 becomes • (A0[0] )2 = (A0[2] )2 = 0, • ∇ E A0[0] = ∇ E A0[2] = 0, and • (∇ E )2 + A0[0] A0[2] + A0[2] A0[0] = 0. In particular, A0[0] defines a differential complex on the fibers of E. Let g T B be a Riemannian metric on B, and let h E be a graded Euclidean inner product on E, meaning that E i is orthogonal to E j if i 6= j. Then there are an adjoint (A0 )∗ to A0 and a Laplacian 4 E = A0 (A0 )∗ + (A0 )∗ A0 on (B; E). Let 4 Ep be the restriction of 4 E to L a b a+b= p (B; E ). Using the C 0 -continuity of the spectrum and the geometric results of Cheeger, Fukaya, and M. Gromov [12], we can reduce our study of collapsing to certain special fiber bundles. As is recalled in Section 3, an infranilmanifold Z has a canonical flat linear connection ∇ aff . Let Aff(Z ) be the group of diffeomorphisms of Z which preserve ∇ aff . •

Definition 1 An affine fiber bundle is a smooth fiber bundle M → B whose fiber Z is an infranilmanifold and whose structure group is reduced from Diff(Z ) to Aff(Z ). A Riemannian affine fiber bundle is an affine fiber bundle with the following: • a horizontal distribution T H M on M whose holonomy lies in Aff(Z ), • a family g T Z of vertical Riemannian metrics that are parallel with respect to the flat affine connections on the fibers Z b , and • a Riemannian metric g T B on B. Fix a smooth connected closed Riemannian manifold B. Fukaya showed that any manifold M that collapses to B, with bounded sectional curvature, is the total space of an affine fiber bundle over B (see [20]). If M → B is an affine fiber bundle, let T H M be a horizontal distribution on M as above. Let T ∈ 2 (M; T Z ) be the curvature of T H M. There is a Z-graded real vector bundle E on B whose fiber over

270

JOHN LOTT

b ∈ B is isomorphic to the differential forms on the fiber Z b which are parallel with respect to the flat affine connection on Z b . The exterior derivative d M induces a flat degree-1 superconnection A0 on E. If M → B is in addition a Riemannian affine fiber bundle, then we obtain a Riemannian metric g T M on M constructed from g T Z , g T B , and T H M. There is an induced L 2 inner product h E on E. Define 4 E as above. Let diam(Z ) denote the maximum diameter of the fibers {Z b }b∈B in the intrinsic metric, and let 5 denote the second fundamental forms of the fibers {Z b }b∈B . Our first result says that the spectrum σ (4 Ep ) of 4 Ep contains all of the spectrum of the p-form Laplacian 4 M p which stays bounded as dG H (M, B) → 0. 1 There are positive constants A, A0 , and C which depend only on dim(M) such that if kR Z k∞ diam(Z )2 ≤ A0 , then for all 0 ≤ p ≤ dim(M), THEOREM

−2 σ (4 M − C(kR M k∞ + k5k2∞ + kT k2∞ ) p ) ∩ 0, A diam(Z ) = σ (4 Ep ) ∩ 0, A diam(Z )−2 − C(kR M k∞ + k5k2∞ + kT k2∞ ) . (1.3) When Z is flat, there is some intersection between Theorem 1 and the adiabatic limit results of [5], [16], [18], and [26]. However, there is the important difference that we need estimates that are uniform with respect to dG H (M, B), whereas the adiabatic limit results concern the asymptotics of the eigenvalues under the collapse of a given Riemannian fiber bundle coming from a constant rescaling of its fibers. We apply Theorem 1 to estimate the eigenvalues of a general Riemannian manifold M which is Gromov-Hausdorff close to B, assuming sectional curvature bounds on M. Of course, we cannot say precisely what σ (4 M ) is, but we can use Theorem 1 to approximate it to a given precision > 0. We say that two nonnegative numbers λ1 and λ2 are -close if e− λ2 ≤ λ1 ≤ e λ2 . We show that for a given > 0, if dG H (M, B) is sufficiently small, then there is a flat degree-1 superconnection A0 on B whose Laplacian 4 Ep has a spectrum that is -close to that of 4 M p , at least up to a high level. THEOREM 2 Let B be a fixed smooth connected closed Riemannian manifold. Given n ∈ Z+ , > 0, and K ≥ 0, there are positive constants A(n, , K ), A0 (n, , K ), and C(n, , K ) with the following property: if M n is an n-dimensional connected closed Riemannian manifold with kR M k∞ ≤ K and dG H (M, B) ≤ A0 (n, , K ), then there are (1) a Z-graded real vector bundle E on B, (2) a flat degree-1 superconnection A0 on E, and (3) a Euclidean inner product h E on E,


271

such that if λ p, j (M) is the jth eigenvalue of the p-form Laplacian on M, λ p, j (B; E) is the jth eigenvalue of 4 Ep , and min λ p, j (M), λ p, j (B; E) ≤ A(n, , K ) dG H (M, B)−2 − C(n, , K ),

(1.4)

then λ p, j (M) is -close to λ p, j (B; E). Using [2], one can also show that the eigenspaces of 4 Ep are L ∞ -close to those of 4M p , with respect to the embedding (B; E) → (M). In the case of the Laplacian on functions, only E 0 is relevant. Although E 0 is the 0 trivial R-bundle on B with a trivial connection, its Euclidean inner product h E need not be trivial and corresponds exactly to the measure in Fukaya’s work. In order to apply Theorem 2, we prove a compactness result for the superconnection and the Euclidean metric. Definition 2 Let S E be the space of degree-1 superconnections on E, let G E be the group of smooth grading-preserving GL(E) gauge transformations on E, and let H E be the space of graded Euclidean inner products on E. We equip S E and H E with the C ∞ topology. Give (S E × H E )/G E the quotient topology. THEOREM 3 In Theorem 2, we may assume that E is one of a finite number of isomorphism classes of real Z-graded topological vector bundles {E i } on B. Furthermore, there are compact subsets D Ei ⊂ (S Ei × H Ei )/G Ei depending on n, , and K , such that we may assume that the gauge equivalence class of the pair (A0 , h E ) lies in D E .

We remark that there may well be a sequence of topologically distinct Riemannian manifolds of a given dimension, with uniformly bounded sectional curvatures, which converge to B in the Gromov-Hausdorff topology (see Example 3). This contrasts with the finiteness statement in Theorem 3. The eigenvalues of 4 Ep are continuous with respect to [(A0 , h E )] ∈ (S E × H E )/G E . One application of Theorem 3 is the following relationship between the spectra of 4 M p and the ordinary differential form Laplacian on B. THEOREM 4 Under the hypotheses of Theorem 2, let λ0p, j (B) be the jth eigenvalue of the Laplacian L p−r on r r (B) ⊗ Rdim(E ) . Then there is a positive constant D(n, , K ) such that

e−/2 λ0p, j (B)1/2 − D(n, , K ) ≤ λ p, j (M)1/2 ≤ e/2 λ0p, j (B)1/2 + D(n, , K ). (1.5)

272

JOHN LOTT

Now consider a flat degree-1 superconnection A0 on a real Z-graded vector bundle E over a smooth manifold B. As (A0 )2 = 0, there is a cohomology H∗ (A0 ) for the action of A0 on (B; E), the latter having the total grading. There is a flat Z-graded “cohomology” vector bundle H∗ (A0[0] ) on B. Furthermore, there is a spectral sequence to compute H∗ (A0 ), with E 2 -term H∗ (B; H∗ (A0[0] )). Suppose that M is a connected closed manifold with at least j small eigenval∞ ues of 4 p for j > b p (M). Consider a sequence of Riemannian metrics {gi }i=1 in M (M, K ) with limi→∞ λ p, j (M, gi ) = 0. There must be a subsequence of ∞ which converges to a lower-dimensional limit space X . That is, we are {(M, gi )}i=1 in the collapsing situation. Suppose that the limit space is a smooth manifold B. From ∞ to obtain a sinTheorems 2 and 3, we can take a further subsequence of {(M, gi )}i=1 ∞ of superconnecgle vector bundle E on B, equipped with a sequence {(Ai0 , h iE )}i=1 tions and Euclidean inner products. Using the compactness result in Theorem 3, we can take a convergent subsequence of these pairs, modulo gauge transformations, to obtain a superconnection A0∞ on E with dim Ker(4 Ep ) ≥ j. Then dim(H p (A0∞ )) ≥ j. It is no longer true that H∗ (A0∞ ) ∼ = H∗ (M; R) for this limit superconnection. How∗ 0 ever, we can analyze H (A∞ ) using the spectral sequence. We obtain X j≤ dim Ha (B; Hb (A0∞,[0] )) . (1.6) a+b= p

This formula has some immediate consequences. The first one is a bound on the number of small eigenvalues of the 1-form Laplacian. COROLLARY 1 Suppose that M has j small eigenvalues of the 1-form Laplacian with j > b1 (M). Let X be the limit space coming from the above argument. Suppose that X is a smooth manifold B. Then

j ≤ b1 (B) + dim(M) − dim(B) ≤ b1 (M) + dim(M).

(1.7)

The second consequence is a bound on the number of small eigenvalues of the pform Laplacian for a manifold that is Gromov-Hausdorff close to a codimension-1 manifold. 2 Let B be a connected closed (n − 1)-dimensional Riemannian manifold. Then for any K ≥ 0, there are δ, c > 0 with the following property: suppose that M is a connected closed smooth n-dimensional Riemannian manifold with kR M k∞ ≤ K and dG H (M, B) < δ. First, M is the total space of a circle bundle over B. Let O be the orientation bundle of M → B, a flat real line bundle on B. Then λ p, j (M, g) > c for j = b p (B) + b p−1 (B; O ) + 1. COROLLARY


273

The rest of our results concern small eigenvalues in collapsing sequences. Definition 3 If M → B is an affine fiber bundle, a collapsing sequence associated to the affine ∞ ∈ M (M, K ) for some K ≥ 0 such that fiber bundle is a sequence of metrics {gi }i=1 limi→∞ (M, gi ) = B in the Gromov-Hausdorff topology, and for some > 0, each (M, gi ) is -bi-Lipschitz to a Riemannian affine fiber bundle structure on M → B. We show that there are three mechanisms to make small positive eigenvalues of the differential form Laplacian on M in a collapsing sequence. Either the differential form Laplacian on the fiber admits small positive eigenvalues, the holonomy of the flat “cohomology” bundle on B fails to be semisimple, or the Leray spectral sequence of M → B does not degenerate at the E 2 -term. THEOREM 5 ∞ be a collapsing sequence associated to an affine fiber bundle M → Let {(M, gi )}i=1 B. Suppose that limi→∞ λ p, j (M, gi ) = 0 for some j > b p (M). Write the fiber Z of the affine fiber bundle as the quotient of a nilmanifold b Z =b 0 \N by a finite group F. Then (1) for some q ∈ [0, p], bq (Z ) < dim(3q (n∗ ) F ); or (2) for all q ∈ [0, p], bq (Z ) = dim(3q (n∗ ) F ), and for some q ∈ [0, p], the holonomy representation of the flat vector bundle Hq (Z ; R) on B fails to be semisimple; or (3) for all q ∈ [0, p], bq (Z ) = dim(3q (n∗ ) F ) and the holonomy representation of the flat vector bundle Hq (Z ; R) on B is semisimple, and the Leray spectral sequence to compute H p (M; R) does not degenerate at the E 2 term.

Examples show that small positive eigenvalues can occur in each of the three cases in Theorem 5. Theorem 5 has some immediate consequences. The first is a characterization of when the 1-form Laplacian has small positive eigenvalues in a collapsing sequence. COROLLARY

3

∞ {(M, gi )}i=1

Let be a collapsing sequence associated to an affine fiber bundle M → B. Suppose that limi→∞ λ1, j (M, gi ) = 0 for some j > b1 (M). Then (1) the differential d2 : H0 (B; H1 (Z ; R)) → H2 (B; R) in the Leray spectral sequence for H∗ (M; R) is nonzero; or (2) the holonomy representation of the flat vector bundle H1 (Z ; R) on B has a nontrivial unipotent subrepresentation; or

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(3)

Z is almost flat but not flat and there is a nonzero covariantly constant section of the flat vector bundle (H 1 (A0∞,[0] ))/(H 1 (Z ; R)∞ ).

The differential d2 : H0 (B; H1 (Z ; R)) → H2 (B; R) can be considered to be a type of Euler class; in the case of an oriented circle bundle over a smooth base, it gives exactly the Euler class. The second consequence is a characterization of when the p-form Laplacian has small positive eigenvalues in a collapsing sequence over a circle. COROLLARY

4

∞ {(M, gi )}i=1

Let be a collapsing sequence associated to an affine fiber bundle M → S 1 . Suppose that limi→∞ λ p, j (M, gi ) = 0 for some j > b p (M). Write the fiber Z of the affine fiber bundle as in Theorem 5. Then (1) for some q ∈ { p − 1, p}, bq (Z ) < dim(3q (n∗ ) F ); or (2) for q ∈ { p − 1, p}, bq (Z ) = dim(3q (n∗ ) F ), and if 8∗ ∈ Aut(H∗ (Z ; R)) denotes the holonomy action on the fiber cohomology, then 8 p or 8 p−1 has a nontrivial unipotent factor in its Jordan normal form. The third consequence is a characterization of when the p-form Laplacian has small positive eigenvalues in a collapsing sequence over a codimension-1 manifold. COROLLARY

5

∞ {(M, gi )}i=1

Let be a collapsing sequence associated to an affine fiber bundle M → B with dim(B) = dim(M) − 1. Suppose that limi→∞ λ p, j (M, gi ) = 0 for some j > b p (M). Let O be the orientation bundle of M → B, a flat real line bundle on B. Let χ ∈ H2 (B; O ) be the Euler class of the circle bundle M → B. Let Mχ be multiplication by χ . Then Mχ : H p−1 (B; O ) → H p+1 (B; R) is nonzero or Mχ : H p−2 (B; O ) → H p (B; R) is nonzero. Finally, we give a class of examples for which the inequality in (1.6) is an equality. THEOREM 6 Suppose that M → B is an affine fiber bundle with a smooth base B and fiber Z = b Z /F, where b Z is a nilmanifold b 0 \N and F is a finite group. Let

n = n0[0] ⊃ n0[1] ⊃ · · · ⊃ n0[S] ⊃ 0

(1.8)

be the lower central series of the Lie algebra n. Let c(n) be the center of n. For 0 ≤ k ≤ S, put n[k] = n0[k] + c(n) (1.9)


275

and put r[k] = n[k] /n[k+1] . Let P be the principal Aff(Z )-bundle such that M = L P ×Aff(Z ) Z . Let G = b G b be the Z-graded flat vector bundle on B with M S F G b = P ×Aff(Z ) 3b r∗[k] .

(1.10)

k=0

Then for any 0 ≤ p ≤ dim(M), M has of the p-form Laplacian.

P

a+b= p

dim(Ha (B; G b )) small eigenvalues

The structure of the paper is as follows. In Section 2 we give examples of collapsing which show that the superconnection formalism is necessary. In Section 3 we give some background information about infranilmanifolds Z and show that the orthogonal projection onto the parallel forms of Z is independent of the choice of parallel Riemannian metric. In Section 4 we give a detailed analysis of the spectrum of the differential form Laplacian on an infranilmanifold. In Section 5 we show that the eigenvalues of the superconnection Laplacian are continuous with respect to the superconnection, the Riemannian metric, and the Euclidean inner product. We then analyze the differential form Laplacian on a Riemannian affine fiber bundle and prove Theorem 1. In Section 6 we consider manifolds M that are Gromov-Hausdorff close to a smooth manifold B and prove Theorems 2, 3, and 4. Section 7 uses the compactness results to prove Theorem 5 and Corollaries 1 – 5. We then prove Theorem 6. More detailed descriptions appear at the beginnings of the sections. 2. Examples For notation in this paper, if G is a group that acts on a set X , we let X G denote the set of fixed points. If B is a smooth manifold and E is a smooth vector bundle on B, we let (B; E) denote the smooth E-valued differential forms on B. If n is a nilpotent Lie algebra on which a finite group F acts by automorphisms, then n∗ denotes the dual space, 3∗ (n∗ ) denotes the exterior algebra of the dual space, and 3∗ (n∗ ) F denotes the F-invariant subspace of the exterior algebra. Example 1 Let N be a simply connected, connected nilpotent Lie group, such as the 3dimensional Heisenberg group. Let n be its Lie algebra of left-invariant vector fields, let g T N be a left-invariant Riemannian metric on N , and let 4 N be the corresponding Laplacian on ∗ (N ). (For simplicity of notation, we omit reference to the form degree p.) The left-invariant differential forms 3∗ (n∗ ) form a subcomplex of ∗ (N ) with differential d n , on which 4 N restricts to a finite-dimensional operator 4n . If 0 is a lattice in N , then the left-invariant forms on N push down to forms on Z = 0\N , giving a subcomplex of ∗ (Z ) which is isomorphic to 3∗ (n∗ ). One knows that H∗ (Z ; R)

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is isomorphic to the cohomology of this subcomplex (see [28, Corollary 7.28]). We see that the spectrum σ (4n ) of 4n is contained in the spectrum σ (4 Z ) of the differential form Laplacian on ∗ (Z ). ∞ is a sequence of lattices in N with quotients Z = 0 \N Suppose that {0i }i=1 i i ∞ obviously converges to a point, with such that limi→∞ diam(Z i ) = 0. Then {Z i }i=1 bounded sectional curvature in the collapse. We see that there are eigenvalues of 4 Z i which are constant in i, namely, those that come from σ (4n ). By Proposition 2, the other eigenvalues go to infinity as i → ∞. If N is nonabelian, then there are positive eigenvalues of 4 Z i which are constant in i. In terms of Theorem 2, B is a point, E ∗ = 3∗ (n∗ ), and A0 = A0[0] = d n . This shows that the term A0[0] does appear in examples. In fact, A0[0] = 0 if and only if N is abelian. By choosing different left-invariant metrics on N , we can make σ (4n ) arbitrarily close to zero while keeping the sectional curvature bounded. (In fact, the sectional curvature goes to zero.) This is a special case of Theorem 6. We see that, in general, there is no nontrivial lower bound on the first positive eigenvalue of 4 Z under the assumptions of bounded sectional curvature and bounded diameter. Example 2 Let M be a compact manifold with a free T k -action. Let g T M be a T k -invariant Riemannian metric on M. Then for > 0, there is a Riemannian metric gT M obtained by multiplying g T M in the direction of the T k -orbit by . Clearly, lim→0 (M, gT M ) = M/T k , the collapse being with bounded sectional curvature (see [13]). This collapsing is an example of the so-called adiabatic limit, for which the eigenvalues of the differential form Laplacian have been studied in [5], [16], [18], and [26]. Let E be the flat “cohomology” vector bundle on M/T k with fiber H∗ (T k ; R); in fact, it is a trivial bundle. The results of the cited references imply that as → 0, the eigenvalues of 4 M which remain finite approach those of the Laplacian on ∗ (M/T k ; E). In particular, the number of eigenvalues of the p-form Laplacian which go to zero as → 0 P is a+b= p dim Ha (M/T k ; E b )) , which is also the dimension of the E 2 -term of the Leray spectral sequence for computing H p (M; R). This is consistent with Theorems 5 and 6. Let A0 be the superconnection on E coming from Theorem 2, using gT M . Then lim→0 A0 = ∇ E . Example 3 Suppose that M is the total space of an oriented circle bundle, with an S 1 invariant Riemannian metric. For k ∈ Z+ , consider the subgroup Zk ⊂ S 1 . Then limk→∞ M/Zk = M/S 1 , the collapse obviously being with bounded sectional curvature. By Fourier analysis, one finds that as k → ∞, the spectrum of 4 M/Zk ap-


277

proaches the spectrum of the Laplacian on S 1 -invariant (not necessarily basic) differential forms on M. In terms of Theorem 2, B = M/S 1 and E is the direct sum of two trivial R-bundles on B. Let T be the curvature 2-form of the fiber bundle M → M/S 1 . Then one finds that the Laplacian acting on S 1 -invariant forms on M is ∗ ∗ isomorphic to the Laplacian 4 E = A0 A0 + A0 A0 , where A0 is the extension of the superconnection on C ∞ (B; E) = C ∞ (B) ⊕ C ∞ (B) given by ! 0 ∇E T 0 A = . (2.1) 1 0 ∇E 0

1

Here ∇ E and ∇ E are product connections. This shows that the term A0[2] does appear in examples. Note that if M is simply connected, then {M/Zk }∞ k=1 are mutually nondiffeomorphic. 3. Infranilmanifolds In this section we first recall some basic facts about infranilmanifolds. Then in Proposition 1 we show that the orthogonal projection onto the parallel differential forms of Z comes from an averaging technique and so is independent of the choice of parallel metric on Z , a result that is crucial in what follows. Let N be a simply connected, connected nilpotent Lie group. Following [12], when N acts on a manifold on the left, we denote it by N L , and when it acts on a manifold on the right, we denote it by N R . As in [12], let us recall the elementary but confusing point that the right action of N on N generates left-invariant vector fields, while the left action of N on N generates right-invariant vector fields. There is a flat linear connection ∇ aff on N which is characterized by the fact that left-invariant vector fields are parallel. The group Aff(N ) of diffeomorphisms of N e Aut(N ). which preserve ∇ aff is isomorphic to N L × Suppose that 0 is a discrete subgroup of Aff(N ) which acts freely and cocompactly on N , with 0 ∩ N L of finite index in 0. Then the quotient space Z = 0\N is an infranilmanifold modeled on N . We have the short exact sequences p

1 −→ N L −→ Aff(N ) −→ Aut(N ) −→ 1 and

p

1 −→ 0 ∩ N L −→ 0 −→ p(0) −→ 1.

(3.1) (3.2)

Put b 0 = 0 ∩ N L and F = p(0). Then F is a finite group. There is a normal cover b Z =b 0 \N of Z with covering group F. The connection ∇ aff descends to a flat connection on T Z , which we again denote by ∇ aff . Let Aff(Z ) denote the affine group of Z , let Aff0 (Z ) denote the connected component of the identity in Aff(Z ), and let aff(Z ) denote the affine Lie algebra of Z . Any element of Aff(Z ) can be lifted to an element of Aff(N ). That is,

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JOHN LOTT

Aff(Z ) = 0\(N0 Aff(N )), where N0 Aff(N ) is the normalizer of 0 in Aff(N ). Similarly, Aff0 (Z ) = C(0)\(C0 Aff(N )), where C0 Aff(N ) is the centralizer of 0 in Aff(N ) and C(0) is the center of 0. There is a short exact sequence 1 −→ Aff0 (Z ) −→ Aff(Z ) −→ Out(0) −→ 1.

(3.3)

As affine vector fields on Z can be lifted to F-invariant affine vector fields on b Z, we have aff(Z ) = aff( b Z ) F . If C(N ) denotes the center of N , then Aff0 ( b Z ) = (b 0∩ C(N R ))\N R . In particular, if n is the Lie algebra of N , then F acts by automorphisms on n and aff(Z ) = n FR . The F-invariant subspace 3∗ (n∗ ) F of 3∗ (n∗ ) is isomorphic to the vector space of differential forms on Z which are parallel with respect to ∇ aff or, equivalently, to e F)-invariant subspace of ∗ (N ). the (N L × Let g T Z be a Riemannian metric on Z which is parallel with respect to ∇ aff . Such metrics correspond to F-invariant inner products on n. Let diam(Z ) denote the diameter of Z , let ∇ Z denote the Levi-Civita connection of Z , and let R Z denote the Riemann curvature tensor of Z . Let P : ∗ (Z ) → 3∗ (n∗ ) F be the orthogonal projection onto parallel differential forms. PROPOSITION 1 The orthogonal projection P is independent of the parallel metric g T Z .

Proof We first consider the case when F = {e}, so that Z is a nilmanifold 0\N . As N is R nilpotent, it has a bi-invariant Haar measure µ. We normalize µ so that 0\N dµ = 1. Given ω ∈ ∗ (Z ), let e ω ∈ ∗ (N ) be its pullback to N . If L g denotes the left action of g ∈ N L on N , then for all γ ∈ 0, L ∗γ g e ω = L ∗g L ∗γ e ω = L ∗g e ω.

(3.4)

Hence it makes sense to define e ω ∈ ∗ (N ) by Z e ω= (L ∗g e ω) dµ(g).

(3.5)

0\N L

For h ∈ N L , L ∗h e ω=

Z 0\N L

Z = 0\N L

(L ∗h L ∗g e ω) dµ(g)

Z =

(L ∗g e ω) dµ(gh −1 )

0\N L

(L ∗gh e ω) dµ(g)

Z = 0\N L

(L ∗g e ω) dµ(g) = e ω.

(3.6)


279

Thus e ω is N L -invariant and, in particular, descends to a form ω ∈ ∗ (Z ). Put P(ω) = ω. Then P is idempotent, with Im(P) being the parallel differential forms. By construction, P is independent of the choice of g T Z . It remains to show that P is self-adjoint. Given η ∈ ∗ (Z ), let e η be its lift to N . Consider the function f : N × N → R given by

f (g, n) = he η, L ∗g e ωin = e η(n), e ω(gn) n . (3.7) For γ ∈ 0, we have f (γ g, n) = f (gγ −1 , γ n) = f (g, n). It follows that we can write Z hη, Pωi Z = he η, L ∗g e ωin dµ(g) dµ(n), (3.8) (0×0)\(N ×N )

where the action of 0 × 0 on N × N is (γ1 , γ2 ) · (g, n) = (γ1 gγ2−1 , γ2 n). Changing variable to g 0 = gn, we have Z Z hη, Pωi Z = he η, L ∗g0 n −1 e ωin dµ(g 0 n −1 ) dµ(n) (3.9) 0\N L 0\N L Z Z = he η, L ∗n −1 L ∗g0 e ωin dµ(g 0 ) dµ(n) 0\N L 0\N L Z Z = hL ∗ne η, L ∗g0 e ωie dµ(g 0 ) dµ(n) 0\N L 0\N L Z Z = ηie dµ(n) dµ(g 0 ) hL ∗g0 e ω, L ∗ne 0\N L

0\N L

= hω, Pηi Z = hPη, ωi Z . Thus P is self-adjoint. In the case of general F, we can apply the above argument equivariantly on b Z with respect to F. As F acts isometrically on b Z , it commutes with the orthogonal projection P on b Z . As F preserves µ, it also commutes with the averaging operator b P on Z . The proposition follows. 4. Eigenvalue estimates on infranilmanifolds In this section we show in Proposition 2 that if an infranilmanifold Z has bounded sectional curvature and a diameter that goes to zero, then all of the eigenvalues of 4 Z go to infinity, except for those that correspond to eigenforms which are parallel on Z . Let N be a simply connected, connected n-dimensional nilpotent Lie group with n a left-invariant Riemannian metric. Let {ei }i=1 be an orthonormal basis of n. DeP fine the structure constants of n by [ei , e j ] = nk=1 cki j ek . Take the corresponding n n . Then the comleft-invariant basis {ei }i=1 of T N , with dual basis of 1-forms {τ i }i=1 P P n ponents ωij = k=1 ωijk τ k of the Levi-Civita connection 1-form ω = k ωk τ k are

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the constant matrices

1 j ωijk = − (cijk − c ik − cki j ). 2

(4.1)

LEMMA 1 Let κ denote the scalar curvature of Z . Then n X

(cijk )2 = −4κ.

(4.2)

i, j,k=1

Proof P As i,n j,k=1 (cijk )2 is independent of the choice of orthonormal basis, we compute it using a special orthonormal basis. Recall the definition of n[k] from (1.9). In particular, n n[S] = c(n). Following the notation of [20, §6], we take an orthonormal basis {ei }i=1 of n such that ei ∈ n[O(i)] for some nondecreasing function O : {1, . . . , n} → {0, . . . , S},

(4.3)

and ei ⊥ n[O(i)+1] . For a general Riemannian manifold, we have the structure equations X dτ i = − ωi j ∧ τ j ,

(4.4)

j

j = dω j + i

i

X

ωim ∧ ωmj .

m

Then i j = d

X

ωijl τ l +

X

ωim ∧ ωmj

(4.5)

m

l

=

X

(ek ωijl )τ k ∧ τ l +

X m

k,l

ωijm dτ m +

X

ωimk ωmjl τ k ∧ τ l .

k,l,m

This gives the Riemann curvature tensor as X R ijkl = ek ωijl − el ωijk + [−ωijm ωmlk + ωijm ωmkl + ωimk ωmjl − ωiml ωmjk ]. (4.6) m

Then κ=

X X (ei ωij j − e j ωiji ) + [−ωijm ωmji + ωijm ωmij + ωimi ωmj j − ωim j ωmji ] i, j

i, j,m

X X = (ei ωij j − e j ωiji ) + [ωijm ωmij + ωimi ωmj j ]. i, j

i, j,m

(4.7)


281

In our case, the components of the connection matrix are constant. Also, as n is nilpotent, j ωij j = c i j = 0. (4.8) Then one obtains κ=−

n X

ωijk ωik j .

(4.9)

i, j,k=1

Separating ωijk into components that are symmetric or antisymmetric in j and k, and using (4.1), we obtain n n 1 X j 1 X i 2 k 2 κ=− (c ik + c i j ) + (c jk ) 4 4

=−

i, j,k=1 n h X

i, j,k=1

(4.10)

i, j,k=1

i 1 j k 1 c ik c i j + (cijk )2 . 2 4

j

As n is nilpotent, it follows that c ik cki j = 0. This proves the lemma. Let Z be an infranilmanifold with an affine-parallel metric. Let 4 Z denote the Laplacian acting on ∗ (Z ). Let 4inv be the finite-dimensional Laplacian acting on 3∗ (n∗ ) F . PROPOSITION 2 There are positive constants A and A0 , depending only on dim(Z ), such that if kR Z k∞ diam(Z )2 ≤ A0 , then the spectrum σ (4 Z ) of 4 Z satisfies σ (4 Z ) ∩ 0, A diam(Z )−2 = σ (4inv ) ∩ 0, A diam(Z )−2 . (4.11)

Proof Recall the definition of P from Proposition 1. It is enough to show that under the hypotheses of the present proposition, the spectrum of 4 Z on Ker(P ) is bounded below by A diam(Z )−2 . As b Z isometrically covers Z with covering group F, the spectrum of 4 Z on b Ker(P ) ⊂ ∗ (Z ) is contained in the spectrum of 4 Z on Ker(P ) ⊂ ∗ ( b Z ). LEMMA 2 There is a function η : N → N such that

diam( b Z ) ≤ η |F| diam(Z ).

(4.12)

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JOHN LOTT

Proof Let b z 1 ,b z2 ∈ b Z be such that diam( b Z ) = d(b z 1 ,b z 2 ). It is easy to see that d(b z 1 , F ·b z2) ≤ diam(Z ). Let z 2 ∈ Z be the projection of b z2 ∈ b Z . Then it is enough to bound k · kgeo from above on π1 (Z , z 2 ) ∼ = F, that is, to bound the minimal lengths of curves in the classes of π1 (Z , z 2 ). From [23, Proposition 3.22], there is a set of generators of π1 (Z , z 2 ) on which k · kgeo is bounded above by 2 diam(Z ). Given r ∈ N, there is a finite number of groups of order r , up to isomorphism, and each of these groups has a finite number of generating sets. The lemma follows. Furthermore, there is a universal bound |F| ≤ const.(dim(Z )) (see [9]). Hence, without loss of generality, we may assume that F = {e}, so that Z is a nilmanifold 0\N . Let E i denote exterior multiplication on ∗ (Z ) by τ i , and let I i denote interior multiplication by ei . From the Bochner formula, if η ∈ ∗ (Z ), then XZ hη, 4 Z ηi Z = h∇ Z η, ∇ Z ηi Z + RiZjkl hE i I j η, E k I l ηi d vol Z . (4.13) i jkl

Z

Using the left-invariant vector fields on N , there is an isometric isomorphism ∗ (Z ) ∼ = C ∞ (Z ) ⊗ 3∗ (n∗ ).

(4.14)

With respect to this isomorphism, X j ∇eTi Z = (ei ⊗ Id) + Id ⊗ ω ki E j I k ,

(4.15)

j,k

where E j and I k now act on 3∗ (n∗ ). It follows that 2 X

X X j h∇ Z η, ∇ Z ηi Z ≥ (ei ⊗ Id)η, (ei ⊗ Id)η Z − ω ki E j I k η . i

i

j,k

Z

(4.16)

Let 40Z be the ordinary Laplacian on C ∞ (Z ). With respect to (4.14), consider the operator 40Z ⊗ Id. We have X

η, (40Z ⊗ Id)η Z = (ei ⊗ Id)η, (ei ⊗ Id)η Z . (4.17) i

Using (4.1), (4.13), (4.16), (4.17), and Lemma 1, we obtain

hη, 4 Z ηi Z ≥ η, (40Z ⊗ Id)η Z − const. kR Z k∞ |η|2Z .

(4.18)

In terms of (4.14), Ker(P ) ∼ = 1⊥ ⊗3∗ (n∗ ), where 1 denotes the constant function on Z . Thus if η ∈ Ker(P ), then hη, (40Z ⊗ Id)ηi Z ≥ λ0,2 |η|2Z , where λ0,2 is the first positive eigenvalue of the function Laplacian on Z . There is a lower bound λ0,2 ≥ diam(Z )−2 f kR Z k∞ diam(Z )2 (4.19)


283

for some smooth function f with f (0) > 0 (see [3]). Thus the spectrum of 4 Z on Ker(P ) is bounded below by diam(Z )−2 f kR Z k∞ diam(Z )2 − const. kR Z k∞ diam(Z )2 . (4.20) Taking A = (3/4) f (0), the proposition follows. 5. Affine fiber bundles In this section we first show that the eigenvalues of a superconnection Laplacian are continuous with respect to the superconnection, the Riemannian metric, and the Euclidean inner product. We then construct the superconnection A0 associated to an affine fiber bundle M → B and prove Theorem 1. L Let B be a smooth connected closed Riemannian manifold. Let E = mj=0 E j be a Z-graded real vector bundle on B. For background information about superconnections, we refer to [4, Chapter 1.4], [7], [8], and [27]. Let A0 be a degree-1 superconnection on E. That is, A0 is an R-linear map from C ∞ (B; E) to (B; E) with a decomposition dim(B) X A0 = A0[k] , (5.1) k=0

where A0[1] is a connection ∇ E on E which preserves the Z-grading; • for k 6 = 1, A0[k] ∈ k (B; Hom(E ∗ , E ∗+1−k )). We can extend A0 to an R-linear map on (B; E) using the Leibniz rule. We assume that A0 is flat in the sense that (A0 )2 = 0. (5.2) •

0

Let h E be a Euclidean inner product on E such that E j is orthogonal to E j if j 6= j 0 . Let (A0 )∗ be the adjoint superconnection with respect to h E , and put 4 E = A0 (A0 )∗ + (A0 )∗ A0 .

(5.3)

Then 4 E preserves the total Z-grading on (B; E) and decomposes with respect to L the grading as 4 E = p 4 Ep . By elliptic theory, 4 Ep has a discrete spectrum. If g1T B and g2T B are two Riemannian metrics on B and ≥ 0, we say that g1T B and g2T B are -close if e− g2T B ≤ g1T B ≤ e g2T B . (5.4) Similarly, if h 1E and h 2E are two Euclidean inner products on E, we say that h 1E and h 2E are -close if (5.5) e− h 2E ≤ h 1E ≤ e h 2E .

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JOHN LOTT

If S1 = {λ1, j } and S2 = {λ2, j } are two countable nondecreasing ordered sets of nonnegative real numbers, then we say that S1 and S2 are -close if for all j, e− λ2, j ≤ λ1, j ≤ e λ2, j .

(5.6)

For simplicity we omit the subscript p, the form degree, in this section when its role is obvious. 3 There is an integer J = J (dim(B)) > 0 such that if g1T B and g2T B are -close, and h 1E and h 2E are -close, then the corresponding Laplacians 41E and 42E have spectra that are J -close. LEMMA

Proof As in [17, Proposition 3.1], using a trick apparently first due to Cheeger, we can write the spectrum of 4 E on Im((A0 )∗ ) as n hη, ηi o λ j = inf sup sup : η = A0 θ , (5.7) V η∈V −{0} θ ∈(B;E) hθ, θi where V ranges over j-dimensional subspaces of Im(A0 ). As the Riemannian metric and Euclidean inner product enter only in defining h·, ·i, the lemma follows as in [17]. We also need a result about how the spectrum of 4 E depends on the superconnection A0 . Given X ∈ (B; End(E)), let kX k be the operator norm for the action of X on the L 2 -completion of (B; E). If A01 and A02 are two superconnections as above, then A01 − A02 ∈ (B; End(E)). Fix g T B and h E . 4 For all j ∈ Z+ , LEMMA

√ λ j (A0 )1/2 − λ j (A0 )1/2 ≤ (2 + 2)kA0 − A0 k. 1 2 1 2

(5.8)

Proof Put x = kA01 − A02 k. If ω ∈ (B; E) is nonzero, then

and

|A0 ω| |A0 ω| |(A0 − A0 )ω| 1 1 2 − 2 ≤ ≤x |ω| |ω| |ω|

(5.9)

|(A0 )∗ ω| |(A0 )∗ ω| |((A0 )∗ − (A0 )∗ )ω| 1 2 1 2 − ≤ x. ≤ |ω| |ω| |ω|

(5.10)


Define vE1 , vE2 ∈ R2 by vEi =

285

|A0 ω| |(A0 )∗ ω| i i , . |ω| |ω|

(5.11)

Then (5.10) and (5.11) imply that √ (kE v2 k − kE v1 k) ≤ kE v2 − vE1 k ≤ 2x.

(5.12)

Hence kE v2 k2 − kE v1 k2 = (kE v2 k − kE v1 k) · (kE v2 k + kE v1 k) ≤

√ √ 2x(2kE v2 k + 2x),

(5.13)

so √ kE v1 k2 ≥ kE v2 k2 − 2 2xkE v2 k − 2x 2 √ = (kE v2 k − 2x)2 − 4x 2 . Thus kE v1 k2 ≥ max 0, (kE v2 k −

√

2x)2 − 4x 2 ,

(5.14)

(5.15)

or equivalently, hω, 4 A0 ωi 1

hω, ωi

hω, 4 0 ωi √ 2 1/2 A2 2 ≥ max 0, − 2x − 4x . hω, ωi

(5.16)

The minmax characterization of eigenvalues λ j (A0 ) = inf V

n hω, 4 0 ωi o A , hω, ωi ω∈V −{0} sup

where V ranges over j-dimensional subspaces of (B; E), implies √ 1/2 λ j (A01 ) ≥ max 0, (λ j (A02 ) − 2x)2 − 4x 2 .

(5.17)

(5.18)

An elementary calculation then gives λ j (A01 )1/2 − λ j (A02 )1/2 ≥ −(2 +

√ 2)x.

(5.19)

Symmetrizing in A01 and A02 , the proposition follows. Let M be a closed manifold that is the total space of an affine fiber bundle, as in Definition 1. Let T H M be a horizontal distribution on M so that the corresponding holonomy on B lies in Aff(Z ). If m ∈ Z b , then using T H M, we can write b3∗ (Tm∗ Z b ). That is, we can compose differential forms on 3∗ (Tm∗ M) ∼ = 3∗ (Tb∗ B)⊗ M into their horizontal and vertical components. Correspondingly, there is an infinitedimensional Z-graded real vector bundle W on B such that ∗ (M) ∼ = (B; W ) (see

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JOHN LOTT

[8, Section III(a)]). A fiber Wb of W is isomorphic to ∗ (Z b ). We call C ∞ (B; W ) the vertical differential forms. The exterior derivative d M : ∗ (M) → ∗ (M), when considered to be an operator d M : (B; W ) → (B; W ), is the extension to (B; W ) of a flat degree-1 superconnection on W . From [8, Proposition 3.4], we can write the superconnection as d Z + ∇ W + iT ,

(5.20)

where d Z ∈ C ∞ (B; Hom(W ∗ ; W ∗+1 )) is vertical differentiation, • ∇ W : C ∞ (B; W ) → 1 (B; W ) comes from Lie differentiation in the horizontal direction, and • i T ∈ 2 (B; Hom(W ∗ ; W ∗−1 )) is interior multiplication by the curvature 2form T ∈ 2 (M; T Z ) of T H M. Acting on ∗ (M), we have •

d M = d Z + d W + iT ,

(5.21)

where d W : ∗ (B; W ) → ∗+1 (B; W ) is exterior differentiation on B using ∇ W . Let E be the finite-dimensional subbundle of W such that E b consists of the elements of ∗ (Z b ) which are parallel on Z b . The fibers of E are isomorphic to 3∗ (n∗ ) F , and C ∞ (B; E) is isomorphic to the vertical differential forms on M whose restrictions to the fibers are parallel. Furthermore, the superconnection (5.20) restricts to a flat degree-1 superconnection A0 on E, as exterior differentiation on M preserves the space of fiberwise-parallel differential forms. From (5.20), A0 = d n + ∇ E + i T ,

(5.22)

where d n is the differential on 3∗ (n∗ ) F and ∇ E comes from T H M through the action of Aff(Z ) on 3∗ (n∗ ) F . Acting on (B; E), we have A0 = d n + d E + i T ,

(5.23)

where d E is exterior differentiation on (B; E) using ∇ E . Remark. The connection ∇ E is generally not flat. As A0 is flat, we have (∇ E )2 = −(d n i T + i T d n ).

(5.24)

Thus the curvature of ∇ E is given by Lie differentiation with respect to the (negative of the) curvature 2-form T . More geometrically, given b ∈ B, let γ be a loop in B starting from b, and let h(γ ) ∈ Aff(Z b ) be the holonomy of the connection T H M around γ . Then the holonomy of ∇ E around γ is the action of h(γ ) on the fiber E b .


287

In particular, the infinitesimal holonomy of ∇ E lies in the image of the Lie algebra aff(Z ) in End(E b ). From the discussion after (3.3), aff(Z ) lies in n R . As the elements of E b are N L -invariant forms on N , they are generally not annihilated by aff(Z ). Suppose in addition that M is a Riemannian affine fiber bundle, as in Definition 1. Then g T Z induces an L 2 inner product h W on W and a Euclidean inner product h E on E. Let diam(Z ) denote the maximum diameter of the fibers {Z b }b∈B in the intrinsic metric, and let 5 denote the second fundamental forms of the fibers. From g T Z , T H M, and g T B , we obtain a Riemannian metric g T M on M. Let 4 M denote the Laplacian acting on ∗ (M), and define 4 E , acting on (B; E), as in (5.3). Let R M denote the Riemann curvature tensor of g T M . Let P fib be fiberwise orthogonal projection from (B; W ) to (B; E). We claim that P fib commutes with d M . Looking at (3.5), P fib clearly commutes with d Z . Using the fact that the holonomy of T H M lies in Aff(Z ), it follows from (3.5) and the proof of Proposition 1 that P fib commutes with ∇ W . As T takes values in parallel vector fields on Z , it follows from (3.5) and the proof of Proposition 1 that P fib commutes with i T . Thus P fib commutes with d M . As the fiberwise metrics are parallel on the fibers, it follows that P fib also commutes with (d M )∗ . Then with respect to the decomposition ∗ (M) = Im(P fib ) ⊕ Ker(P fib ), 4 M is isomorphic to 4 E ⊕ 4 M Ker(P fib ) . Proof of Theorem 1 From Proposition 2, there is a constant A > 0 such that for all b ∈ B, the spectrum of 4 Z b Ker(P ) is bounded below by A · diam(Z b )−2 . It suffices to show that there is a constant C as in the statement of the theorem such that σ 4 M Ker(P fib ) ⊂ A diam(Z )−2 − C(kR M k∞ + k5k2∞ + kT k2∞ ), ∞ . (5.25) We use the notation of [8, Section III(c)] to describe the geometry of the fiber bundle M. In particular, lowercase Greek indices refer to horizontal directions, lowercase italic indices refer to vertical directions, and uppercase italic indices refer to dim(Z ) dim(B) either. Let {τ i }i=1 and {τ α }α=1 be a local orthonormal basis of 1-forms as in dim(Z ) dim(B) [8, Section III(c)], with dual basis {ei }i=1 and {eα }α=1 . Let E J be exterior mulJ J tiplication by τ , and let I be interior multiplication by e J . The tensors 5 and T P i j P i β are parts of the connection 1-form component ωiα = j ω αj τ + β ω αβ τ with symmetries ωαk j = ωα jk = −ω jαk = −ωkα j , ωβα j = −ωαβ j = −ωα jβ = ω jαβ = ωβ jα = −ω jβα .

(5.26)

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Given η ∈ ∗ (M), the Bochner formula gives X Z hη, 4 M ηi M = h∇ M η, ∇ M ηi M + R PMQ R S hE P I Q η, E R I S ηi d vol M . M

P Q RS

(5.27) Here ∇ M : C ∞ (M; 3∗ T ∗ M) → C ∞ (M; T ∗ M ⊗ 3∗ T ∗ M)

(5.28)

∇V

+ ∇ H , where

∇ V : C ∞ (M; 3∗ T ∗ M) → C ∞ (M; T vert M ⊗ 3∗ T ∗ M)

(5.29)

is, of course, the Levi-Civita connection on

M. We can write ∇ M

=

denotes covariant differentiation in the vertical direction and ∇ H : C ∞ (M; 3∗ T ∗ M) → C ∞ (M; T hor M ⊗ 3∗ T ∗ M)

(5.30)

denotes covariant differentiation in the horizontal direction. Then hη, 4 M ηi M = h∇ V η, ∇ V ηi M + h∇ η, ∇ ηi M + H

H

Z

X M P Q RS

R PMQ R S hE P I Q η, E R I S ηi d vol M

≥ h∇ V η, ∇ V ηi M − const. kR M k∞ hη, ηi M Z Z V 2 = |∇ η| (z) − const. kR M k∞ |η(z)|2 d vol Z b d vol B . (5.31) B

Zb

Let ∇ T Z : C ∞ M; 3∗ (T ∗,vert M) → C ∞ M; T ∗ M ⊗ 3∗ (T ∗,vert M)

(5.32)

denote the Bismut connection acting on 3∗ (T ∗,vert M) (see [4, Proposition 10.2], [7, Definition 1.6]). On a given fiber Z b , there is a canonical flat connection on T hor M Z . Hence we can use ∇ T Z to vertically differentiate sections of 3∗ (T ∗ M) = b b3∗ (T ∗,hor M). That is, we can define 3∗ (T ∗,vert M)⊗ ∇ T Z : C ∞ M; 3∗ (T ∗ M) → C ∞ M; T ∗,vert M ⊗ 3∗ (T ∗ M) . (5.33) Explicitly, with respect to a local framing, ∇eTi Z η = ei η +

X

j

ω ki E j I k η

(5.34)

j,k

and ∇eVi η = ∇eTi Z η +

X jα

ω αi E j I α η + j

X αk

ωαki E α I k η +

X αβ

ωαβi E α I β η.

(5.35)


289

Then from (5.34) and (5.35), h∇ V η, ∇ V ηi M Z Z TZ 2 ≥ |∇ η| (z) − const.(kTb k2 + k5b k2 )|η|2 (z) d vol Z b d vol B . (5.36) B

Zb

On a given fiber Z b , for η Z b ∈ ∗ (Z b ), we have Z Zb |∇ T Z b η Z b |2 (z) d vol Z b hη Z b , 4 η Z b i Z b = Zb XZ b + RiZjkl hE i I j η, E k I l ηi d vol Z b . i jkl

(5.37)

Zb

If η Z b ∈ Ker(P ), then hη, 4 Z b ηi Z b ≥ A diam(Z b )−2 hη, ηi Z b . Hence Z Zb

|∇ T Z b η Z b |2 (z) d vol Z b ≥ A diam(Z b )−2 − const. kR Z b k∞ hη, ηi Z b .

(5.38)

(5.39)

From (5.31), (5.36), and (5.39), if η ∈ Ker(P fib ), then hη,4 M ηi M ≥ A diam(Z )−2 − const.(kR M k∞ + kT k2∞ + k5k2∞ + kR Z k∞ ) hη, ηi M . (5.40) Using the Gauss-Codazzi equation, we can estimate kR Z k∞ in terms of kR M k∞ and k5k2∞ . The theorem follows. 6. Collapsing to a smooth base In this section we prove Theorem 2, concerning the spectrum of the Laplacian 4 M on a manifold M which is Gromov-Hausdorff close to a smooth manifold B. We prove Theorem 3, showing that the pairs (A0 , h E ) that appear in the conclusion of Theorem 2 satisfy a compactness property. We then prove Theorem 4, relating the spectrum of 4 M to the spectrum of the differential form Laplacian on the base space B. Proof of Theorem 2 For simplicity, we omit reference to p. Let g0T M denote the Riemannian metric on M. From [17] or Lemma 3, if a Riemannian metric g1T M on M is -close to g0T M , then the spectrum of 4 M , computed with g1T M , is J -close to the spectrum computed with

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JOHN LOTT

g0T M . We use the geometric results of [12] to find a metric g2T M on M which is close to g0T M and to which we can apply Theorem 1. First, as in [12, (2.4.1)], by the smoothing results of U. Abresch and others (see [12, Theorem 1.12]), we can find metrics on M and B which are -close to the original metrics such that the new metrics satisfy k∇ i Rk∞ ≤ Ai (n, ) for some appropriate ∞ . By rescaling, we may assume that kR M k B sequence {Ai (n, )}i=0 ∞ ≤ 1, kR k∞ ≤ 1, and inj(B) ≥ 1. Let g1T M denote the new metric on M. We now apply [12, Theorem 2.6], with B fixed. It implies that there are positive constants λ(n) and c(n, ), so that if dG H (M, B) ≤ λ(n), then there is a fibration f : M → B such that (1) diam( f −1 (b)) ≤ c(n, ) dG H (M, B), (2) f is a c(n, )-almost Riemannian submersion, (3) k5 f −1 (b) k∞ ≤ c(n, ). As in [20], the Gauss-Codazzi equation, the curvature bound on M, and the second −1 fundamental form bound on f −1 (b) imply a uniform bound on {kR f (b) k∞ }b∈B . Along with the diameter bound on f −1 (b), this implies that if dG H (M, B) is sufficiently small, then f −1 (b) is almost flat. From [12, Propositions 3.6 and 4.9], we can find another metric g2T M on M which is -close to g1T M so that the fibration f : M → B gives M the structure of a Riemannian affine fiber bundle. Furthermore, by [12, Proposition 4.9], there is a sequence ∞ , so that we may assume that g T M and g T M are close in the sense that {Ai0 (n, )}i=0 1 2 k∇ i (g1T M − g2T M )k∞ ≤ Ai0 (n, ) dG H (M, B),

(6.1)

where the covariant derivative in (6.1) is that of the Levi-Civita connection of g2T M (see also [30, Theorem 1.1] for an explicit statement). In particular, there is an upper bound on kR M (g2T M )k∞ in terms of B, n, , and K . We now apply Theorem 1 to the Riemannian affine fiber bundle with metric g2T M . It remains to estimate the geometric terms appearing in (1.3). We have an estimate on k5k∞ as above. Applying O’Neill’s formula (see [6, (9.29c)]) to the Riemannian affine fiber bundle, we can estimate kT k2∞ in terms of kR M k∞ and kR B k∞ . Putting this together, the theorem follows. The vector bundles E and Euclidean inner products h E which appear in Theorem 2 are not completely arbitrary. For example, E 0 is the trivial R-bundle on B. More substantially, if E is a real Z-graded topological vector bundle on B, let C E be the space of grading-preserving connections on E, let G E be the group of smooth gradingpreserving GL(E) gauge transformations on E, and let H E be the space of graded Euclidean inner products on E. We equip C E and H E with the C ∞ -topology. Give (C E × H E )/G E the quotient topology. Let ∇ E denote the connection part A0[1] of A0 .


291

PROPOSITION 3 In Theorem 2 we may assume that E is one of a finite number of isomorphism classes of real Z-graded topological vector bundles {E i } on B. Furthermore, there are compact subsets C Ei ⊂ (C Ei × H Ei )/G Ei depending on n, , and K , such that we may assume that the gauge-equivalence class of the pair (∇ E , h E ) lies in C E .

Proof As the infinitesimal holonomy of the connection T H M lies in aff(Z ) = n FR , its action on n, which is through the adjoint representation, is nilpotent. Hence its action on 3∗ (n∗ ) F is also nilpotent. Given b ∈ B, it follows that the local holonomy group of T H M at b acts unipotently on E b . Then the structure group of E can be topologically reduced to a discrete group, and so E admits a flat connection. The rank of E is at most 2dim(M)−dim(B) . By an argument of G. Lusztig, only a finite number of isomorphism classes of real topological vector bundles over B of a given rank admit a flat connection (see [22, p. 22]). This proves the first part of the proposition. To prove the second part of the proposition, we first reduce to the case F = {e}. Recall that b Z is a nilmanifold that covers Z , with covering group F. Given g ∈ Aff(Z ), we can lift it to some b g ∈ Aff( b Z ). There is an automorphism αb g ∈ Aut(F) such that for all f ∈ F and b z∈b Z, αb z = (b g fb g −1 )(b z). g ( f ) ·b

(6.2)

Considering the different possible liftings of g, we obtain a well-defined homomorphism Aff(Z ) → Out(F). Then there is an exact sequence 1 −→ Aff( b Z ) F −→ Aff(Z ) −→ Out(F).

(6.3)

b = Let P be the principal Aff(Z )-bundle such that M = P ×Aff(Z ) Z . Put M b b P ×Aff( b Z ) F Z . Then M is an affine fiber bundle that regularly covers M, with the order of the covering group bounded in terms of |F|. Again, there is a uniform upper bound on |F| in terms of dim(Z ) (see [9]). Instead of considering M, it suffices to consider b and work equivariantly with respect to the covering group. Thus we assume that Z M is a nilmanifold with 0 ⊂ N L and F = {e}. As the fiber of E j is 3 j (n∗ ), it suffices to prove the second part of the proposition for E 1 with fiber n∗ . Let us consider instead for a moment (E 1 )∗ with fiber n. With respect to the lower central series (1.8) of n, let (E 1 )∗[k] be the vector bundle associated to P with fiber n0[k] . Then there is a filtration (E 1 )∗ = (E 1 )∗[0] ⊃ (E 1 )∗[1] ⊃ · · · ⊃ (E 1 )∗[S] ⊃ 0.

(6.4)

Let Spl be the set of splittings of the short exact sequences 0 −→ (E 1 )∗[k+1] −→ (E 1 )∗[k] −→ (E 1 )∗[k] /(E 1 )∗[k+1] −→ 0.

(6.5)

292

JOHN LOTT

Put V[k] = (E 1 )∗[k] /(E 1 )∗[k+1] and V =

S M

V[k] .

(6.6)

k=0

Let HV be the set of graded Euclidean inner products on the Z-graded real vector 1 ∗ S−1 bundle V . A Euclidean inner product h (E ) determines splittings {sk }k=0 of (6.5) 1 )∗ V (E and a Euclidean inner product h ∈ HV . Conversely, one recovers h from the S−1 splittings {sk }k=0 and h V . Thus there is an isomorphism H(E 1 )∗ ∼ Spl × H = V. 1 ∗ Let Cfil denote the set of connections on (E ) which preserve the filtration (6.4). Let CV be the set of connections on V which are grading-preserving with respect to (6.6). Let End< (V ) be the set of endomorphisms of V which are strictly lower trian1 ∗ 1 ∗ gular with respect to (6.6). Given (∇ (E ) , h (E ) ) ∈ Cfil × H(E 1 )∗ , let i : (E 1 )∗ → V

be the isomorphism induced by h (E ) . Then i ◦∇ (E In this way there is an isomorphism 1 ∗

1 )∗

◦i −1 ∈ CV ×1 (B; End< (V )).

Cfil × H(E 1 )∗ ∼ = CV × 1 B; End< (V ) × Spl ×HV .

(6.7)

Let Gfil be the set of filtration-preserving gauge transformations of (E 1 )∗ , and let GV be the set of grading-preserving gauge transformations of V . Note that the set of splittings of (6.5) is acted upon freely and transitively by the gauge transformations of (E 1 )∗[k] which preserve (E 1 )∗[k+1] and act as the identity on (E 1 )∗[k] /(E 1 )∗[k+1] . It follows that Ker(Gfil → GV ) acts freely and transitively on Spl. Then Cfil × H(E 1 )∗ / Ker (Gfil → GV ) ∼ (6.8) = CV × 1 B; End< (V ) × HV , and so Cfil × H(E 1 )∗ /Gfil ∼ = CV × 1 (B; End< (V )) × HV /GV .

(6.9)

There is an obvious continuous map (Cfil × H(E 1 )∗ )/Gfil → (C(E 1 )∗ × H(E 1 )∗ )/G(E 1 )∗ . As Aff(Z ) preserves the lower-central-series filtration of n, in our case the dual 1 connection to ∇ E lies in Cfil . Then considering dual spaces in (6.9), it is enough for us to show that there is a compact subset of CV ∗ × 1 (B; End> (V ∗ )) × HV ∗ /GV ∗ (6.10) 1

1

in which we may assume that the gauge equivalence of the pair (∇ E , h E ) lies. We can then map the compact subset into (C E 1 × H E 1 )/G E 1 . As the local holonomy of E 1 comes from an N R -action, it factors through the ∗ LS ∗ 1 coadjoint action of N on n∗ . Letting ∇ V = k=1 ∇ V[k] be the component of ∇ E in ∗ ∗ CV ∗ , it follows that the local holonomy of ∇ V[k] is trivial and so ∇ V[k] is flat. We first


293

claim that there is a compact subset CV[k]∗ ⊂ (CV[k]∗ × HV[k]∗ )/GV[k]∗ such that we may

assume that the gauge equivalence class of the pair (∇ V[k] , h V[k] ) lies in CV[k]∗ . ∗ . Let F be the For simplicity of notation, fix k ∈ [0, S] and let E denote V[k] E space of flat connections on E , with the subspace topology from CE . We show that there is a compact subset of (FE × HE )/GE in which we may assume that the gauge equivalence class of the pair (∇ E , h E ) lies. Then the claim follows from mapping the compact subset into (CE × HE )/GE . Let Ee be the lift of E to the universal cover e B of B. Fix a basepoint e b0 ∈ e B with e over e projection b0 ∈ B, and let Eee be the fiber of E b . Then a flat connection ∇E 0 b0 e ) be the holonomy gives a trivialization Ee = e B × Eee b0 . Let ρ : π1 (B, b0 ) → Aut(Ee b0 E E of ∇ . Then a Euclidean inner product h on E can be identified with a Euclidean e inner product h E on Ee which satisfies ∗

∗

e e b) = ρ(γ )T h E (e b)ρ(γ ) h E (γ −1e

(6.11)

for all γ ∈ π1 (B, b0 ) and e b∈e B. In short, we can identify (FE × HE )/GE with the e E e )-action to pairs (ρ, h ) satisfying (6.11), modulo Aut(Eee b0 ). We can use the Aut(Ee b0 N N R identify Eee with R , with the standard inner product h . If we put b0

e Ee e RN X E = (ρ, h E ) ∈ Hom π1 (B, b0 ), GL(N , R) × H e B×R N : h (b0 ) = h e e and for all γ ∈ π1 (B, b0 ) and e b∈e B, h E (γ −1e b) = ρ(γ )T h E (e b)ρ(γ ) , (6.12) then we have identified (FE × HE )/GE with X E /O(N ). Let {γ j } be a finite generating set of π1 (B, b0 ). The topology on X E comes from the fiber bundle structure X E → Hom π1 (B, b0 ), GL(N , R) ,

(6.13)

whose fiber over ρ ∈ Hom(π1 (B, b0 ), GL(N , R)) is

e Ee e RN hE ∈ He and for all γ ∈ π1 (B, b0 ) and e b∈e B, B×R N : h (b0 ) = h

e e h E (γ −1e b) = ρ(γ )T h E (e b)ρ(γ ) . (6.14) Here Hom(π1 (B, b0 ), GL(N , R)) has a topology as a subspace of GL(N , R){γ j } , and e the fiber (6.14) has the C ∞ -topology. Thus it suffices to show that (ρ, h E ) lies in a predetermined compact subset CE of X E . By [20, (1) – (7)], we may assume that we have uniform bounds on the second fundamental form 5 of the Riemannian affine fiber bundle M, along with its covariant derivatives. As 5 determines how the Riemannian metrics on nearby 1 fibers vary (with respect to T H M) and h E comes from the inner product on the

294

JOHN LOTT

parallel differential forms on the fibers {Z b }b∈B , we obtain uniform bounds on (h E 1 )−1 (∇ E 1 h E 1 ) and its covariant derivatives. In particular, we also have a uniform e e bound on (h E )−1 (∇ E h E ) and hence on (h E )−1 (dh E ). For the finite generating set N e {γ j }, using the fact that h E (e b0 ) = h R , we obtain in this way uniform bounds on e −1e E {h (γ j b0 )}. The equivariance (6.11) then gives uniform bounds on {ρ(γ j )T ρ(γ j )} and hence on {ρ(γ j )}. Thus ρ lies in a predetermined compact subset of the representation space Hom(π1 (B, b0 ), GL(N , R)). Given ρ, the uniform bounds on the e covariant derivatives of h E over a fundamental domain in e B show that we have compactness in the fiber (6.14). As these bounds can be made continuous in ρ, the claim follows. ∗ ∗ ∗ Fix a Euclidean inner product h V0 on V ∗ . Given a pair (∇ V , h V ) ∈ CV ∗ × HV ∗ , we can always perform a gauge transformation to transform the Euclidean inner ∗ ∗ product to h V0 . Let OV ∗ be the orthogonal gauge transformations with respect to h V0 . Then we can identify (CV ∗ × HV ∗ )/GV ∗ with CV ∗ /OV ∗ . Similarly, CV ∗ × 1 (B; End> (V ∗ )) × HV ∗ /GV ∗ ∼ = CV ∗ × 1 (B; End> (V ∗ )) /OV ∗ . (6.15) There is a singular fibration p : CV ∗ × 1 (B; End> (V ∗ )) /OV ∗ → CV ∗ /OV ∗ . ∗ The fiber over a gauge equivalence class [∇ V ] is 1 (B; End> (V ∗ ))/G, where G is ∗ the group of orthogonal gauge transformations that are parallel with respect to ∇ V . In particular, upon choosing a basepoint b0 ∈ B, we can view G as contained in the finite-dimensional orthogonal group O(Vb∗0 ). From what we have already shown, we know that we are restricted to a compact subset of the base (CV ∗ × HV ∗ )/GV ∗ ∼ = CV ∗ /OV ∗ of the singular fibration p. Let 1 T 1 1 E E (∇ ) be the adjoint connection to ∇ with respect to h E . The uniform bounds 1 1 1 1 on (h E )−1 (∇ E h E ) and its derivatives give uniform C ∞ -bounds on the part of ∇ E 1 1 1 which does not preserve the metric h E , that is, on ∇ E −(∇ E )T ∈ 1 (B; End(E 1 )). 1 In particular, using the upper triangularity of ∇ E , we obtain uniform C ∞ -bounds on 1 the part of ∇ E in 1 (B; End> (V ∗ )). As the bounds can be made continous with ∗ respect to [∇ V ] ∈ CV ∗ /OV ∗ , we have shown that there is a fixed compact subset of CV ∗ × 1 (B; End> (V ∗ )) × HV ∗ /GV ∗ in which we may assume that the pair 1 1 (∇ E , h E ) lies. To summarize, we have shown that the topological vector bundle E 1 has a flat ∗ structure V ∗ , with flat connection ∇ V . We showed that there are bounds on the ∗ 1 holonomy of ∇ V which are uniform in n, , and K . We then showed that h E and 1 ∗ ∇ E − ∇ V are C ∞ -bounded in terms of n, , and K . (More precisely, we showed that these statements are true after an appropriate gauge transformation is made.) The proposition follows. Let S E be the space of degree-1 superconnections on E, with the C ∞ -topology.


295

PROPOSITION 4 With reference to Proposition 3, there are compact subsets D Ei ⊂ (S Ei × H Ei )/G Ei depending on n, , and K such that we may assume that the gauge-equivalence class of the pair A0 , h E lies in D E .

Proof Let E be as in Proposition 3. As in the proof of Proposition 3, upon choosing h 0E , we have identifications (C E × H E )/G E ∼ = = C E /O E and (S E × H E )/G E ∼ S E /O E . There is a singular fibration p : S E /O E → C E /O E coming from the projection A0 → A0[1] . The fiber of p over a gauge equivalence class [∇ E ] is L a ∗ ∗+b ))/G, where G is the group of orthogonal gauge transa+b=1 (B; End(E , E formations that are parallel with respect to ∇ E . From Proposition 3 we know that we are restricted to a compact subset of the base (C E × H E )/G E ∼ = C E /O E . The superconnection on E has the form (5.22). We measure norms on (B; End(E)) using h 0E . The differential d n comes from exterior differentiation on the parallel forms on the fibers of the Riemannian affine fiber bundle. Note that as A0 is flat, d n is parallel with respect to ∇ E . As we have a uniform n-, - and K -dependent bound on the curvatures of the fibers Z , Lemma 1 gives a uniform bound on the structure constants {cijk } and hence a uniform bound on kd n k∞ . The operator i T is also parallel with respect to ∇ E . From O’Neill’s formula (see [6, (9.29)]), we obtain a uniform bound on ki T k∞ . Thus we have uniform C ∞ -bounds on A0 − ∇ E ∈ (B; End(E)), and so we have compactness in the fibers of p. As the bounds can be made continuous with respect to [∇ E ] ∈ C E /O E , the proposition follows. Proof of Theorem 3 Propositions 3 and 4 prove the theorem. We need certain eigenvalue statements. Let E be a Z2 -graded real topological vector bundle on a smooth closed manifold B. Let SE be the space of superconnections on E , let GE be the GL(E ) gauge group of E , and let HE be the space of Euclidean metrics on E . Fix a Euclidean metric h E0 ∈ HE . Given a pair (A0 , h E ) ∈ SE × HE , we can always perform a gauge transformation to transform the Euclidean metric to h E0 . Let OE be the group of orthogonal gauge transformations of E with respect to h E0 . Then we can identify (SE × HE )/GE with SE /OE . Given A0 ∈ SE , let (A0 )∗ be its adjoint with respect to h E0 , and put 4 A0 = A0 (A0 )∗ + (A0 )∗ A0 , acting on (B; E ). For j ∈ Z+ , let µ j (A0 ) be the jth eigenvalue of 4 A0 , counted with multiplicity. It is OE -invariant. Equivalently, µ j is GE -invariant as a function of the pair (A0 , h E ).

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Proof of Theorem 4 As E p admits a flat connection, there is some r ∈ N such that for all p, E p ⊗ Rr is p topologically isomorphic to the trivial vector bundle B × Rr ·r k(E ) . Hence E ⊗ Rr is topologically isomorphic to the Z-graded trivial vector bundle E = B × Rr ·r k(E) . For simplicity, we omit reference to p. In view of Theorem 2, it suffices to show that there is a positive constant D(n, , K ) such that |λ j (B; E)1/2 − λ0j (B)1/2 | ≤ D(, n, K ). The operator 4 E ⊗ Id on (B; E) ⊗ Rr has a spectrum that is the same as that of 4 E , but with multiplicities multiplied by r . Hence it is enough to compare the spectrum of 4 E ⊗ Id, acting on (B; E ⊗ Rr ), with that of the standard Laplacian on (B; E ). From Theorem 3, we may assume that the gauge equivalence class of the pair (A0 , h E ) lies in a predetermined compact subset D ⊂ (S E × H E )/G E . Put A01 = r A0 ⊗ Id, acting on (B; E ⊗ Rr ), and put h 1 = h E ⊗ h R on E ⊗ Rr . Using the isomorphism E ∼ = E ⊗ Rr , we may assume that the gauge equivalence class of the 0 pair (A1 , h 1 ) lies in a predetermined compact subset D1 ⊂ (SE × HE )/GE . Let A02 be the trivial flat connection on E , and let h 2 be the product Euclidean inner product on E . With an appropriate gauge transformation g ∈ GE , we can transform (A01 , h 1 ) to (g · A01 , h 2 ) without changing the eigenvalues. Under the identification (SE × HE )/GE = SE /OE , we can assume that the equivalence class of g · A01 in SE /OE lies in a predetermined compact subset D2 ⊂ SE /OE . The eigenvalues of the Laplacian associated to the superconnection A01 and the Euclidean inner product h 2 are unchanged when the group of orthogonal gauge transformations OE acts on A01 . Consider the function l : SE × SE → R given by l(A01 , A02 ) = inf kg 0 · A01 − A02 k. g 0 ∈OE

(6.16)

An elementary argument shows that l is continuous. Hence it descends to a continuous function on (SE /OE ) × (SE /OE ). When Lemma 4 is applied to g 0 · (g · A01 ) and A02 , the compactness of D2 and the finiteness statement in Theorem 3 give the desired eigenvalue estimate. The theorem follows. 7. Small positive eigenvalues In this section we characterize the manifolds M for which the p-form Laplacian has small positive eigenvalues. We first describe a spectral sequence that computes the cohomology of a flat degree-1 superconnection A0 . We use the compactness result of Theorem 3 to show that if M has j small eigenvalues of the p-form Laplacian with j > b p (M), and M collapses to a smooth manifold B, then there is an associated flat degree-1 superconnection A0∞ on B with dim(H p (A0∞ )) ≥ j. We then use the spectral sequence of A0∞ to characterize when this can happen. In Corollary 1 we give


297

a bound on the number of small eigenvalues of the 1-form Laplacian. In Corollary 2 we give a bound on the number of small eigenvalues of the p-form Laplacian when one is sufficiently close to a smooth limit space of dimension dim(M) − 1. Theorem 5 describes when a collapsing sequence can have small positive eigenvalues of the pform Laplacian, in terms of the topology of the affine fiber bundle M → B. Corollary 3 gives a precise description of when there are small positive eigenvalues of the 1form Laplacian in a collapsing sequence. In Corollaries 4 and 5 we look at collapsing sequences with limit spaces of dimension 1 or dim(M)−1, respectively. Finally, given an affine fiber bundle, in Theorem 6 we give a collapsing construction that produces small eigenvalues of the p-form Laplacian. In the collapsing arguments in this section, when the limit space is a smooth manifold, we can always assume that its Riemannian metric is smooth. At first sight the smoothness assumption on the metric may seem strange as the limit space of a bounded sectional curvature collapse, when a smooth manifold, generally only has a C 1,α -metric. The point is that we are interested in the case when an eigenvalue goes to zero, which gives a zero eigenvalue of 4 E in the limit. The property of having a zero eigenvalue is essentially topological in nature and so is also true for a smoothed metric. For this reason, we can apply smoothing results to the metrics and so ensure that the limit metric is smooth. L Let B be a smooth connected closed manifold. Let E = mj=0 E j be a Z-graded P real vector bundle on B, and let A0 = i≥0 A0[i] be a flat degree-1 superconnection on E. Let H p (A0 ) denote the degree- p cohomology of the differential A0 on (B; E), where the latter has the total grading. Given a, b ∈ N, we write ωa,b for an element of a (B; E b ). In order to compute H p (A0 ), let us first consider the equation A0 ω = 0. Putting ω = ω p,0 + ω p−1,1 + ω p−2,2 + · · · ,

(7.1)

(A0[0] + A0[1] + A0[2] + · · · )(ω p,0 + ω p−1,1 + ω p−2,2 + · · · ) = 0

(7.2)

we obtain

or A0[0] ω p,0 = 0, A0[0] ω p−1,1 + A0[1] ω p,0 = 0, A0[0] ω p−2,2 + A0[1] ω p−1,1 + A0[2] ω p,0 = 0, .. . We can try to solve these equations iteratively.

(7.3)

298

JOHN LOTT

Formalizing this procedure, we obtain a spectral sequence to compute H p (A0 ). Put E 0a,b = a (B; E b ), and define d0 : E 0a,b → E 0a,b+1 by d0 ωa,b = A[0] ωa,b . For r ≥ 1, put Era,b =

P −1 {{ωa+s,b−s }rs=0 : for 0 ≤ s ≤ r − 1, st=0 A0[s−t] ωa+t,b−t = 0} . P −1 −1 {{ωa+s,b−s }rs=0 : ωa+s,b−s = st=0 A0[s−t] b ωa+t,b−t−1 for some {b ωa+s,b−s−1 }rs=0 } (7.4)

Define a differential dr : Era,b → Era+r,b−r +1 by −1 dr {ωa+s,b−s }rs=0 =

r −1 nX

A0[r +s−t] ωa+t,b−t

t=0

or −1 s=0

.

(7.5)

∗,∗ ∗,∗ Then Er +1 ∼ = Ker(dr )/ Im(dr ). The spectral sequence {Er }r∞=0 has a limit E ∞ with

H p (A0 ) ∼ =

M

a,b E∞ .

(7.6)

a+b= p

From [8, Proposition 2.5], for each b ∈ N, Hb (A0[0] ) is a flat vector bundle on B. Then E 0a,b = a (B; E b ), E 1a,b = a B; Hb (A0[0] ) , E 2a,b = Ha B; Hb (A0[0] ) .

(7.7)

Example 4 If M → B is a fiber bundle, E is the infinite-dimensional vector bundle W of vertical differential forms (see [8, Section III(a)]), and A0 is the superconnection arising from exterior differentiation on M, then we recover the Leray spectral sequence to compute H∗ (M; R). Example 5 If M → B is an affine fiber bundle, E is the vector bundle of parallel differential forms on the fibers, and A0 is as in (5.22), then it follows from [28, Corollary 7.28] that Er∗,∗ is the same as the corresponding term in the Leray spectral sequence for H∗ (M; R) if r ≥ 1. Suppose that M is a connected closed manifold with at least j small eigenvalues of ∞ in 4 p for some j > b p (M). Consider a sequence of Riemannian metrics {gi }i=1 M (M, K ) with limi→∞ λ p, j (M, gi ) = 0. As in the proof of Theorem 2, for any 0 > 0 there is a sequence {Ak (n, )}∞ k=0 , so that for all i we can find a new metric gi 0 k M on M which is -close to gi , with k∇ R (gi )k∞ ≤ Ak (n, ). Fix to be, say, 1/2. From [17] or Lemma 3, we have that λ p, j (M, gi0 ) is J -close to λ p, j (M, gi ) for some


299

fixed integer J . Thus, without loss of generality, we may replace gi by gi0 . We relabel gi0 as gi . ∞ which GromovAs j > b p (M), there must be a subsequence of {(M, gi )}i=1 Hausdorff converges to a lower-dimensional limit space X . That is, we are in the collapsing situation. Suppose that the limit space is a smooth manifold B. From [12, Section 5], the regularity of the metrics on M implies that B has a smooth Riemannian metric g T B . (We are in the situation in which the limit space Xˇ of the frame bundles, a smooth Riemannian manifold, has an O(n)-action with a single orbit type.) From Theorem 2, for large i there are vector bundles E i on B, flat degree-1 superconnections Ai0 on E i , and Euclidean inner products h Ei on E i such that λ p, j (M, gi ) is -close to λ p, j (B; E i ). From Theorem 3, after taking a subsequence, we may assume that all of the E i ’s are topologically equivalent to a single vector bundle E on B and that the pairs (Ai0 , h Ei ) converge after gauge transformation to a pair (A0∞ , h E ∞ ). Then from Lemmas 3 and 4, the Laplacian associated to (A0∞ , h E ∞ ) satisfies dim Ker(4 Ep ) ≥ j. Applying standard Hodge theory to the superconnection Laplacian 4 E , we obtain dim(H p (A0∞ )) ≥ j. On the other hand, looking at the E 2 -term of the spectral sequence gives dim(H p (A0∞ )) ≤ P a b 0 a+b= p dim H (B; H (A∞,[0] )) . Thus X j≤ dim Ha (B; Hb (A0∞,[0] )) . (7.8) a+b= p

Proof of Corollary 1 In the case p = 1, we obtain j ≤ dim H1 (B; H0 (A0∞,[0] )) + dim H0 (B; H1 (A0∞,[0] )) . (7.9) As H0 (A0∞,[0] ) is the trivial R-bundle on B, dim H1 (B; H0 (A0∞,[0] )) = b1 (B). As A0∞,[0] acts by zero on E 0 , there is an injection H1 (A0∞,[0] ) → E 1 . Then dim H0 (B; H1 (A0∞,[0] )) ≤ dim H1 (A0∞,[0] ) ≤ dim(E 1 ) ≤ dim(M) − dim(B). (7.10) Thus j ≤ b1 (B) + dim(M) − dim(B). On the other hand, the spectral sequence for H∗ (M; R) gives H1 (M; R) = H1 (B; R) ⊕ Ker H0 (B; H1 (Z ; R)) → H2 (B; R) . (7.11) In particular, b1 (B) ≤ b1 (M). The corollary follows. Remark. Using heat equation methods (see [3]), one can show that there is an increasing function f such that if Ric(M) ≥ −(n − 1)λ2 and diam(M) ≤ D, then the number of small eigenvalues of the 1-form Laplacian is bounded above by f (λD).

300

JOHN LOTT

This result is weaker than Corollary 1 when applied to manifolds with sectional curvature bounds, but it is more general in that it applies to manifolds with just a lower Ricci curvature bound. Proof of Corollary 2 From Fukaya’s fibration theorem, if a manifold M n with kR M k∞ ≤ K is sufficiently Gromov-Hausdorff close to B, then M is the total space of a circle bundle over B. Suppose that the claim of the corollary is not true. Then there is a ∞ sequence of connected closed n-dimensional Riemannian manifolds {(Mi , gi )}i=1 M i with kR (gi )k∞ ≤ K and limi→∞ Mi = B which provides a counterexample. As there is a finite number of isomorphism classes of flat real line bundles on B, after passing to a subsequence, we may assume that each Mi is a circle bundle over B with a fixed orientation bundle O and that limi→∞ λ p, j (Mi , gi ) = 0 for j = b p (B) + b p−1 (B; O ) + 1. Following the argument before the proof of Corollary 1, we obtain E = E 0 ⊕ E 1 on B, with E 0 a trivial R-bundle and E 1 = O , and a limit superconnection A0∞ on E with A0∞,[0] = 0 and A0∞,[1] = ∇ E , the canonical flat connection. Then, as in (7.8), we obtain j ≤ b p (B) + b p−1 (B; O ),

(7.12)

which is a contradiction. Proof of Theorem 5 As in the proof of Theorem 2, without loss of generality we may assume that each (M, gi ) is a Riemannian affine fiber bundle structure on the affine fiber bundle M → B. Suppose that for each q ∈ [0, p], bq (Z ) = dim(3q (n∗ ) F ) and the holonomy representation of the flat vector bundle Hq (Z ; R) on B is semisimple. Let E → B be the real vector bundle associated to the affine fiber bundle M → B as in Section 5. Then E ∼ = H∗ (Z ; R). The superconnection A0E on E, from Section 5, has 0 0 A E,[0] = 0 and A E,[1] = ∇ E , the canonical flat connection on E ∼ = H∗ (Z ; R). As the 0 affine fiber bundle is fixed, each E i equals E and each Ai equals A0E . However, the ∞ on E vary. There is a sequence of gauge transformations Euclidean metrics {h iE }i=1 ∞ E ) {gi }i=1 , so that after passing to a subsequence, limi→∞ gi · (Ai0 , h iE ) = (A0∞ , h ∞ 0 E ). Clearly, A0 E for some pair (A0∞ , h ∞ ∞,[0] = 0 and A∞,[1] = limi→∞ gi · ∇ . As the q holonomy representation of H (Z ; R)q is semisimple for q ∈ [0, p], the connection A0∞,[1] E q is gauge equivalent to ∇ E . That is, the connection does not degenerate. (In the complex case this follows from [25, Theorem 1.27], and the real case follows from [29, Theorem 11.4].) Equation (7.8) now implies X j≤ dim Ha (B; Hb (Z ; R)) . (7.13) a+b= p


301

If the Leray spectral sequence to compute H∗ (M; R) degenerates at the E 2 -term, then b p (M) =

X

dim Ha (B; Hb (Z ; R)) ,

(7.14)

a+b= p

which contradicts the assumption that j > b p (M). Example 6 Let Z be an almost flat manifold as in Example 1. Put M = Z × B. If there is a sequence of affine parallel metrics on Z which give it r p small eigenvalues of the P p-form Laplacian, then M has a+b= p ra · bb (B) small eigenvalues of the p-form Laplacian. This gives an example of Theorem 5.1. Example 7 ([24]) Let N be the Heisenberg group of upper-diagonal unipotent (3 × 3)-matrices, and let 0 be the integer lattice in G. Put M = 0\N . Then M fibers over S 1 , the fiber being T 2 and the monodromy being given by the matrix 10 11 . One has b1 (M) = 2, but for any K > 0, a1,3,K = 0. That is, one can collapse M to a circle by a sequence of affine parallel metrics, while producing 3 small eigenvalues of the 1-form Laplacian. This gives an example of Theorem 5.2. Example 8 Consider M as in Example 2. If the Leray spectral sequence to compute H p (M; R) does not degenerate at the E 2 -term, then there are small positive eigenvalues of the p-form Laplacian on M. This gives an example of Theorem 5.3. Proof of Corollary 3 The affine fiber bundle M → B induces a vector bundle E → B and a flat degree-1 superconnection A0E , as in Section 5. As in Example 5, the spectral sequence associated to A0E is the same as the Leray spectral sequence for computing H∗ (M; R). Let A0∞ denote the limit superconnection arising as in the proof of Theorem 5. The spectral sequence for H∗ (A0∞ ) gives H1 (A0∞ ) = H1 (B; R) ⊕ Ker H0 (B; H1 (A0∞,[0] )) → H2 (B; R) . (7.15) In particular, dim H1 (A0∞ ) = b1 (B) + dim Ker(H0 (B; H1 (A0∞,[0] )) → H2 (B; R)) . (7.16) We wish to compare this with the corresponding spectral sequence for H∗ (A0E ), that is, (7.11).

302

JOHN LOTT

Suppose that the differential d2 : H0 (B; H1 (Z ; R)) → H2 (B; R) vanishes. Then from (7.11), b1 (M) = b1 (B) + dim H0 (B; H1 (Z ; R)) . (7.17) By assumption, dim(H1 (A0∞ )) = j > b1 (M). This implies that dim H0 (B; H1 (A0∞,[0] )) > dim H0 (B; H1 (Z ; R)) .

(7.18)

In terms of the original superconnection A0E , we have that H1 (Z ; R) is a flat subbundle of H1 (A0E,[0] ). After taking limits, we obtain a flat subbundle H1 (Z ; R)∞ of H1 (A0∞,[0] ). Here the fibers of H1 (Z ; R)∞ are again isomorphic to the first real cohomology group of Z , but the flat structure could be different from that of the bundle which we denoted by H1 (Z ; R). In particular, dim H0 (B; H1 (Z ; R)∞ ) ≥ dim H0 (B; H1 (Z ; R)) . (7.19) Clearly, dim H0 (B; H1 (A0∞,[0] )) ≥ dim H0 (B; H1 (Z ; R)∞ ) .

(7.20)

Then from (7.18) we must have dim H0 (B; H1 (Z ; R)∞ ) > dim H0 (B; H1 (Z ; R))

(7.21)

dim H0 (B; H1 (A0∞,[0] )) > dim H0 (B; H1 (Z ; R)∞ ) .

(7.22)

or If (7.21) holds, then the holonomy representation of the flat vector bundle H1 (Z ; R) must have a nontrivial unipotent subrepresentation (see [29, Theorem 11.4 and Proposition 11.14]). If (7.22) holds, then there is a nonzero covariantly constant section of the vector bundle (H 1 (A0∞,[0] ))/(H 1 (Z ; R)∞ ) on B, where the flat connection on (H 1 (A0∞,[0] ))/(H 1 (Z ; R)∞ ) is induced from the flat connection on H1 (A0∞,[0] ). This proves the corollary. Proof of Corollary 4 Suppose that for q ∈ { p − 1, p}, bq (Z ) = dim(3q (n) F ). From the Leray spectral sequence, H p (M) ∼ = Ker(8 p − I ) ⊕ Coker(8 p−1 − I ). Let H∗ (Z ; R)∞ denote the p limiting flat vector bundle on S 1 , as in the proof of Corollary 3, with holonomy 8∞ ∈ p p ∗ p 0 0 Aut(H (Z ; R)). The spectral sequence for H (A∞ ) gives H (A∞ ) ∼ = Ker(8∞ − p−1 p I ) ⊕ Coker(8∞ − I ). We have dim(Ker(8∞ − I )) ≥ dim(Ker(8 p − I )) and p dim(Coker(8∞ − I )) ≥ dim(Coker(8 p − I )). By assumption, j = dim(H p (A0∞ )) > p p dim(H (M; R)) = b p (M). If dim(Ker(8∞ − I )) > dim(Ker(8 p − I )), then 8 p p must have a nontrivial unipotent subfactor. Similarly, if dim(Coker(8∞ − I )) > dim(Coker(8 p − I )), then 8 p−1 must have a nontrivial unipotent subfactor.


303

Example 9 Suppose that the affine fiber bundle M → S 1 has fiber Z = T 2 . If M has a Sol geometry or an R3 -geometry, then Corollary 4 implies that there are no small positive eigenvalues in a collapsing sequence associated to M → S 1 . On the other hand, if M has a Nil geometry, then Example 7 shows that there are small positive eigenvalues of the 1-form Laplacian. (See [24] for further examples of homogeneous collapsings.) Proof of Corollary 5 p,0 The E 2 -term of the spectral sequence for computing H∗ (M; R) consists of E 2 = p,1 H p (X ; R) and E 2 = H p (X ; O ). The differential is Mχ . The corollary now follows from Theorem 5. Proof of Theorem 6 As in [20, §6], we can reduce the structure group of the fiber bundle P → B so that the local holonomy lies in a maximal connected compact subgroup of Aff(Z ), a torus group. Choose a horizontal distribution T H M on M whose local holonomy lies in this torus group. Add vertical Riemannian metrics g T Z , parallel along the fibers, and a Riemannian metric g T B on B to give M → B the structure of a Riemannian affine fiber bundle. We use Theorem 1 to make statements about the eigenvalues of the differential form Laplacian on M. LS There is a vector space isomorphism n∗ ∼ = k=0 r∗[k] . Define a number operator on n∗ to be multiplication by 3k on r∗[k] . Extend this to a number operator on 3∗ (n∗ ) F and to a number operator N on the vector bundle E ∗ over B. k For > 0, rescale g T Z to a new metric gT Z by multiplying it by 3 on r[k] ⊂ n. Let gT M be the corresponding Riemannian metric on M. The rescaling does not affect d M . The adjoint of d M with respect to the new metric is (d M )∗ = N (d M )∗ −N . Putting C0 = −N /2 d M N /2 , C00 = N /2 (d M )∗ −N /2 ,

(7.23)

we have that C0 is a flat degree-1 superconnection, with C00 being its adjoint with respect to g T Z . The Laplacian 4 M coming from gT M is conjugate to C0 C00 + C00 C0 . By [20, §6], lim→0 (M, gT M ) = B with bounded sectional curvature in the k limit. (The proof in [20, §6] uses a scaling by 2 , but the proof goes through for a k scaling by 3 . The phrase “The element Yi of g, through the right action of G, . . .” in [20, p. 349, line b9] should read “. . . the left action of G, . . .”.) Let A0 denote the superconnection on E constructed by restricting C0 to the fiberwise-parallel forms. We show that lim→0 A0 = ∇ G , the flat connection on G. The theorem then follows from Theorem 1 and Lemma 4.

304

JOHN LOTT

Consider A0,[0] . It acts on a fiber of E by −N /2 d n N /2 . Consider first its action LS on a fiber (n∗ ) F ∼ = k=0 (r∗[k] ) F of E 1 . As d n acts on 31 (n∗ ) by the dual of the Lie L L ∗ ∗ n ∗ bracket and [r[k] , r[l] ] ⊂ m>max(k,l) r[m] , we have d r[m] ⊂ k,l<m r[k] ∧ r[l] . It follows that −N /2 d n N /2 : (r∗[m] ) F → (r∗[k] ∧ r∗[l] ) F (7.24) is O( (3 −3 −3 )/2 ). We obtain that the action of A0,[0] on E 1 is O( 1/2 ) as → 0. A similar argument shows that the action of A0,[0] on E ∗ is O( 1/2 ). Now consider A0,[1] = −N /2 ∇ E N /2 . Put Fk∗ = P ×Aff(Z ) (r∗[k] ) F , so that E 1 ∼ = LS ∗ ∗ k=0 Fk . (Here the ∗ in Fk denotes an adjoint, not a Z-grading.) Consider first the 0 action of C,[1] on C ∞ (B; E 1 ). As the holonomy of ∇ E comes from an Aff(Z ) action, L we have ∇ E : C ∞ (B; Fk∗ ) → l≤k C ∞ (B; Fl∗ ) (see the proof of Proposition 3). If l < k, then the component m

k

l

−N /2 ∇ E N /2 : C ∞ (B; Fk∗ ) → C ∞ (B; Fl∗ )

(7.25)

of −N /2 ∇ E N /2 is O( 1/2 ). On the other hand, the component ∇ E : C ∞ (B; Fk∗ ) → C ∞ (B; Fk∗ ) is the restriction of the flat connection ∇ G from G 1 to Fk∗ . A similar argument applies to all of E to show that as → 0, A0,[1] = ∇ G + O( 1/2 ). Finally, consider A0,[2] = −N /2 i T N /2 . The curvature T of the fiber bundle M → B is independent of . As T acts by interior multiplication on the fibers of E, k the action of −N /2 i T N /2 on (r∗[k] ) F ⊂ E 1 is O( 3 /2 ). A similar argument applies to all of E to show that as → 0, A0,[2] = O( 1/2 ). The theorem follows. Note. After this paper was finished, we learned of the preprint version of [15] which, among other things, contains proofs of Corollaries 2 and 5 in the case when M and B are oriented. Paper [24] is also related to the present paper. Acknowledgments. I thank Bruno Colbois, Gilles Courtois, and Pierre Jammes for corrections to an earlier version of this paper. I thank the referee for a very careful reading of the manuscript and many useful remarks, among these suggesting a simplification of the proof of Proposition 2. References [1]

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Academic Press, Boston, 1990, 143 – 238. MR 92k:53076 268 [22] [23] [24] [25] [26]

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´ M. GROMOV, Volume and bounded cohomology, Inst. Hautes Etudes Sci. Publ. Math. 56 (1982), 5 – 99. MR 84h:53053 291 , Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math. 152, Birkhäuser, Boston, 1999. MR 2000d:53065 268, 282 P. JAMMES, Note sur le spectre des fibrés en tores sur le cercle qui s’effondrent, preprint, 2000. 301, 303, 304 A. LUBOTZKY and A. MAGID, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985), no. 336. MR 87c:20021 300 R. MAZZEO and R. MELROSE, The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration, J. Differential Geom. 31 (1990), 185 – 213. MR 90m:58004 268, 270, 276 D. QUILLEN, Superconnections and the Chern character, Topology 24 (1985), 89 – 95. MR 86m:58010 269, 283 M. RAGHUNATHAN, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. 68, Springer, New York, 1972. MR 58:22394a 276, 298 R. RICHARDSON, Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988), 1 – 35. MR 89:20061 300, 302 X. RONG, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. (2) 143 (1996), 397 – 411. MR 97a:53067 290

Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor, Michigan 48109-1109, USA; [email protected]


UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS IN CHARACTERISTIC p > 0 PHILIPPE GILLE

Abstract We prove two conjectures of J. Tits on the unipotent elements of semisimple algebraic groups defined over a field with positive characteristic. Résumé Nous démontrons deux conjectures de J. Tits sur les e´ léments unipotents des groupes algébriques semi-simples définis sur un corps de caractéristique positive.

1. Introduction 1.1. Good finite subgroups Let k be a field of characteristic p ≥ 0, and then let ks be a separable closure of k. Let G/k be a reductive connected algebraic group. For the notions on algebraic groups used here, we refer to the paper by A. Borel and Tits [BT1] and the book of M. Demazure and P. Gabriel [DG]. If G is semisimple and absolutely almost simple, we denote by S(G) the finite set of torsion primes of G as defined by J.-P. Serre in [Se2]. In the case of G simply connected, the Dynkin index is denoted by dG (cf. [LS, §2]). Note that the prime factors of dG lie in S(G). We recall Tits’s definition of good unipotent elements. Definition 1 ([T2]) Let u be a unipotent element of G(k). The element u is k-good if u lies in the unipotent radical of a k-parabolic subgroup of G; otherwise, u is k-bad. In characteristic zero, all unipotent elements are k-good, so the definition is relevant only if char(k) = p > 0; in that case, a unipotent element u has finite order q = pr DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 2, Received 8 December 2000. Revision received 6 September 2001. 2000 Mathematics Subject Classification. Primary 20G15; Secondary 12G05.

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and can be viewed as a morphism u ] : Z/qZ → G sending 1 to u. This viewpoint leads to the following definition of goodness for morphisms M → G, where M/k is a finite e´ tale group. Definition 10 Let M/k be a finite e´ tale group, and let φ : M/k → G be a morphism. The morphism φ is k-good if φ(M(ks )) lies in the radical of a k-parabolic subgroup of G; otherwise, φ is k-bad. If φ is injective, we also say that M is a k-good subgroup of G. We say that φ : M → G is unipotent if φ(M(ks )) consists of unipotent elements. Let us prove that the two definitions are compatible. 1 Assume that char(k) = p > 0. Let u be a unipotent element of order q, and let u ] : Z/qZ → G be the associated morphism. Then u is k-good if and only if u ] is k-good. LEMMA

Proof If u is k-good, it is obvious that the morphism u ] is k-good. Conversely, assume that the morphism u ] : Z/qZ → G is k-good. Then u = u ] (1) lies in the radical R(P) of a k-parabolic subgroup P of G, which is an extension of a k-torus S by the unipotent radical Ru P of P; that is, we have an exact sequence 1 → Ru (P) → R(P) → S → 1. As S(k) has no element of order p, u lies in Ru (P)(k), and u is k-good. Definition 2 ([T2]) Let α be a k-automorphism of G of finite order. The morphism α is k-isotropic if α normalizes some proper k-parabolic subgroup of G; otherwise, φ is k-anisotropic. As above, we can extend this definition to isotropic morphisms M → Aut(G) which yield the previous one in the case when M = Z/nZ. Definition 20 Let φ ∈ Homk−gr (M, Aut(G)). The morphism φ is k-isotropic if φ(M(ks )) normalizes some proper k-parabolic subgroup of G; otherwise, φ is k-anisotropic. We say that φ ∈ Homk−gr (M, G) is k-isotropic if the composite Ad ◦ φ : M → G is k-isotropic. Obviously, k-anisotropic implies k-bad.

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1.2. The conjectures of Tits Our first result is the proof (given in §2.3) of the following conjecture. CONJECTURE 1 ([T3]) Assume that p > 0, [k : k p ] ≤ p, and that G/k is semisimple and simply connected. Then every unipotent subgroup of G(k) (i.e., a subgroup consisting of unipotent elements) is k-embeddable into the unipotent radical of a k-parabolic subgroup of G.

The case when p = 2 is due to Tits (see [T2, §4.5]). Further known cases are when p is not a torsion prime of G and when G is of type An or Cn (see [T2, §§3.5, 4.4]). In Conjecture 1, the condition [k : k p ] ≤ p is necessary, as the following theorem of Tits shows. 1 ([T4, Prop. S1], [T5, Th. 7]) Assume that G is split simply connected and almost simple. If [k : k p ] ≥ p 2 and p divides dG , then the group G(k) contains a k-bad unipotent element of order p. THEOREM

Our proof of Conjecture 1 is based on the reduction to a problem in Galois cohomology for the group G k((t)) , which is done in Proposition 3 (§2.2), and on known cases of Serre’s conjecture II in Galois cohomology (see [Gi2]). Our second result is the proof of the following conjecture (see Theorem 3 of §4), which is related to the shape of conjugacy classes of k-anisotropic automorphisms. CONJECTURE 2 ([T2]) Assume that p > 0 and that G/k is split and almost simple. Let α be a k-anisotropic automorphism of order p. Then α normalizes a maximal k-split torus of G.

Let us recall first the known cases handled by Tits: (1) p = 2, all types, (2) p = 3, G of type D4 or E 6 , using a triality argument due to G. Harder [H], (3) type An . It turns out that Conjecture 2 and a large part of the present paper make sense in a broader setting, including the characteristic zero case. To this end, we formulate the following conjecture, which extends the preceding one. CONJECTURE 20

Assume that G/k is split almost simple. Let l be a prime of S(G). We assume either that l = char(k) or that k contains a primitive l-root of unity. Let α be a k-anisotropic automorphism of order l. Then α normalizes a maximal k-split torus of G.

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Such anisotropic automorphisms are called special automorphisms. In Section 4, we prove Conjecture 20 , except for the E 8 -case. However, the case of E 8 and l = p = 5 is proven in [Gi3]. Tits’s original conjecture is thus true. The proof of Conjecture 20 requires several steps. First, we work in characteristic zero and prove by a case-by-case analysis Conjecture 20 (except for the E 8 -case) by using methods inspired by the proof of the Hasse principle (§4.2). The tame case of l 6= p is then not hard, and we can concentrate on the wild case of l = p = char(k). We consider a complete discrete valuation ring A with residue field k and fraction field FA of characteristic zero. We remark that special elements lift in characteristic zero; that is, if α ∈ Aut(G)(k) is a special automorphism of order p, then there exists e α ∈ Aut(G)(A) of order p specializing to α. We show that anisotropic automorphisms lift in characteristic zero (§4.3). The proof concludes by a specialization argument. 1.3. Complements on finite subgroups In this section, we remark on the extension of some known results to our present setting. The argument as used in [T2, proof of Prop. 3.2] yields in fact the following. PROPOSITION 1 Let φ ∈ Homk−gr (M, G), and let P/k be a k-parabolic subgroup normalized by φ. Let L be a Levi subgroup of P defined over k, so that P = Ru (P) o L, and let φ 0 be the composition of M → P → L. (a) The following are equivalent: (i) φ is k-good in G, (ii) φ 0 is k-good in G, (iii) φ 0 is k-good in L. (b) If P is minimal among all k-parabolic subgroups normalized by φ, then φ 0 is k-anisotropic in L.

Similarly, we can extend [T2, Lem. 3.5]. 2 Let H/k be a reductive subgroup of G/k. Let φ : M → Aut(G, H ) ⊂ Aut(G) be a k-anisotropic morphism. Then the morphism φ : M → Aut(G, H ) → Aut(H ) is k-anisotropic. LEMMA

We also recall the following. PROPOSITION 2 ([BT2, Prop. 3.6]) Let U ⊂ G(k) be a unipotent subgroup such that every element of U is ks -embeddable

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in a Borel subgroup. Then U is k-embeddable in the unipotent radical of a k-parabolic subgroup of G. In particular, and in contrast to the tame case, a unipotent element u is k-good if and only if it is ks -good. LEMMA 3 Let g ∈ G(k) be an element of finite order invertible in k. Then g lies in a maximal k-torus of G. In particular, g is ks -good.

Proof Let g ∈ G(k) be an element of finite order invertible in k. Then g is semisimple, and g lies in some maximal torus T /k. As T ×k ks is a split maximal torus, g is ks -good. More generally, if char(k) = 0, A. Borel and G. Mostow’s theorem asserts that an automorphism of G of finite order normalizes a maximal torus of G (see [BM]; see also [P]).

2. Finite unipotent subgroups and Galois cohomology 2.1. A Galois cohomology class Let us set K = k((t)) and denote by K mod a maximal tamely ramified extension of K . Let I = Iw o µm be a k-étale group, semidirect product of µm (m ∈ k × ) by a constant p-group Iw . Throughout this paper we assume the following: There exists a totally ramified field extension L/K such that Spec(L) → Spec(K ) is an I -torsor. This hypothesis is satisfied in the two extreme cases when I = µm (obviously) and I = Iw since the maximal pro- p quotient of Gal(K s /K ) is a free pro- p group (see [Se1, §II.2.2, Cor. 1]). We denote e L = L · K (µm ), which is a Galois extension of K with Galois group e I . So there is a diagram of field extensions e L

K (µm )

L

L ∩ K (µm ) | K

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and the residue field of e L is k(µm ). We denote by f θ ∈ Z 1 (e I , I (e L)) the 1-cocycle defined by the I -torsor Spec(L) → Spec(K ) and denote by θ ∈ H 1 (K , I ) its class. For simplicity the reader can think of I = Z/ pZ; this case, which is of main interest for us, was treated in [Gi1]. We consider the set Homk−gr (I, G) and pick φ ∈ Homk−gr (I, G). The key point, due to Serre, is to associate to φ : I → G the 1-cocycle I , G(e L) f φ = φ∗ ( f θ ) in Z 1 e and the cohomology class γ (φ) = φ∗ (θ ) = [ f φ ] in H 1 (K , G). Remark 1. We use functoriality for a surjective base change λ : I 0 → I and an extension L 0 /L/K of monodromy group I 0 : if there exists θ 0 such that λ∗ (θ 0 ) = θ , then φ∗ (θ ) L 0 = λ∗ (φ ◦ λ) in H 1 (L 0 , G). 2.2. How to see if a finite unipotent subgroup is k-good PROPOSITION 3 (a) If φ is k-good, then γ (φ) = 1 in H 1 (K , G). (b) Assume that φ is unipotent. Then the following assertions are equivalent: (i) φ is k-good, (ii) γ (φ) = 1 in H 1 (K , G), (iii) γ (φ) K mod = 1 ∈ H 1 (K mod , G). Proof (a) Let P/k be a parabolic subgroup of G/k such that the radical R(P) contains Im(φ). The group R(P) is split, and we know that H 1 (K , R(P)) = 1. Thus γ (φ) = 1. (b) As we consider unipotent morphisms, we can assume that I = Iw , that is, that I is a finite p-group. There is only the implication (iii) =⇒ (i) to prove. By hypothesis, there exists a tamely ramified extension K 0 /K of valuation ring O 0 /O and residue extension k 0 /k such that γ (φ) K 0 = 1 in H 1 (K 0 , G). One can assume that K 0 = k 0 ((t 0 )), (t 0 )e = t, and (e, p) = 1 and that G k 0 is split. Let B/k 0 be a Borel subgroup of G/k 0 . We consider the extension L 0 = L · K 0 /K 0 , which is Galois of group I = Gal(L 0 /K 0 ) and valuation ring O L 0 . Let us denote by X/k 0 = G/B the variety of Borel subgroups of G and by π : G/k 0 → X/k 0 the canonical map. This variety is projective, the morphism π is smooth, and one knows that the map on k 0 points G(k 0 ) → X (k 0 ) is surjective. Using Hensel’s lemma and the valuative criterion of properness, one obtains easily that the map π : G(O L 0 ) → X (O L 0 ) is surjective

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and the map G(O L 0 )/B(O L 0 ) −→ G(L 0 )/B(L 0 ) is bijective. We have the following commutative diagram of pointed sets (cf. [Se1, §1.5.4, Prop. 36]): I G(O L 0 )/B(O − → H 1 I,  B(O L 0 ) − → H 1 I, G(O L0 )   L0 )    yo y y I G(L 0 )/B(L 0 ) − → H 1 I, B(L 0 ) − → H 1 I, G(L 0 ) Taking the fixed points of the twisted I -sets by the cocycle f φ , one gets I ∼ I G(O L 0 )/B(O L 0 ) −→ f G(L 0 )/B(L 0 ) , f φ

φ

so the class [ f φ ] ∈ H 1 (I, G(O L 0 )) has a reduction in H 1 (I, B(O L 0 )). Thus there exists h ∈ G(O L 0 ) such that hφ(σ )(σh−1 ) ∈ B(O L 0 ) for any σ ∈ I . Let h be the image of h in G(k 0 ). Then hφ(σ )(h)−1 ∈ B(k 0 ) for any σ ∈ I (we use here that L/K is totally ramified), and φ is k 0 -good. By Proposition 2, because k 0 /k is separable, φ is k-good. The wild inertia group Gal(K s /K mod ) is a pro- p group. If p is a good prime for G, that is, if p is not a torsion prime of G, we know that H 1 (K mod , G) = 1 (see [Se2, Th. 400 ]). The last proposition together with Proposition 2 then gives another proof of Tits’s result [T2, Cor. 2.6]. COROLLARY 1 Assume that p = char(k) 6 ∈ S(G). Then any unipotent subgroup U of G(k) is kembeddable in the unipotent radical of a k-parabolic subgroup of G.

2.3. The case when [k : k p ] ≤ p THEOREM 2 Assume that char(k) = p > 0, that [k : k p ] ≤ p, and that G is semisimple simply connected. Then every unipotent subgroup of G(k) is k-embeddable in the unipotent radical of a k-parabolic subgroup of G. We recall K. Kato’s definition in [K] of the p-dimension dim p (k) of k by means of cohomology groups H pi (k). Let k be the k-vector space of the 1-differential forms of V the Z-algebra k. For any nonnegative integer i, we set ik = i k , and the exterior differential d maps ik to i−1 k . There exists a unique additive p-linear application γ : ik → ik /di−1 such that γ (xω) = x p w for any logarithmic differential form k ω = dy1 /y1 ∧ · · · ∧ dyi /yi . The operator ω is the inverse of the Cartier operator. We set H pi+1 (k) = Coker(γ − 1 : ik → ik /di−1 k ). If [k : k p ] = ∞, one sets dim p (k) = ∞. If [k : k p ] = pr , one sets

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dim p (k) = r if H pr +1 (k 0 ) = 0 for any finite extension k 0 /k, dim p (k) = r + 1 otherwise.

Proof Assume that [k : k p ] ≤ p. By Proposition 2, one can assume that k = ks , that G is split and almost simple, and that U is generated by a unipotent element of order q. One has p Br (k 0 ) = H p2 (k 0 ) = 0 for any finite extension k 0 /k because k is separably closed, so dim p (k) ≤ 1. Then dim p (K ) ≤ 2 by [K, cor. to Th. 3], where K = k((t)) as in §2. By definition, one has H p3 (K 0 ) = 0 for any finite field extension K 0 /K . As the wild inertia group Gal(K s /K mod ) is a pro- p group, the main result of [Gi2] then gives H 1 (K mod , G) = 1. Then Proposition 3(b) shows that u is k-good. 2.4. Bad unipotents and torsors on the affine line M. Raghunathan has conjectured that if G is semisimple simply connected, then Gtorsors on the affine line A1k are constant, that is, come from G-torsors on Spec(k) (see [R, p. 189]). It turns out that bad unipotent elements yield counterexamples to this conjecture. 4 Assume that G/k admits a bad unipotent element u of order p. Let P : A1k → A1k be the Artin-Schreier covering, which is Galois of group F p and defined by t = P (x) = x p − x. Let us define the 1-cocycle h = (h σ )σ ∈F p for this covering by h σ = u σ ∈ G k[x] (σ ∈ F p ). PROPOSITION

Then the cocycle h defines a G-torsor on A1k which is not isomorphic to a constant torsor. Proof One can assume that k = ks . So H 1 (k, G) = 1, and we have to show that the torsor defined by h is not trivial. We lift our cocycle h in G k((1/x)) , and we observe that k(x)/k(t) is wildly ramified at ∞. As u is a bad unipotent element, Proposition 3(b) shows that [h]k((1/t)) is not trivial in H 1 k((1/t)), G , and a fortiori [h] is not trivial in H 1 (A1k , G). Remark 2. Theorem 1 (§1.2) yields nonconstant G-torsors on the affine line for G split semisimple simply connected, provided p divides dG and [k : k p ] ≥ p 2 .

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3. Conjugacy classes of k-anisotropic automorphisms of split groups From now on, we assume that G/k is split and semisimple. So G is the extension of Z to k of the corresponding Chevalley group scheme G / Spec(Z) . For any ring A, we denote for simplicity G ×Z A = G ×Spec(Z) Spec(A). The goal of this section is to give a powerful cohomological criterion analogous to Proposition 3 for distinguishing conjugacy classes of k-anisotropic automorphisms. We denote by T /Z a maximal split torus of G/Z and by B/Z a Borel subgroup containing T . Let J ⊂ Aut(G, B, T ) be a subgroup, and let us consider the semidirect product G J /Z = G/Z o J . We have a natural map G J /Z → Aut(G)/Z with kernel the center of G/Z, and we note that if G/Z is adjoint and J = Aut(G, B, T ), one has G J /Z = Aut(G)/Z. Let us introduce the following notation: • W = N G (T )/T and W J = N G J (T )/T = W o J (the Weyl groups); b = HomZ−gr (T, Gm ), T b0 = HomZ−gr (Gm , T ). • T PROPOSITION 5 Let φ ∈ Homgr (I, G J (k)). (a) The following assertions are equivalent: (i) φ is k-anisotropic, e I (ii) [ fφ G J (Oe L )] = ( f φ G J )(K ), (iii) the group fφ G J /K is K -anisotropic. (b) Assume that φ is k-anisotropic. Let φ 0 : I → G J be another group morphism. The following assertions are equivalent: (i) φ 0 is conjugate to φ under G J (k), (ii) γ (φ) = γ (φ 0 ) in H 1 (K , G J ). 0 (c) Let k /k be a Galois extension such that φ/k 0 : I → G J is a k 0 -anisotropic morphism. With the notation Z G (φ) = Z G (Im(φ)), one has an injection H 1 k 0 /k, Z G J (φ) ,→ H 1 (K · k 0 /K , fφ G J ).

Proof Let us begin with the second assertion. (b) The implication (i) =⇒ (ii) is obvious. Conversely, assume that γ (φ) = γ (φ 0 ) in H 1 (e L/K , G). Then there exists g ∈ G(L) such that g −1 φ(σ ) σ g = φ 0 (σ )

(σ ∈ e I ).

Let B be the Bruhat-Tits building of the group G e L (see [BrT, §I.7.4]) which is a metric space. This building is equipped with an (isometric) left action of the group G J (e L) and an action of the Galois group e I = Gal(e L/K ) denoted by x 7→ σx for any σ ∈ e I . We denote by c the center of the building, that is, the vertex of B fixed by

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e G(Oe L ). One defines a twisted action of I on B by σ · x = φ(σ ) ·σ x

(σ ∈ e I ).

This action is isometric and simplicial (i.e., maps a facet on a facet). Clearly, because Im(φ) ⊂ G J (k), the center c is fixed under the twisted action of e I . Because σ · (g · c) = φ(σ )(σg) · c = gφ 0 (σ ) · c = g · c, we see that g · c is also fixed by e I for the twisted action. Assume that g · c 6= c. As the segment [c, g · c] is fixed pointwise by e I , there exists a facet of B containing strictly c which is stabilized by the twisted action of e I . Hence there exists (by the isomorphism between the link of c in B and the spherical building of G J /k(µm )) a facet of the spherical building of G J /k(µm ) which is stabilized by e I . In other words, there exists a proper k-parabolic subgroup P of G such that fφ

e I G J (k(µm ))/N G J (P)(k(µm )) 6 = ∅,

that is, such that Gal(k(µm )/k) [ϕ] ∈ G J (k(µm ))/N G J (P)(k(µm )) ϕ −1 Im(φ)ϕ ∈ N G J (P)(k(µm )) 6 = ∅, which means that φ normalizes the k-parabolic subgroup ϕ(P). This contradicts the anisotropy of φ. We conclude that g · c = c, that is, that g ∈ G J (O L ). Reducing the identity g −1 φ(σ )(σg) = φ 0 (σ ) to G J (k(µm )), one gets g −1 φg = φ 0 with g ∈ G J (k). (a) We have the following. (i) =⇒ (ii). Assume that φ is k-anisotropic. The same argument as before with φ 0 = φ shows I ⊂ G J (Oe ( fφ G J )(K ) = g ∈ G J (e L) φ(σ )(σg)φ(σ )−1 = g, ∀σ ∈ e L ), I so ( fφ G J )(K ) = [ fφ G J (Oe L )] . e

(ii) =⇒ (iii). As ( fφ G)(K ) ⊂ G J (Oe L ) is bounded, the group any nontrivial K -split torus; hence it is K -anisotropic.

f φ G/K

does not contain

(iii) =⇒ (i). We prove not (i) =⇒ not (iii). Assume that φ is k-isotropic, that is, that there exists a k-parabolic subgroup P such that Im(φ) ⊂ N G J (P)(k). Then the class γ (φ) in H 1 (K , G J ) comes from H 1 (K , N G J (P)), and the twisted group fφ G/K is isotropic.

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(c) We set K 0 = K ⊗k k 0 , O 0 = L ⊗k k 0 , and so on. The first assertion shows e I 0 0 that ( fφ G J )(K 0 ) = [ fφ G J (Oe L 0 )] . But the map of Gal(k /k)-groups Z G J (φ)(k ) →

I 1 0 1 0 [ fφ G J (Oe L 0 )] is split, so the map H (k /k, Z G J (φ)) → H (K · k /K , injective. e

fφ G J )

is

First, we give the following corollary of the proposition and of Lemma 3 (§1.3). COROLLARY 2 Assume that the adjoint group G ad of G is simple. Assume that m is an integer such that k contains a primitive m-root of unity and such that there exists a k-anisotropic automorphism of order m. Then m ∈ S(G).

Proof One can assume that G = G ad . Let α be a k-anisotropic automorphism of order m, and assume that m 6∈ S(G). The exact sequence 1 → G → Aut(G) → Out(G) → 1 and the fact that Out(G) is a finite group such that the prime divisors of ] Out(G) are bad primes show that α is an inner automorphism. By Lemma 3, there exists a maximal torus S which contains α. The choice of a primitive m-root of unity ζ induces an √ ∼ isomorphim Z/mZ −→ µm ; taking L = k( m t)/K and θ = (t) ∈ H 1 (K , µm ) as in Section 2, we see that γ (α) ∈ Im(H 1 (L/K , S) → H 1 (L/K , G)). But H 1 (L/K , S) is an S(G)-primary torsion group, so H 1 (L/K , S) = 1 and γ (α) = 1. By Proposition 5(a), α is then k-isotropic. This is a contradiction. Questions Assume that p > 0. We lack an analogue of Proposition 2 (§3.1) for k-anisotropic automorphisms of order p. More precisely, we ask the following. (1) Let α be a k-anisotropic automorphism of G of order p. Is αks ks -anisotropic? (2) Let α, α 0 be k-anisotropic automorphisms of G of order p such that α and α 0 are conjugate under Aut(G)(ks ). Are α and α 0 conjugate under Aut(G)(k)? 4. Anisotropic automorphisms are special In this section we prove Conjecture 20 ; as mentioned earlier, this proves that Tits’s original conjecture is proven, as the case of E 8 and p = 5 has already been done in [Gi3]. THEOREM 3 Conjecture 20 is true for all types, excepting possibly the case of E 8 and l = 5 6= p.

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The steps have been described in Section 1.2, and we take into account cases done by Tits in the wild case. Their proof works also in the tame case, which is much simpler. 4.1. Special automorphisms Definition 3 A k-automorphism f of a torus S/k is anisotropic if the k-group S f of fixed points of S by f is finite. Definition 30 (a) An element w of W J is special if the automorphism w : T → T defined by t 7→ w · t is anisotropic. (b) An element of G J (k) of order n is special (relative to the split torus T ) if it lies in N G J (T )(k) and its image in W J is a special element of order n. Some special elements of W J are listed in [Sp] for any simple type. 4 Let w be a special element of W J of order n. (a) The homomorphism N (w) = 1 + w + · · · + wn−1 : T → T is trivial. (b) Let g be a special element relative to T mapping to w. Then for any τ ∈ T (k), τ g is a special element of G J (k) of order n and is N G (T )(k)-conjugate to g. (c) Let g be an element in N G (T )(k) such that g is k-anisotropic in G. Then g acts anisotropically on T . Moreover, if char(k) = p > 0 and g has order pr , then g is special (relative to T ). (d) Assume that n is odd. Let A be a complete discrete valuation ring with residue field k and fraction field FA . Let g be a special element of G J (k) (relative to T ) mapping to w. Then there exists e g ∈ N G J ×Z A (T )(A) of order n mapping to g in G J (k) such that e g is a special element of G J (FA ) (relative to T ) and mapping to w in W J . LEMMA

Proof b ⊗Z Q, then (a) As (1 − w) · N (w) = 0 and 1 − w is an invertible automorphism of T b ⊗Z Q; hence N (w) : T → T is trivial. N (w) = 0 as an automorphism of T (b) Let us denote by w the image of g in W J . We have to show that τ g has order n; it is given by the computation (τ g)n = τ × gτ g −1 × g 2 τ g −2 × · · · × g n−1 τ g (−n+1) = τ × (w · τ ) × · · · × (wn−1 · τ ) = N (w) · τ = 1.

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So the order of τ g divides n, and as w has order n, the order of τ g is exactly n. We consider the equation τ g = xgx −1 x ∈ T (k) or, equivalently, τ = xgx −1 g −1 = x × w · x −1

x ∈ T (k) .

As the group morphism T → T given by t 7→ t × w · t −1 has finite kernel, it is surjective, so there exists x ∈ T (k) such that τ g = xgx −1 . (c) Let g be an element of N G J (T )(k) such that g is k-anisotropic. Assume that g acts isotropically on T ; then T hgi = Ker(1 − g : T → T ) has dimension greater than 1, so there exists a k-split subtorus S of T such that g ∈ Z G (S)(k). As Z G (S) is a k-Levi subgroup of a proper k-parabolic subgroup, g is k-isotropic. So g acts anisotropically on T . Assume, moreover, that g has order pr ; we have to show that r −1 the image of g in W J , say w0 , has order pr . If not, g p is an element of order p of T (k), and as µ p (k) = 1, this leads to a contradiction. (d) As G J is a Chevalley group defined over Z, the sequence 1 → T → N G J (T J ) → W J → 1 ∼

can be defined over Z, and as He´1t (Z, T ) −→ (Pic(Z))rank(G) = 1, one has the exact sequence 1 → T (Z) → N G J (T J )(Z) → W J → 1 and an isomorphism T (Z) ≈ (Z/2Z)rank(G) (see [T1]). So the class of the preceeding extension is killed by 2. Let g ∈ N G J (T J )(k) be of odd order n. Then there exists g0 ∈ N G J (T J )(Z) of order n with same image in W J as g. There thus exists τ ∈ T (k) such that α = τ α 0 , where α 0 is the reduction in k of α0 via the map N G J (T J )(Z) → N G J (T J )(A) → N G J (T J )(k). As T ×Z A is a smooth group scheme, the map T (A) → T (k) is surjective by Hensel’s lemma. We pick e τ ∈ T (A) mapping to τ . The element e g := e τ α0 of N G J (T J )(A) lifts g = τ α 0 and maps to w in W J , and by assertion (b), e g is a special element of G J (FA ) relative to T . 4.2. Triality In this section we handle the when case l = 3, assuming then that k contains a primitive third root of unity. First, let us remark that we can assume that G is adjoint, so the group Aut(G) = G o Aut(G, B, T ) is of the type we studied before. Let α ∈ Aut(G)(k) be a k-anisotropic automorphism of order 3. We use the following triality argument.

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LEMMA 5 Assume that G has type D4 (resp., F4 , E 6 , E 7 , E 8 ). Then α normalizes a k-Levi subgroup of a k-parabolic subgroup of G of type A2 (resp., A2 , A2 × A2 , E 6 , E 7 ).

Proof Let P be a k-parabolic subgroup of G such that Pred has type A3 (resp., C3 , D5 , E 6 , E 7 ). Then codimG (P) = 6 (resp., 15, 16, 27, 57). Let us define the k-group C = P ∩ α(P) ∩ α 2 (P). Then codimG (C) ≤ 3 × codimG (P), so dimG (C) ≥ 28 − 18 − 10 (resp., 7, 30, 52, 77) (cf. [PR, p. 380]). We claim that C is a k-Levi subgroup of some k-parabolic subgroup of G. Let Q be a k-parabolic subgroup such that C ⊂ Q ⊂ P and minimal for this property. Then the k-parabolic subgroup Ru (Q) · (Q ∩ α(Q)) contains C and is contained in Q, so Ru (Q) · (Q ∩ α(Q)) = Q and M := Q ∩ α(Q) is a k-Levi subgroup of G (see [BT1, Prop. 4.10]). But C = M ∩α 2 (Q) is a k-parabolic subgroup of M; if C 6= M, then α normalizes the split k-unipotent group Ru (C), and by [BT2, Prop. 3.1], α normalizes a k-proper parabolic subgroup of G, which contradicts the assumption of anisotropy. So C is a k-Levi subgroup of Q/k, and the restriction of α is still k-anisotropic (Lemma 2). As dimk (C) ≥ 10, the only possibility is that C has type A2 . Other cases are considered on a case-by-case basis using Corollary 2, and we leave the details of the argument to the reader. Let us recall that Theorem 3 is true for A2 and prove it inductively in the cases considered in Lemma 5. Lemma 5 gives a k-Levi subgroup C of a k-parabolic proper subgroup of G normalized by α. So the connected center Z G (C)0 is a k-split torus normalized by α and, by induction, the derived group D C contains a k-split maximal torus S normalized by α. It follows that Z G (C)0 · S is a maximal k-split torus of C (and G) normalized by α. 4.3. Lifting in characteristic zero. LEMMA 6 Assume that p = char(k) > 0. Let A be a complete discrete valuation ring with residue field k and fraction field FA . Let φ : I → G J be a group homomorphism that e : I → G J ×Spec(Z) Spec(A). lifts to φ e : I → G J ×Z FA is an FA -isotropic morphism, then φ is a k-isotropic (a) If φ morphism. e normalizes some maximal FA -split torus (b) Assume that φ is k-anisotropic. If φ of G J,FA , then φ normalizes some maximal k-split torus of G J /k. Moreover, e : I → G J ×Spec(Z) let φ 0 : I → G J be a group homomorphism that lifts to φ

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e and φ e0 are conjugate under G J (FA ). Then φ and φ 0 Spec(A) such that φ are conjugate under G J (k), and φ 0 normalizes some maximal k-split torus of G J /k. Proof e is a k-isotropic element of G J (FA ). This means that there exists a (a) Assume that φ e) has a fixed point in standard parabolic subgroup P/Z of G/Z such that I (through φ X (FA ), where X = G J /N G J (P) is the projective Z-scheme of parabolic subgroups of G J of the same type as P. As X (A) = X (FA ), then I has a fixed point in X (k), and φ is k-isotropic. e normalizes an FA -maximal split (b) Assume that φ is k-anisotropic and that φ torus, that is, that there exists g ∈ G J (FA ) such that e(σ )g = e g −1 φ f (σ ) ∈ N G J (T )(FA )

(σ ∈ I ),

(*)

where e f : I → N G J (T )(FA ) is a group homomorphism. We consider the BruhatTits building B of the group G J,FA with center c, which is the vertex stabilized by e), which is convex and G J (A). We denote by B I the fixed point of X by I (through φ b ⊗Z R ⊂ X . From (∗), we get contains c. The torus T defines an apartment A = T I g · A ⊂ B . Let us denote by π(c) the projection of c on the apartment g · A . CLAIM

We have the following: π(c) = g · c = c. Proof We write c = g · a with a ∈ A . For any σ ∈ I , one has g · a = φ(σ )g · a = g e f (σ ) · a, b0 ⊗Z R) I = A I . But so e f (σ ) · a = a. As I acts on A through e f , we get a ∈ (T 0 I 0 I b b (T ⊗Z R) = (T ) ⊗Z R, so if a 6 = c, it means then that e f is k-isotropic, which gives a contradiction to the anisotropy of φ by (a). We conclude that π(c) = g · c. As B I is convex, the segment [c, π(c)] is fixed pointwise by α. If c 6= π(c), there exists a facet F of B strictly containing c which is stabilized by I . By the isomorphism between the link of c and the spherical building of G J /k, there exists a proper k-parabolic subgroup P of G such that φ(P) = P, which again gives a contradiction to the anisotropy of φ. We conclude that π(c) = c, that is, that c ∈ g · A . So g · c = c; that is, g ∈ G J (A) and e f (σ ) ∈ N G J (T )(A). Reducing (∗) in k yields g −1 φ(σ )g ∈ N G J (T )(k), so φ normalizes the maximal k-split torus g · T . One can e normalize T . then assume that φ and φ

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e : I → G J ×Z A Now, let φ 0 : I → G J be a group homomorphism that lifts to φ 0 e e and such that φ and φ are conjugate under G J (FA ). Let g ∈ G J (FA ) be an element e0 = g φ eg −1 . Then φ e0 normalizes the FA -torus gT g −1 . But φ e is anisotropic such that φ 0 e by assertion (a), so φ is anisotropic and the proof of assertion (b) shows that g ∈ e0 = g φ eg −1 with g ∈ G J (A), so φ = gφ(g)−1 with g ∈ G J (A). We have then φ 0 G J (k), and φ normalizes the maximal k-split torus gT g −1 . LEMMA 7 Let φ ∈ Homgr (I, G J (k)) be a k-anisotropic morphism. Then there exists g ∈ G J (ks ) such that the following hold: (i) g −1 Im(φ)g ⊂ N G J (T )(ks ), (ii) g−1sg ⊂ N G J (T )(ks ), ∀s ∈ Gal(ks /k). The torus g · T is defined over k and normalized by φ. Moreover, in the case when I is cyclic, one can replace condition (i) with (i0 ) g −1 Im(φ)g ∈ N G J (T )(k).

Remark 3. Lemma 7 yields Conjecture 2 of Tits by a uniform argument in the case when k is separably closed. Proof It is well known that the map H 1 K , N G J (T ) → H 1 (K , G J ) is surjective. In other words, there exist a finite Galois extension M/K such that K ⊂ L ⊂ M ⊂ K s , an element g ∈ G J (M), and a cocycle h ∈ Z 1 (Gal(M/K ), N G J (T )(K s )) such that g−1 φ(s)sg = h s s ∈ Gal(M/K ) . (*) Let us denote by K 0 /K the maximal unramified subextension of M, by O 0 its valuation ring, and by k 0 its residue field. Set L 0 = L · K . We may assume that K 0 = k 0 ((t)). We have the following: M | L0 K0 L K

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Let B be the Bruhat-Tits building of the group G M (see [BrT, §I.7.4]). This building is equipped with an action of the group G J (M) and an action of the Galois group Gal(M/K ) denoted by x 7 → sx for any s ∈ Gal(M/K ). We denote by c the center of the building, that is, the vertex of B fixed by G J (O M ), where O M is the valuation ring b0 ⊗Z R ⊂ B , which contains of M. The torus T ×k M defines an apartment A = T the center c and which is fixed pointwise by Gal(M/K ). One defines a twisted action of Gal(M/K ) on B by s · x = φ(s) · sx s ∈ Gal(M/K ) . This action is isometric and simplicial (i.e., maps a facet on a facet). Clearly, since Im(φ) ⊂ G J (k 0 ), the center c is fixed under the twisted action of Gal(M/K ). We claim that the apartment g A is stabilized by Gal(M/K ) for the twisted action. This is a result of the following computation: for any x = g · a ∈ A , one has s · x = φ(s) · s(g · a) = g · (h s · a) ∈ g A . Let us denote by π(c) = g · a the projection of c to the apartment g A . CLAIM

We have the following: π(c) = g · c = c. Proof As c is fixed under the twisted action of Gal(L/K ), the projection π(c) is also fixed. Moreover (observe that sa = a), one has π(c) = g · a = s · (g · a) = φ(s) · sg · sa = gh s · a (s ∈ Gal(L/K )), so a = h s · a s ∈ Gal(M/K ) . Now, we consider the maximal torus n T /K

⊂ n G/K ≈

f φ G/K .

The group n G/K ≈ fφ G/K is anisotropic by Proposition 5, so the torus n T is 0 anisotropic and 0 = nc T (K ) ⊗Z R = {x ∈ A | h s · x = x}. One deduces that a = c, that is, π(c) = h · c. Let us assume that π(c) 6= c. Then the segment [c, π(c)] is fixed pointwise by the twisted action of Gal(M/K ). So there exists a facet of B containing strictly c which is stabilized by the twisted action of Gal(M/K ), and there exists (by the isomorphism between the link of c in B and the spherical building of G/k) a facet of the spherical

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building of G/k which is pointwise fixed by Gal(M/K ). In other words, there exists a proper k-parabolic subgroup of G such that Gal(M/K ) G J (k 0 )/N G J (P)(k 0 ) 6 = ∅. f φ

The group Gal(M/K ) acts on the previous set through Gal(L 0 /K ) = I × Gal(k 0 /k). One has I G J (k 0 )/N G J (P)(k 0 ) fφ = [d] ∈ G J (k 0 )/N G J (P)(k 0 ) d −1 Im(φ)d ∈ N G J (P)(k 0 ) , Gal(k 0 /k) G J (k 0 )/N G J (P)(k 0 ) fφ = [d] ∈ G J (k 0 )/N G J (P)(k 0 ) d −1 (sd) ∈ N G J (P)(k 0 ), ∀s ∈ Gal(k 0 /k) . Let d ∈ N G J (P)(k 0 ) be a coset in the intersection of the two previous sets, and let us denote by z ∈ Z 1 (k 0 /k, N G J (P)) the cocycle defined by z s = d −1 (sd). The element ∼ d gives rise to a trivialization βd : z G J −→ G J such that Im(φ) ⊂ βd (z N G J (P)), which gives a contradiction to the anisotropy of φ. We conclude that π(c) = c. Let us denote by a 7→ a the map G J (O M ) → G J (k 0 ). The identity (∗) holds in G J (O M ), and replacing d by d ∈ G J (k 0 ), one can assume that g ∈ G J (k 0 ) ⊂ G J (L 0 ), that L 0 = M, and that h s ∈ G J (k 0 ) ⊂ G J (L 0 ) for every s ∈ Gal(L 0 /K ). So we get (i) and (ii). Assume now that I is cyclic with a generator σ . Identity (∗) for σ yields g −1 φ(σ )g = h σ ∈ N G J (T )(k 0 ).

(**)

s Since Gal(k 0 /k) acts trivially on the Weyl group W J , the map s 7 → h −1 σ ( h σ ) defines 0 0 a Gal(k /k)-cocycle in T (k ); and by Hilbert’s Theorem 90, there exists τ ∈ T (k 0 ) s −1 (sτ ). Replacing g with gτ in (∗∗), one can assume that such that h −1 σ ( hσ ) = τ h σ ∈ N G J (k), and taking h = h σ , the proposition is proved.

LEMMA 8 Assume that p = char(k) is an odd prime; let A be a complete discrete valuation ring with residue field k and fraction field FA . Let H/k be a k-form of G J , and let H be an A-form of G J ×Z A with special fiber H/k. Let S be a maximal k-torus of H , and let S be a maximal A-torus of H with special fiber S. Let h ∈ N H (S)(k) be an element of finite order q = pr . Then h lifts to an element e h ∈ NH (S)(A) of order q.

Remark 4. Such an H exists by Hensel isomorphism ∼ He´1t A, Aut(G J ) −→ H 1 k, Aut(G J ) , and such an A-torus S exists by [SGA3, exp. XV, §8].

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Proof Let As /A be the e´ tale extension associated to ks /k. We remark first that we can assume that h generates H/H 0 . Let us denote by w the image of h in (N H (S)/S)(k), which has order q. First case: w acts anisotropically on S ×k ks . First, remark that the lifting assumption in NH (S)(As ) is satisfied by Lemma 4(d) (see §4.1; we use here that p is odd). Let us denote the Weyl group scheme NH (S)/S by W (S)/A, that is, a twisted finite group, so isomorphic to N G (S)/S. We also denote by hwi the cyclic subgroup of W (S) generated by w, and by Nw /A the preimage of hwi in NH (S). By Hensel’s lemma, one has the following two exact commutative diagrams of pointed sets: 1 − → S(A) − →   y 1 − →

S(k)

− →

Nw(A) − → hwi − →



y

Nw (k)

− → hwi − →

H 1 (A,  S)  yo H 1 (k, S)

and 1 ↓ S = ↓ 1 − → S(A) − →   sp y 1 − →

S(k) ↓ 1

− →

1 ↓ S ↓ Nw(A) − → hwi − → 1



sp y

Nw (k) ↓ 1

− → hwi − → 1

where S := Ker(S(A) → S(k)). So the obstruction to lifting h to an element of order q in NH (S)(A) is the class, say, η in H 2 (hhi, S), of the restriction to the cyclic group hhi of the vertical extension. We put µ/A = Ker(1 − w : S → S) which is a finite A-multiplicative group, and we have H 2 hhi, S = (S)hwi /N (w) · S = Ker µ(A) → µ(k) because N (w) = 0 (Lemma 4(a)). The same approach for k = ks gives an obstruction ηs in µ(As ) which vanishes. As µ(A) ⊂ µ(As ), one has η = 0, and h lifts to NH (S)(A). Second case: h acts isotropically on S ×k ks . This means that dim(S hhi ) ≥ 1, that is, that there exists a nontrivial subtorus S0 of S such that h ∈ Z H (S0 )(k). We have Z H 0 (S0 ) = Z H (S0 )0 . As h generates H/H 0 , we have an exact sequence

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1 → Z H 0 (S0 ) → Z H (S0 ) → H/H 0 → 1 which is split by h. The element h acts trivially on the coradical torus (cf. [SGA3, exp. XXII, §6.2]) C =: corad(Z H 0 (S0 )) of Z H 0 (S0 ), and it defines an extension of the canonical morphism Z H 0 (S0 ) → C to a morphism Z H (S0 )0 → C. So one can define the group H 0 /k by the following diagram: 1 ↑ C x  

1 − → 1 − →

Z H 0x(S0 )  

= − →

D Z H 0 (S0 ) − → ↑ 1

1 ↑ C x  

Z Hx(S0 ) − →   H0 ↑ 1

− →

0 H/H

− → 1

H/H 0

− → 1

So H 0 is a k-form of some G 0J 0 and h ∈ H (k) because C(k) has no k-points of p-power order. Denoting S 0 = Ker(S → C) ⊂ H 0 , h acts anisotropically on the maximal torus S 0 ×k ks of H 0 . Defining H0 /A and S0 /A by Hensel’s lift, the first case shows that h lifts in NH0 (S0 )(A) = NH (S)(A) ∩ H0 (A), so h lifts in NH (S)(A). 4.4. Proof of Theorem 3 in the positive characteristic case Recall that char(k) = p > 0; we consider a k-anisotropic automorphism α of prime order l of G. We can assume that l is odd. Let A be a complete discrete valuation ring with residue field k and with fraction field FA such that FA contains a primitive p-root of unity. First, we show that α lifts to Aut(G)(A), that is, that there exists e α ∈ Aut(G)(FA ) of order l lifting α. If l = p, this is a consequence of Lemmas 7 and 8. If l 6= p, then Z/lZ is a finite group of multiplicative type; as Aut(G) is smooth, the Grothendieck rigidity theorem [SGA3, exp. VI, Cor. 7.3] says that the map Hom A−gr Z/lZ, Aut(G) → Homk−gr Z/lZ, Aut(G) is surjective, so α lifts in characteristic zero. By Lemma 6(a), e α is an FA -anisotropic automorphism of G/FA , the characteristic zero case shows that e α is special, and, finally, Lemma 6(b) shows that α is special. Acknowledgments. It is a pleasure to thank Tamás Szamuely for his careful reading of a preliminary version of this paper. I wish to express my hearty thanks to Jean-Pierre Serre and Jacques Tits, who introduced me to this subject.

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Département de Mathématiques, Centre National de la Recherche Scientifique, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay CEDEX, France; [email protected]


WAVE KERNELS RELATED TO SECOND-ORDER OPERATORS PETER C. GREINER, DAVID HOLCMAN, and YAKAR KANNAI

Abstract The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Gruˇsin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study propagation of the singularities. The results are expressed in terms of the real values of a complex function extending the Carnot-Caratheodory distance, and the geodesics of the associated sub-Riemannian geometry play a crucial role in the analysis. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1. Sub-Riemannian (Carnot-Caratheodory) metrics . . 1.2. A curve in the complex plane and the complex action 1.3. Lorentz-Carnot-Caratheodory metric . . . . . . . . 1.4. Separation of variables . . . . . . . . . . . . . . . 2. The wave kernel for the Gruˇsin operator . . . . . . . . . 2.1. The boundary of the forbidden set . . . . . . . . . 2.2. The zero of the phase and the integration path . . . 2.3. Explicit formula for the integral . . . . . . . . . . 2.4. Analysis of the wave kernel near the singularities . . 2.5. The operator satisfies the wave equation . . . . . . 2.6. Source not at the origin . . . . . . . . . . . . . . 3. Wave kernels in one dimension . . . . . . . . . . . . . . 3.1. Wave kernel for the harmonic oscillator . . . . . . 3.2. Wave kernel for the Klein-Gordon operator . . . . . 4. The Heisenberg wave kernel via the heat kernel . . . . . . 4.1. Deforming the path of integration . . . . . . . . . DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 2, Received 10 April 2001. Revision received 7 September 2001. 2000 Mathematics Subject Classification. Primary 35L80, 53C17; Secondary 35H20. Holcman and Kannai’s work supported by the Minerva Foundation, Germany. 329

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330 333 337 340 341 343 346 348 349 353 355 357 358 358 360 361 363

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4.2. Computation of the wave kernel . . Wave kernels via the continuation method 5.1. The Heisenberg Laplacian . . . . . 5.2. Degenerate elliptic operators . . . . 6. Directions for further studies . . . . . . . References . . . . . . . . . . . . . . . . . . 5.

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

365 371 373 376 385 386

1. Introduction We study in this paper properties of fundamental solutions of wave equations associated with several subelliptic second-order self-adjoint operators L. We give an explicit expression for the Gruˇsin operator, the Heisenberg Laplacian, and the harmonic oscillator. Recall that the general solution of the wave equation ∂ 2wL = −Lw L , ∂t 2 w L (0) = f, ∂w L (0) = g, ∂t

(1.1)

has the formal expression w L (t) =

sin(t L 1/2 ) g + cos(t L 1/2 ) f. L 1/2

(1.2)

It suffices to compute the fundamental solution w L that satisfies equation (1.1) and w L (0) = 0, ∂w L (0)/∂t = δ0 , where δ0 denotes the Dirac distribution at the origin zero, that is, sin(L 1/2 t) , (1.3) w L (t) = L 1/2 or to compute ∂w L /∂t = cos(t L 1/2 )—the solution of (1.1) where w L (0) = δ0 , ∂w L (0)/∂t = 0. We consider the wave kernel for a number of second-order operators. We obtain in some cases an explicit representation for the kernel and some information about propagation of singularities. The wave kernel for the standard wave equation in the general n-dimensional Euclidean space Rn was computed first by J. Hadamard (see R. Courant and D. Hilbert [5]), who considered more generally the case where L is elliptic. The kernel is of the form δ ( p) (t 2 − |x|2 ) for the usual (Euclidean) Laplacian. In the general elliptic case, the solution is represented as a sum of terms, starting from the less regular to the more regular (see [5]). Once again the leading term is proportional to δ ( p) (t 2 − |x|2 ), where

WAVE KERNELS RELATED TO SECOND-ORDER OPERATORS

331

now |x| is the distance between x and the origin in the Riemannian metric associated with the second-order elliptic operator. The wave equation for a subelliptic L has been considered in [10]. The finite speed of propagation was established in [12]. The first systematic analysis of the wave kernel for the Heisenberg Laplacian was performed in [13], where the “light cone” was described and propagation of singularities discussed. The computations are somewhat complicated. In [15] the broad features of the propagation of singularities for the Heisenberg Laplacian are also described using a different method based on deforming the path of integration in the complex plane. The geometry of the light cone is rather different from the standard Euclidean case. It turns out that finite speed of propagation is associated with the so-called Carnot-Caratheodory distance defined as the sub-Riemannian length of a minimizing geodesic (see [4], [14]); the formula for the wave kernel, and the full light cone, involve all geodesics. Sub-Riemannian geometry differs substantially (even locally) from usual Riemannian geometry. On the other hand, a complex-valued function f , appearing in the integral representation for the heat kernel on Heisenberg group (see [6]), was shown in [1] to satisfy a Hamilton-Jacobi equation with the symbol of the Heisenberg Laplacian as Hamiltonian. Critical points and critical values of this function f (extended analytically to the complex plane) correspond to sub-Riemannian geodesics and their lengths, respectively. A curve on which the function f is real is constructed in [15] and in [1]. Our formulas involve integration along this curve. Moreover, adding a time-dependent term to f , we obtain a complex phase satisfying a Hamilton-Jacobi equation with the symbol of the wave operator as Hamiltonian. Observe that in both [13] and [15] propagation of singularities is studied without actually computing the wave kernel. In [13] the kernel is given as a limit of expressions containing integrals (or an infinite series); one could presumably get a closed form with extra effort. No attempt at calculating the kernel is made in [15]; the appearance of fractional powers in [15, (8.7)] makes explicit computations difficult. One of the main purposes of the present paper is to obtain a more explicit formula for the Heisenberg wave kernel. Known properties of singularities (such as propagation) are then easily obtained. Moreover, the relationship between the sub-Riemannian geometry and complex integration formulas (such as in [6] and [1]) is put into context. Explicit formulas for model operators (such as the wave kernels for the Heisenberg Laplacian or the Gruˇsin operator), while interesting in their own right, may also offer new insights into the problem and may serve as principal terms in approximations for more general cases. Three methods are applied in this paper for explicit computation of the wave kernels. The first involves separation of variables, summation of series containing Hermite polynomials, and deformation of integration path in the complex plane. This

332


approach is utilized in Section 2 for solving (1.1) if −L/2 is the Gruˇsin operator in R2 , 2 L 1 ∂2 2 ∂ − = + x , (1.4) 2 2 ∂x2 ∂ y2 and in Section 3 for the case where −L/2 is the Harmonic oscillator in R1 , −

L 1 ∂2 2 − x . = 2 2 ∂x2

(1.5)

As an additional illustration, we solve in Section 3 the Klein-Gordon equation. The integral representing the wave kernel for the Gruˇsin operator may be evaluated explicitly by the residue theorem (Proposition 4), and all properties (geometry of the light cone, behavior near this cone, band structure) may be read off the resulting (rather explicit) formula. On the other hand, the formula for the harmonic oscillator (Proposition 7) involves an integral over a path where the phase is purely imaginary, and it seems that the integral may not be easily evaluated. To the best of our knowledge, wave kernels for the Gruˇsin operator and for the harmonic oscillator were not calculated before, nor was the propagation of singularities for the Gruˇsin operator studied in detail. The second method involves inversion of the transmutation formula (Proposition 9; see [8]) and deforming the integration path used in the integral formula for the heat kernel from the real axis to a path in the complex plane where the exponent is real. This approach is described in Section 4 and applied for the case where −L/2 is the Heisenberg Laplacian 1 H defined on R2n+1 = {(x1 , . . . , x2n , x0 )} by 1H =

n n 1 X ∂ ∂ 2 1 X ∂ ∂ 2 + 2α j x2 j + − 2α j x2 j−1 , (1.6) 2 ∂ x2 j−1 ∂ x0 2 ∂ x2 j ∂ x0 j=1

j=1

where α1 , . . . , αn are positive constants. Throughout most of the paper we consider the isotropic case in which all the α j ’s are equal to a constant α (see, e.g., (4.11), (5.15)). (We comment on the extension to the general anisotropic case at various points in the paper.) The expression for the wave kernel using a complex contour is given in Theorem 2 (formula (4.14)). One may rewrite the formula using integration over real intervals (Theorem 3). The role of the geodesics emerges clearly, as the 2 2 ), where d is the length integration is performed over intervals of the form (d2k−1 , d2k j of the jth geodesic. One may obtain a closed form (not involving integrations) if x = (x1 , . . . , x2n ) = 0 (Theorem 4). The leading singularities of the wave kernel are calculated (and are compatible with the results of [13]). Observe that in Section 4 we deal with the kernel of cos(t L 1/2 ), unlike the rest of the paper where sin(t L 1/2 )/L 1/2 is treated.


333

The third method is based on an analytic continuation of the Green function of the operator −L + ∂ 2 /∂ y 2 and uses an idea due to M. Taylor [15]. This method is described in Section 5 and applied to the Heisenberg Laplacian, as well to the case where −L is a degenerate elliptic operator of the type studied in [2]. This class contains the Gruˇsin operator and the Baouendi-Goulaouic operator as special cases. While the results for the Heisenberg Laplacian parallel those of Section 4, new phenomena occur for degenerate operators—if certain dimensions are odd (as is the case for the Gruˇsin operator), the integral representing the kernel is computable (by the residue theorem or otherwise) and yields a simple expression for the kernel. The results coincide with those of Section 2. There is of course a certain amount of redundancy in rederiving the same results by different methods. We feel, however, that each method has its advantages. Thus separation of variables is directly applicable to the harmonic oscillator; using the heat kernel, we may compute cos(t L 1/2 ) directly; and analytic continuation of the Green function enables a straightforward calculation for degenerate elliptic operators without prior computation of the associated heat kernel. Moreover, one should not forget that separation of variables underlies the computation of heat kernels in [15], as well as that of the Green kernels in [3]. An entirely different method for computing wave kernels for certain second-order operators was suggested in [9]. The method is based on transmutation formulas and on the Trotter product formula. Some kind of a “Feynman integral representation” is obtained, and the expression for a wave kernel involves differentiating to a high order a very high-dimensional integral. In [9] expressions were obtained for the wave kernel cos(t L 1/2 ) where −L/2 is the harmonic oscillator and when −L/2 is the Heisenberg Laplacian. A direct proof of the identity of the expressions from [9] with the expression obtained here appears to be nontrivial. In the remainder of this section we collect some preliminary material concerning sub-Riemannian geometry, complex action, and separation of variables. 1.1. Sub-Riemannian (Carnot-Caratheodory) metrics Recall the definition of sub-Riemannian (Carnot-Caratheodory, also known as C-C) metrics (see [4, pp. 4 – 7]): Let X 1 , . . . , X m be smooth vector fields on a manifold M. For x ∈ M and v ∈ Tx M, kvk2x = inf u 21 + · · · + u 2m s.t. u 1 X 1 (x) + · · · + u m X m (x) = v . (1.7) In particular, kvk2x = ∞ if v is not contained in sp(X 1 , . . . , X m ). The length l(c) of an absolutely continuous curve c(t) (a ≤ t ≤ b) contained in M (absolute continuity is well defined in terms of local charts) is given by the integral Rb Rb 2 ˙ ˙ c(t) dt, and the energy of c is equal to a kc(t)k c(t) dt. a kc(t)k

334


The distance between two points is defined by d( p, q) = inf l(c), where the infimum is taken over all absolutely continuous curves joining p and q. In this paper we always assume that the vector fields X 1 , . . . , X m and their brackets [X i , X j ], 1 ≤ i, j ≤ m, span the tangent space Tx M at every point of M, and M is connected. By Chow’s theorem (see [4, p. 15]), any two points in M can be joined by an absolutely continuous curve with finite length. Hence d( p, q) < ∞ for any points p, q ∈ M. (Note that we consider here only the so-called step-two case.) We can define the Hamiltonian associated with the sub-Riemannian metric by m

H (x, ξ ) =

1X hX i , ξ i2. 2

(1.8)

i=1

Note that we do not have a finite metric defined on the tangent bundle; we are forced to study the cotangent bundle. It is well known that any two points p, q may be joined by a curve whose length equals d( p, q). Thus the distance between p and q is attained as the length of a minimizing geodesic joining p and q. Moreover, the geodesic curves are projections onto M of bicharacteristics of Hamiltonian H (see [14], [1]). Observe that if we normalized the “time” to be equal to 1, then d 2 ( p, q) is equal to the energy of the minimizing geodesic joining p with q and is also equal to twice the action S computed along the corresponding bicharacteristic. Perhaps the simplest example of a sub-Riemannian metric is the metric associated with the Gruˇsin operator. In the Gruˇsin plane, R2 , the sub-Riemannian metric is given by the vectors 1 0 X1 = , X2 = . 0 x The vector fields span the tangent plane everywhere, except along the line x = 0. But since 0 [X 1 , X 2 ] = , 1 Chow’s conditions are satisfied and it follows that the sub-Riemannian distance between any two points is finite (see [4, p. 24]). In the complement of the line x = 0, the sub-Riemannian metric is Riemannian, G = (R2 , ds), where ds 2 = d x 2 + dy 2 /x 2 . The Hamiltonian is given by H (x, y, η, ξ ) =

1 2 (ξ + x 2 η2 ) 2

(1.9)

and is equal to the symbol of the Gruˇsin operator. The distance between two points is Z d(P, Q) =

inf

c(t)∈C 1 ([0,1],G), c(0)=P, c(1)=Q 0

1

kc(t)k ˙ c(t) dt.

(1.10)


335

A simple computation yields the Euler-Lagrange equations y˙ = b, x2 y˙ 2 x¨ + 3 = 0, x

(1.11)

where b is a constant. All geodesics may be computed explicitly from (1.11). In particular, the geodesics starting at the origin are given by c sin(bt), b c2 t sin(2bt) y(t) = − , b 2 4b

x(t) =

(1.12)

where b and c are arbitrary real parameters. It is easy to see that these geodesics are projections of certain bicharacteristics—the solutions of the system ∂ H (x, y, η, ξ ) = ξ, ∂ξ ∂ H (x, y, η, ξ ) ξ˙ = − = −xη2 , ∂x ∂ H (x, y, η, ξ ) = x 2 η, y˙ = ∂ξ ∂ H (x, y, η, ξ ) η˙ = − = 0, ∂y x˙ =

(1.13) with the initial conditions x(0) = y(0) = 0, ξ(0) = c, η(0) = b. A similar system was studied in [6] and in [13] for the Heisenberg group. Observe that cos(bt) 1 y b t − = = µ(θ), (1.14) 2 sin2 (bt) b sin(bt) 2 x2 where µ(θ) =

θ sin2 θ

− cot(θ),

(1.15)

and θ = bt. It follows that if x 6= 0, then for every solution of the equation 2

y = µ(θ ) x2

(1.16)

there corresponds a geodesic joining the origin with the point (x, y). The graph of the function µ is portrayed in Figure 1. The Hamiltonian is constant along any bichar-

336


18 16 14 12 10 8 6 4 2

0

2

4

6

8

10

12

14

x

Figure 1. φ → φ/sin2 φ − cot φ

acteristic. For a geodesic starting at the origin, H is equal to (1/2)ξ(0)2 = (1/2)c2 . Hence the energy is equal to Z t Z t c2 θ 2S(x, y) = 2H (t) dt = c2 dt = c2 t = , (1.17) b 0 0 and a similar computation shows that the length of the geodesic is equal to cθ/b. (Note that once θ is found from (1.16), c/b is obtained from the first equation in (1.12).) If x 6= 0, then the number of geodesics joining (x, y) with the origin is finite and grows from 1 to ∞ as y/x 2 varies from 0 to ∞. If x = 0, then µ(θ) = ∞ so that θ = kπ for any integer k > 0. Correspondingly, there exist infinitely many geodesics joining the origin to (0, y) with lengths satisfying d 2 = 2πk|y|. (Here c/b is calculated from the second equation of (1.12).) Another example is the Heisenberg group Hn . The Carnot-Caratheodory metric associated with the left-invariant vector fields has been discussed in detail (see [13], [1], [4]). In particular, the Hamiltonian is the symbol of the Heisenberg Laplacian. If x = (x1 , . . ., x2n ) 6= 0, then there exist (in the isotropic case) finitely many geodesics joining the origin with (x1 , . . ., x2n , x0 ) parametrized by the solutions of the equation P2n 2 µ(θ ) = 2x0 /r 2 , where r 2 = |x|2 = i=1 xi . If x = 0, then there exist infinitely


337

many geodesics joining the origin with (0, x0 ) parametrized by the Cartesian product of S 2n−1 with the set of nonzero integers. Otherwise, the computations are similar to the case of the Gruˇsin plane. Degenerate elliptic operators of the kind studied in Section 5.2 form a generalization of the Gruˇsin operator. A subclass consists of operators of the form ∂ 2 ∂ 2 ∂2 + + |x1 |2 2 , (1.18) L=− ∂ x1 ∂ x2 ∂y where xi ∈ Vi (i = 1, 2), V1 , V2 are real Euclidean vector spaces, and ∂/∂ xi denotes the gradient in Vi (i = 1, 2). In order to cover both the Heisenberg Laplacian and the degenerate elliptic operators, let us replace y by x0 /a, where a is a positive constant. In the case of operators described by (1.18), we set r 2 = |x1 |2 . The Hamiltonian is the symbol of the operator. Once again we treat here only geodesics starting at the origin. The following propositions hold in all cases. PROPOSITION 1 There is a finite number of geodesics joining the origin with (x, x0 ) if and only if r 6= 0. They are parametrized by the solutions θ of equation (1.16). Their lengths increase with θ. With φ1 denoting the first critical value of µ, there is only one geodesic if and only if 2|x0 |/r 2 < µ(φ1 ). The number of geodesics increases to ∞ with |x0 |/r 2 . The C-C distance dc (x, x0 ) between the origin and the point (x, x0 ) is given by the length of the shortest geodesic joining these points and dc2 (x, x0 ) = 2S(x, |x0 |), where S is the action along the shortest geodesic. PROPOSITION 2 There is an infinite number of geodesics that join the origin to the point (0, x0 ) of length |x0 | dk2 = 2πk , k = 1, 2, . . . . (1.19) a The Carnot-Caratheodory distance from the origin to (0, x0 ) is given by dc (0, x0 ) = √ 2π(|x0 |/a).

These propositions are proved in [1] for the Heisenberg case; the degenerate elliptic case is very similar. 1.2. A curve in the complex plane and the complex action We continue to use the notation introduced in Section 1.1. Thus let x denote either the vector (x1 , . . ., x2n ) (the Heisenberg group case) or the vector (x1 , x2 ) ∈ V1 , V2 (the generalized Gruˇsin case), and let x0 ∈ R1 , r = |x| (Heisenberg) or r = |x1 | (Gruˇsin),

338


z ∈ C, and a ∈ R+ . Consider the function a 2 r z coth(az) − i x0 z. 2

(1.20)

az a 2 r coth(az) − − i x0 2 sinh2 (az)

(1.21)

f (x, x0 , z; a) = Then f 0 (x, x0 , z; a) = and f 00 (x, x0 , z; a) =

a 2r 2

sinh2 (az)

−1+

az , tanh(az)

(1.22)

where 0 denotes differentiation with respect to z. The function f (x, x0 , z; a), which appears in the heat kernel of the Heisenberg Laplacian (where a = 2α) and other degenerate operators (for the Gruˇsin operator a = 1), has been studied in, for example, [15] and [1], and it may be regarded as a complex action, associated with complex Hamiltonian mechanics and extending the C-C metric to the complex plane. In fact, f satisfies the following analog of the Hamilton-Jacobi equation (cf., e.g., [1]): H (x, x0 , ∇ f ) + z

∂f = f. ∂z

(1.23)

In the sequel we sometimes suppress the parameter a. Let 00,x,x0 denote the set (besides the imaginary axis) in the complex plane where f is real; that is, let 00,x,x0 = z ∈ C, Im f (x, x0 , z) = 0, Re z 6= 0 . (1.24) We recall the main properties of the curve 00,x,x0 (see [1], [15]). Here we assume, without loss of generality, that x0 ≥ 0 and that geodesics are understood with respect to a Carnot-Caratheodory metric associated with H (see Fig. 2). 3 If f (x, x0 , z) is real and ∂ f (x, x0 , z)/∂z = 0, then z is purely imaginary. If r > 0, then the number N of purely imaginary solutions of f 0 (x, x0 , z) = 0 is finite depending on λ = 2x0 /r 2 . More precisely, the set Z f 0 of purely imaginary zeros of f 0 is given by n θ o θ Z f 0 = i , θ ∈ R − πZ s.t. − cot θ = λ (1.25) a sin2 θ

PROPOSITION

(1) (2)

(see Fig. 1). Let pk denote the k th positive root of the equation tan θ = θ; the elements (iθ j ) j=1,...,N of Z f 0 are such that θ1 < π/a < θ2 ≤ p1 /a < θ3 < 2π/a < · · · < θ2K ≤ p K /a ≤ θ2K +1 < (K + 1)π /a, where K =[N/2]. (It may happen that θ2K = θ2K +1 . Then the three curves intersecting at iθ N form angles of π/3 radians with each other, one of them being the imaginary axis; see Fig. 3 and paragraph (5) below.)


339

Figure 2. The curve 0 in the complex plane for a = 2

(3)

(4)

(5)

(6)

00,x,x0 is symmetric with respect to the imaginary axis. If x0 = 0, then 00,x,x0 coincides with the real axis. If r > 0, then the curve branches off to ∞ in both directions from iθ N and the branches are asymptotic to the lines λ| Re z| = Im z. Between iθ2k−1 and iθ2k , 00,x,x0 encircles the pole ik(π/a) of f . Let 0x,x0 denote the union of 00,x,x0 and [0, iθ1 ] ∪ [iθ2 , iθ3 ] ∪ · · · ∪ [iθ2K , iθ2K +1 ]. The real function f is strictly increasing along the path 0x,x0 assuming all values between ar 2 /2 and ∞. Let iθ be a zero of f 0 (x, x0 , z). It is also a zero of f 00 (x, x0 , z) if and only if θ is equal to one of the real numbers pk , and in this case, f (3) (x, x0 , i pk ) 6= 0 and f (iθk ) = (a/2)r 2 (θk2 /(sin θk )2 ). If r > 0, then there exist N geodesic curves joining (x, x0 ) with the origin. The length of the jth geodesic is given by q d j (x, x0 ) = 2 f (x, x0 , iθ j ), 1 ≤ j ≤ N ; and d1 (x, x0 ) ≤ d2 (x, x0 ) ≤ · · · ≤ d N (x, x0 ).

Observe that the equation defining Z f 0 , (1.25), is equivalent to (1.16). The case r = 0 is degenerate. In that case, the function f (0, x0 , z) is real if and only if z is purely imaginary. 0 coincides with the ray Im(τ ) > 0, traversed twice in

340


Figure 3. 0 near a double root of f

opposite directions. The numbers θk tend to πk/a as r → 0 (λ → ∞) (cf. also Taylor [15, pp. 82–83]). √ We thus see from Proposition 3 that 2 f can be interpreted as a “distance” along 0x,x0 . In several applications (nonisotropic Heisenberg Laplacian, certain degenerate elliptic operators), we have to use a more general form of the function f . Let a1 , . . . , am be positive numbers, and let x ∈ Rm . Set f (x, x0 , z) =

m a x2 X j j j=1

2

z coth(a j z) − i x0 z.

(1.26)

Assume, without loss of generality, that a1 ≤ a2 ≤ · · · ≤ a p < a p+1 = · · · = am . Set x 00 = (x p+1 , . . . , xm ), r = |x 00 |. It is well known (see [1]) that Proposition 3 is valid, mutatis mutandis, in this case as well. 1.3. Lorentz-Carnot-Caratheodory metric In analogy to the standard case, we introduce a Lorentz-Carnot-Caratheodory Hamiltonian on M × R defined at a point (m, ξ, t, τ ) ∈ T ∗ (M × R) by Q(m, ξ, τ ) = τ 2 /2 − H (m, ξ ), where H (m, ξ ) is the Hamiltonian defined in formula (1.8). (Q is


341

independent of t.) For the model cases discussed in Sections 1.1 and 1.2, m = (x, x0 ). In these cases, consider the functions F(x, x0 , t, z, a) = i x0 z +

a t2 − r 2 z coth(az) 2 2

(1.27)

and

a t2 φ(x, x0 , t, z) = − r 2 coth(az) + i x0 + . (1.28) 2 2z We need these functions to analyze the wave kernel for all operators discussed in the sequel. In our applications, the function φ satisfies the equation ∂φ(x, x0 , t, z) ∂φ + Q ∇m φ, = 0. (1.29) ∂z ∂t As an example, note that in the particular case of the Gruˇsin plane we have Q x,y,t (ξ, η, τ ) =

τ 2 − ξ 2 − x 2 η2 , 2

(1.30)

and we can check by computation that φ is a solution of ∂φ ∂φ ∂φ ∂φ(x, y, t, z) +Q , , = 0. ∂z ∂ x ∂ y ∂t

(1.31)

In the general case, it follows from Proposition 3 that f (x, x0 , z) = t 2 /2 − F(x, x0 , t, z) = zφ(x, x0 , t, z) − t 2 /2 is a complexification of the action computed along the bicharacteristics of H , and for z = iθ1 , 2 f (x, x0 , iθ1 ) is exactly the square of Carnot-Caratheodory distance from zero to the point (x, x0 ). Hence 2F(x, x0 , t, iθ1 ) is the square of the associated Lorentz indefinite metric. Note that equation (1.29) may have other solutions, not of the type (1.28). 1.4. Separation of variables The general solution of the wave equation for the Gruˇsin operator may be found using separation of variables. Writing u(x, t) = eikt h(x)g(y), we obtain two families of solutions: √

u a,n (x, y, t) = eiat

2n+1

cos(a 2 y)Hn (ax)e−a

2n+1

sin(a 2 y)Hn (ax)e−a

2 x 2 /2

and √

va,n (x, y, t) = eiat

2 x 2 /2

,

(1.32)

where a is a real parameter, n is a nonnegative integer, and Hn is the nth Hermite polynomial, n 2 d 2 Hn (x) = (−1)n e x e−x , (1.33) n dx

342


so that z(x) = Hn (x)e−x

2 /2

is a solution of the ordinary differential equation (ODE) z 00 + (2n + 1 − x 2 )z = 0.

(1.34)

The Hermite polynomials Hn are orthogonal, and Z 2 Hn (x)Hm (x)e−x d x = δnm π 1/2 2n n!.

(1.35)

R

Recall the Mehler formula (see W. Magnus, F. Oberhettinger, and R. Soni [11, p. 252]), according to which for all real x, y and complex z (|z| < 1), +∞ X Hn (x)Hn (y)z n 1 2 2 2 =√ e y −(y−zx) /(1−z ) . n 2 2 n! 1−z n=0

(1.36)

We want to express the Dirac distribution δ((x, y); (0, 0)) using the family u a,n . Recall that in the distribution sense Z Z 1 2 +∞ −iwy δy = e dw = a cos(a 2 y) da; (1.37) 2π R π 0 the last equality in (1.37) follows from a change of variable w = a 2 . Using the base induced by the Hermite polynomials, we have in the distribution sense δ(x1 − x2 ) =

2 2 ∞ X Hn (x1 )e−x1 /2 Hn (x2 )e−x2 /2

kHn k2

0

where kHn k2 =

,

(1.38)

√ πn!2n . Replacing x1 , x2 by ax1 , ax2 , we get

δ(x1 − x2 ) =

2 2 2 2 ∞ X Hn (ax1 )e−a x1 /2 Hn (ax2 )e−a x2 /2

kHn ka2

0

,

(1.39)

√ where kHn ka2 = πn!2n /a. Hence in two-dimensional space, the Dirac distribution at (0, 0) has the form Z ∞ X 2 +∞ a2 2 2 δ (x, y); (0, 0) = cos(a 2 y)Hn (ax)e−a x /2 Hn (0) da. √ n π 0 πn!2 0 (1.40) 2 Applying (1.40) to the function φ(x, y) = f (x)g(y) ∈ D(R ), we get Z Z 1 2 +∞ −iw0 g(0) = g(w)e ˆ dw = g(a ˆ 2 )a da (1.41) 2π R π 0 and f (0) =

∞ X Hn (0)( f, h n,a ) , kHn ka 0

(1.42)


343

R 2 2 2 where h n,a (x) = Hn (ax)e−(ax) /2 and ( f, h n,a ) = R f (x)Hm (ax)e−a x /2 d x. In performing the computations connected with the separation of variables, we also use the following formula (see [11, p. 83]) for the Bessel function Jν : 2iπ Jν (αz) = z

ν

Z

0+

e(α/2)(t−z

2 /t)

t −ν−1 dt,

(1.43)

−∞

where Re(α) > 0 and | arg(t)| ≤ π , and the integral is extended over a contour starting at ∞, going clockwise around 0, going back to ∞, and never cutting the semiaxis x < 0. This contour can be deformed so that it becomes parallel to the x-axis and Z 2iπ Jν (αz) = z ν

c+i∞

e(α/2)(t−z

2 /t)

t −ν−1 dt,

(1.44)

c−i∞

where c, α > 0 and Re ν > 0. Also, it is well known that sin z . J1/2 (z) = √ π z/2

(1.45)

2. The wave kernel for the Gruˇsin operator In this chapter, we study the properties of the fundamental solution of the wave equation associated to the Gruˇsin operator L/2 = −(1/2)(∂ 2 /∂ x 2 + x 2 (∂ 2 /∂ x02 )) using separation of variables. We are interested in computing the fundamental solution that satisfies the initial condition u(x, x0 , 0) = 0 and u t (x, x0 , 0) = δ(0, 0)), where δ(0, 0)) denotes the Dirac distribution at (0, 0) for the variable (x, x0 ). The kernel can be expressed as sin(t L 1/2 ) K w (x, x0 , t) = δ(0, 0). (2.1) L 1/2 Applying formula (1.40), we obtain √ Z ∞ X 2 +∞ sin( 2n + 1at) a 2 K w (x, x0 , t) = √ √ π 0 πn!2n a 2n + 1 0 × cos(a 2 x0 )Hn (ax)e−a

2 x 2 /2

Hn (0) da.

(2.2)

We wish to sum the series (2.2) so as to obtain a more manageable form for the kernel. The situation is summed up in the following theorem. THEOREM 1 The wave kernel K w (x, x0 , t) defined by K w (x, x0 , t) = (sin(t L 1/2 )/L 1/2 )δ(0, 0) is given by K (x, x0 , t) + K (x, −x0 , t) . (2.3) K w (x, x0 , t) = 2

344


Here K (x, x0 , t) = i K 00

Z s C

1 1 du, u sinh(u) 8(x, x0 , t, u)

(2.4)

where K 00 = 1/(2(π)2 ) is a constant, 8(x, x0 , t, z) is the phase given by the expression t2 x 2 coth(z) 8(x, x0 , t, z) = i x0 + − , (2.5) 2z 2 and C (the precise description is given in the following subsections) is a closed contour of integration lying outside the set where Re 8 > 0 and avoiding the zeros of sinh(z). The phase 8 satisfies the Hamilton-Jacobi equation 2

∂φ 2 ∂φ 2 ∂φ 2 ∂φ = − − x2 . ∂z ∂t ∂x ∂ x0

(2.6)

The wave kernel satisfies the finite speed property: it vanishes identically for t 2 < dc2 (x, x0 ). Equivalently, the kernel is zero before the first geodesic of the C-C metric arrives at the point (x, x0 ). Moreover, the kernel vanishes when the time satisfies the conditions 2 f (iθ2k ) < t 2 < 2 f (iθ2k+1 ), (2.7) where the points θk are introduced in Proposition 3. Remark. The singularities of the wave kernel are computed using the zeros of the function 8(x, x0 , t, z) defined by (1.28) with a = 1 (or by (2.5)). When time inp creases to the value t = 2 f (iθ2k+1 ), this means that a new geodesic hits the point (x, x0 ) and then the kernel becomes singular. Proof All the computations in this paragraph are to be understood in the distribution sense. √ From the identity J1/2 (z) = sin z/ π z/2 and from formula (1.44), we obtain for arbitrary c0 > 0, r Z c0 +i∞ π 1 sin αz 2 = e(α/2)(u−z /u) u −3/2 du. (2.8) αz 2 2iπ c0 −i∞ √ Let us choose α = 1 and z = at 2n + 1; then c0 = a 2 t 2 c, where c > 0. Then √ r Z a 2 t 2 c+i∞ sin at 2n + 1 π 1 2 = e(1/2)(u−(2n+1)(at) /u) u −3/2 du, (2.9) √ 2 2iπ a 2 t 2 c−i∞ at 2n + 1


345

and using the change of variable u = (at)2 v for at > 0, √ r Z c+i∞ sin at 2n + 1 π 1 2 2 = e(1/2)(a t v−(2n+1)/v) v −3/2 dv, √ 2 2iπa c−i∞ a 2n + 1

(2.10)

we see that the kernel takes the form Z ∞X ∞ Z c+i∞ a 2 2 e(1/2)(a t v−(2n+1)/v) v −3/2 dv n cos(a 2 x0 ) K w (x, x0 , t) = −i K 0 n!2 0 c−i∞ 0

· Hn (ax)e

−a 2 x 2 /2

Hn (0) da

with

(2.11)

r

1 1 2 1 (2.12) = √ . 2 2 2π π π 2 To sum the series, we use the Mehler formula (1.36) (see [11, p. 252]). Set z = e−1/v so that |z| < 1 since Re(−1/v) < 0 to get K0 =

e

−1/2v

+∞ X Hn (ax)Hn (0)e−n/v e−1/2v −(ax)2 e−2/v /(1−e−2/v ) = e . √ 2n n! 1 − e2/v n=0

(2.13)

√ √ 2 2 Using e−1/2v / 1 − e2/v = 1/ 2 sinh(1/v) and cos(a 2 x0 ) = (1/2)(eia x0 +e−ia x0 ), the kernel can be expressed as K w (x, x0 , t) = (1/2)(K (x, x0 , t) + K (x, −x0 , t)), where Z +∞ Z c+i∞ a K (x, x0 , t) = −i K 0 v −3/2 √ 2 sinh(1/v) 0 c−i∞ 2 e−2/v /(1−e−2/v )−a 2 x 2 /2+ia 2 x

· e−(ax)

0 +(1/2)(a

2 t 2 v)

dv da. √ √ Changing to a new variable z = 1/v, dv/v = −dz/z, and v = 1/ z, the contour Re v = c is transformed to the circle C(1/2c, 1/2c), centered at (1/2c, 0) with radius 1/2c. The previous integral becomes √ Z Z +∞ az −1 z K (x, x0 , t) = i K 0 √ 2 sinh(z) C 0 · e−(ax)

2 e−2z /(1−e−2z )−a 2 x 2 /2+ia 2 x

0 +(1/2)(a

2 t 2 /z)

dz da,

(2.14)

and the term in the exponential can be rewritten as x 2 coth(z) (ax)2 e−2z a2 x 2 1 a2t 2 t2 2 2 − + ia x + = a − + i x + . 0 0 2 2 z 2 2z 1 − e−2z Recall that the phase is given by 8(x, x0 , t, z) = −

x 2 coth(z) t2 + i x0 + . 2 2z

(2.15)

346


An elementary computation proves that Hamilton-Jacobi equation (2.6) is satisfied. Now for Re 8 = Re(−x 2 coth(z)/2 + t 2 /(2z)) < 0, Z ∞ −1 2 aea 8 da = , (2.16) 28 0 so that the kernel is i K0 K (x, x0 , t) = 2

s

Z dz C

1 1 , 2z sinh(z) 8(x, x0 , t, z)

(2.17)

which is equivalent to i K0 K (x, x0 , t) = 2

r

Z dz C

z 1 , 2 sinh(z) F(x, x0 , t, z)

(2.18)

where we recall

t2 x2 − z coth(z). 2 2 It follows from here that if t < |x|, then the kernel is zero (unless x = x0 = 0). Indeed, Re 8 < 0 if Re z > 0. Now since 8 is a continuous function of the complex variable z, 8(x, x0 , t, 0) 6= 0, there exists a neighborhood V of zero such that 8(x, x0 , t, z) does not vanish for z ∈ V . Hence the constant c can be chosen large enough so that the circle C(1/2c, 1/2c) is small enough and contained in V . Finally, a simple application of the residue theorem implies that K (x, x0 , t) = 0. We complete the proof in the following subsections, where we also describe the contour of integration. F(x, x0 , t, z) = i x0 z +

2.1. The boundary of the forbidden set In this paragraph we describe the contours of integration which are used in the proof of Theorem 1. From the discussion leading to (2.17) and (2.18), we see that the contours have to be contained in the region Re(8) < 0. By analytic continuation, we may enlarge the region to Re(8) ≤ 0. Let us denote λ = x 2 /t 2 . Then the set of points 2x,t , where Re(8) = 0, is defined by n 2x,t = z = u + iv ∈ C,

u sinh u cosh u o = λ . u 2 + v2 sin2 v + sinh2 u

(2.19)

The curve 2x,t depends on the time and can be described as follows. It is symmetric with respect to v → −v. The v-axis is always contained in 2x,t since u = 0 is a solution. There are the following two cases.


(1) (2)

347

If λ ≥ 1, the curve has no intersection with the axis v = 0 except at the origin. In this case, there exists no bifurcation point on the v-axis. If λ < 1, then there are several branches that intersect the imaginary axis (see Fig. 4). First by continuity, there exists a curve starting at the point (u λ , 0), where tanh(u λ )/u λ = λ with a vertical tangent, connecting to a point (0, vλ ), where vλ is a solution of the following equation: sin v 2 v

= λ < 1.

(2.20)

Equation (2.20) has a finite number of solutions depending on the relative sizes of (sin pk / pk )2 and λ, where pk is the k-root of tan x = x. Note first that for t, x fixed the non-purely-imaginary part of the curve 2x,t is bounded. This follows from the fact that u cannot go to ∞ in the expression of 2x,t . Now Re φ > 0 on the real interval [0, u λ ], and for v large enough Re(8) < 0 uniformly in u ∈ [0, u λ ]. By continuity, we deduce that there exists a continuous curve that joins the point (0, vλ ) to the point (u λ , 0). By the inverse function theorem, we have a unique tangent at a neighborhood of the point (u λ , 0). Hence the curve starting at this point is unique. More precisely, consider first the case where 1 < λ < 1, 1 + p12

(2.21)

so that equation (2.20) has only one solution. Hence there exists one curve connecting (u λ , 0) to (0, vλ ), and the tangent at (0, vλ ) is parallel to the real axis. In general, at the point (0, vλ ), the curve 2x,t has a horizontal tangent except when vλ = ( pk ) for a certain k. At these points, a bifurcation appears. This results from a simple perturbation analysis: writing v0 = vλ + δ near the point (0, vλ ) in (2.19) gives for u small 2vλ 1 −

1 vλ δ= − 1 u2. tan(vλ ) λ

(2.22)

Now, except at points vλ = pk , the perturbation δ is of second order and only one horizontal branch can start from this point. At the bifurcation points pk , we obtain, after some computations,

1−

cos(2vλ ) 2 1 δ = − 1 u2. λ λ

(2.23)

Thus δ is linear in u, and the curve has two tangents that are not horizontal. 2 ) < λ < 1/(1 + p 2 ), we obtain 2n + 1 solutions For 1/(1 + pn+1 n v1 (t), . . . , v2n+1 (t), and when λ = 1/(1 + pn2 ), a double solution appears.

348

GREINER, HOLCMAN, and KANNAI u/(u 2 + v 2 ) − 1/25 sinh(u) cosh(u)/(sin(v)2 + sinh(u)2 ) = 0 15

10

5

v 0

−5

−10

−15 −25

−20

−15

−10

−5

0 u

5

10

15

20

25

Figure 4. Curve 2x,t for λ = 1/25

At the first time when λ = 1/(1 + p12 ), the curve 2x,t jumps to reach the upper point (0, v2 ), where π < v2 < 2π. Indeed, otherwise the branch of the curve starting at (0, v2 ) would return to itself and could not be connected to the rest of the curve. Thus (u λ , 0) is connected to (0, v2 ), and then an arc joins (0, v2 (t)) to (0, v1 (t)) in the region u > 0. After some time all tangents of the curve near the imaginary axis become horizontal until we reach the second bifurcation point 1/(1 + p22 ) = λ. As time goes on, the part of the curve near the imaginary axis turns around the point (0, kπ), and the two points v2 p+1 (t) < ( p + 1)π < v2 p+2 (t) converge to ( p + 1)π for p ∈ N when t converges to ∞. 2.2. The zero of the phase and the integration path We see in this subsection that the relevant pole of the function 1/8 is located at the intersection of the curves 0 and 2. It has been proved in Section 2.1 that the curve 2 has a unique branch starting on the positive u-axis and intersecting the positive v-axis. This curve cuts the curve 0


349

only once. This intersection point is exactly the pole. Recall that by Proposition 3, the function f is strictly increasing along the curve 0. Moreover, x2 t2 8(x, x0 , t, z) = − coth(z) + i x0 + =0 2 2z is equivalent to zx 2 t2 = − 2i x0 z = 2 f (x, x0 , z), (2.24) tanh(z) so that if z is a zero of 8, then Im f (z) = 0. If t 2 > x 2 , then t 2 is bigger than the minimum of 2 f along 0. This proves that there is precisely one solution z on 0 of equation (2.24). This point is at the intersection of the two curves 0 (Im f = 0) and 2 (Re 8 = 0). The converse holds as well. If t 2 /z − x 2 /tanh(z) is purely imaginary (equals iα, say), then multiplying by z, we get 28 = i(α + 2x0 )z, so that the equation Im 8 = 0 implies that if Im z > 0, then α + 2x0 = 0 and 8 = 0. If z is purely imaginary, the statement remains correct due to the strictly increasing property of f along 0. Indeed, no jump occurs as Im(z) tends to zero due to the continuity of the zero with respect to the arguments. The other purely imaginary zeros of 8 do not contribute to the integral. Hence we may deform the path of integration C and choose it starting in the region where Re 8 < 0 (taking into account the singularity), going along the imaginary axis, and avoiding the poles ikπ for k ∈ N. This completes the statement of Theorem 1 and the proof of formula (2.4). We finish the proof of the theorem in the next subsection, after obtaining an explicit formula. 2.3. Explicit formula for the integral We obtain an explicit expression for the wave kernel when we perform the integration along the contour described in the previous paragraph and apply the residue theorem. 4 The wave kernel K w for the Gruˇsin operator is given at a point (x, x0 ), where x0 6= 0, PROPOSITION

350


by the expression K w (x, x0 , t) s 1 z(x, x0 , t) 1 = 2 2 2π 2 sinh(z(x, x0 , t)) −t /(z(x, x0 , t)) + x z(x, x0 , t)/sinh2 (z(x, x0 , t)) s z¯ (x, x0 , t) 1 , + 2 sinh(¯z (x, x0 , t)) −t 2 /(¯z (x, x0 , t)) + x 2 z¯ (x, x0 , t)/sinh2 (¯z (x, x0 , t)) (2.25) where z(x, x0 , t) is the unique solution on 0 of F(x, x0 , t, z) F 0 (x, x0 , t, z) 6= 0; that is, t2 = x2

z(x, x0 , t) − 2i x0 z(x, x0 , t), tanh z(x, x0 , t)

=

0, and

(2.26)

and the denominator in formula (2.25) does not vanish. The kernel is analytic except at points where F 0 (x, x0 , t, z(x, x0 , t)) = 0. Put differently, s ∂z(x, x0 , t) 1 z(x, x0 , t) K w (x, x0 , t) = − 2 sinh(z(x, x0 , t)) ∂t 4πt s z¯ (x, x0 , t) ∂ z¯ (x, x0 , t) + , 2 sinh(¯z (x, x0 , t)) ∂t where an elementary computation yields that 2t ∂z(x, x0 , t) = . 2 2 ∂t t /(z(x, x0 , t)) − x z(x, x0 , t)/ sinh2 (z(x, x0 , t))

(2.27)

When x = 0 and x0 6= 0, the wave kernel is given for t > 0 by K w (0, x0 , t) =

(−1) j t p χ , 2 2π|x0 | 4x0 sin(−t 2 /(2|x0 |)) {sin(t /(2|x0 |)) 0, s 1 t . (2.39) K w (0, x0 , t) = 2x0 π −4x0 sin(t 2 /(2x0 )) If x0 < 0, then t K w (0, x0 , t) = − 2x0 π

s

1 . −4x0 sin(t 2 /(2x0 ))

(2.40)

In formulas (2.39) and (2.40), the sign of the square root alternates. It follows from this expression that the singularities are located at the points t 2 = 2k|x0 |π. Remark. We could have derived the formula (2.28) of Proposition 4 from formula (2.4) of Theorem 1. For in this case the integration contour is contained in the imaginary axis where the pole z 0 = it 2 /(2x0 ) is located. More precisely, the contour 0 starts at the origin, continues along the imaginary axis encircling the pole z 0 , and returns to the origin, oriented clockwise. When x tends to zero, the curve 0x,x0 converges pointwise to the imaginary axis and F(x, x0 , z, t) converges to F(0, x0 , t, z),


353

except at the points iπk for k ∈ Z. Since K w (x, x0 , t) = (1/2)(K (x, x0 , t) + K (x, −x0 , t)), we need to evaluate the integral Z r i u 1 K (0, x0 , t) = 2 du, (2.41) π 0 sinh(u) 2ui x0 + t 2 where 0 is the part of the contour contained in the imaginary axis. The points iπk do not really contribute a singularity to the integral. In fact, the computation shows that the integral along the integration contour vanishes and only the singularity at the point z 0 is relevant. We may now prove the statements of Theorem 1 about the finite speed of propagation, using the expression of the wave kernel obtained above. Note that it is possible to deform the path of integration below the u-axis. Using this remark, it is possible to prove that the kernel is vanishing before it reaches dc2 (x, x0 ). Indeed, for x 2 < t 2 < dc2 (x, x0 ), the zero of the function 8(x, x0 , t, z) is exactly situated on the imaginary axis between 0 and iθ1 with θ1 < π. Deforming the contour of integration in order to include the pole of 1/8, we obtain a certain residue. The pole where the residue is computed is of the form iα, where α > 0. Using exactly the same argument, we see that the residue for the kernel K (x, −x0 , t) is computed at the pole −iα. Since the kernel K w is an odd function of z, the two residues cancel. This result is valid each time that f (x, x0 , t, z) = 0 has a solution iα in the imaginary axis such that α ∈ [2kπ, (2k +1)π ], k ∈ N. Applying the description of the curve 0x,x0 , given in Proposition 3, this situation appears exactly when 2 f (iθ2k ) < t 2 < 2 f (iθ2k+1 ), where θk are defined in the same proposition and, as above, the terms K (x, x0 , t) and K (x, −x0 , t) cancel. 2.4. Analysis of the wave kernel near the singularities We describe here the wave kernel near the singularities. We have found in the last paragraph an explicit expression for the wave kernel. We are going to focus on the set Z (x, x0 , t) = z ∈ C s.t. F(x, x0 , t, z) = 0 and F 0 (x, x0 , t, z) = 0 .

(2.42)

The characteristic set may be parametrized by a real parameter θ. After some elementary computations, we find that n o sin2 θ 2x0 θ − cos θ sin θ x2 S = (x, x0 , t) ∈ R3 s.t. 2 = and = , θ ∈ R − π Z , t θ2 x2 sin2 θ S = S− ∪ S+ , (2.43)

354


where sin θ n S± = (x, x0 , t) ∈ R3 s.t. x = ±t θ

o 2x0 θ − cos θ sin θ , θ ∈ R − πZ . = x2 sin2 θ The main result of this subsection is the following. and

PROPOSITION 5 The singular set of the wave kernel is S. This set is the disjoint union of the sets S+ and S− . Moreover, near the singular set, the main singularity of the kernel is given by the following expression: s θk 1 1 K w (x, x0 , t) ∼ 2 2 2 sin(θk ) (x /sin θk )(−1 + θk /tan θk ) 2π s s H (x, −x0 , t, −iθk ) H (x, x0 , t, iθk ) , + · t 2 − dk2 (x, −x0 ) t 2 − dk2 (x, x0 )

where dk is the length of the k th geodesic joining (x, x0 ) with the origin, k = 1, . . . , N , H is an analytic function, nonzero on S and θk is a solution of the equation F 0 (x, x0 , t, iθ) = 0 with F real, and θk is not a root of tan θ = θ. When θk is one of the points pk , K w (x, x0 , t) grows like 1/(t 2 − dk2 (x, x0 ))3/2 . Proof The singular sets Z and S are essentially the same as the singular sets analyzed and shown graphically in [13]. Here we study the behavior of K w near the set S and determine the singularity there. In the set Z (x, x0 , t), the points z = i pk = i tan pk are isolated and F 00 (x, x0 , t, i pk ) = 0, but the third derivative is not zero. At points of S corresponding to the latter, the singularity of the wave kernel is of higher order. For z in a neighborhood of Z (x, x0 , t) − {i pk , k ∈ N}, F(x, x0 , t, z) has the following expansion: F(x, x0 , t, z) = F(x, x0 , t, iθk ) + F 0 (x, x0 , t, iθk )(z − iθk ) + (z − iθk )2 H (x, x0 , t, z) = (z − iθk )2 H (x, x0 , t, z),

(2.44)

where H (x, x0 , t, z) does not vanish in a neighborhood of iθk , and by Taylor expansion we have x 2 z(x, x0 , t) θk t2 x2 − + −1+ (z − iθk ) + o(z − iθk ) = 2 z(x, x0 , t) sinh (z(x, x0 , t)) sin θk tan θk


355

and K w (x, x0 , t) s 1 θk 1 1 1 ∼ + . 2π 2 sin(θk ) (x 2 /sin2 θk )(−1 + θk /tan θk ) (z − iθk ) (¯z + iθk ) (2.45) In order to express (2.45) in terms of the distance to the set S, note that F(x, x0 , t, z) =

t2 − f (x, x0 , z) = (z − iθk )2 H (x, x0 , t, z), 2

(2.46)

and using the length of the k th geodesic, f (x, x0 , z) ∼ dk2 (x, x0 )/2, we see that s θk 1 1 K w (x, x0 , t) ∼ 2π 2 sin(θk ) (x 2 /sin2 θk )(−1 + θk /tan θk ) s s H (x, x0 , t, z) H (x, −x0 , t, z¯ ) · + , t 2 − dk2 (x, x0 ) t 2 − dk2 (x, −x0 ) where H does not vanish. When θk is one of the points pk , the same type of analysis shows that K grows like 1/(t 2 − dk2 (x, x0 ))3/2 . 2.5. The operator satisfies the wave equation Using integration by parts, we prove that the wave kernel K w satisfies the wave equation. Starting with the fact that K w (x, x0 , t) = (K (x, x0 , t) + K (x, −x0 , t))/2, we only need to show that K satisfies the wave equation. Recall that Z 1 w(z) dz, (2.47) K (x, x0 , t) = K 0 C 8(x, x 0 , z, t) √ where w(z) = z/(2 sinh(z)) and 8(x, x0 , z, t) = i x0 z + t 2 /2 − x 2 z coth(z)/2, K 0 is constant, and C is a closed contour that may enclose singularities but not pass through them. We have already proved that K w is zero for t small enough. Note that w satisfies 1 w(z) 1 w0 (z) = − , (2.48) 2 z tanh(z) and recall the formula for the derivatives of 8: 80 (z) = i x0 −

z x2 1 − . 2 tanh z sinh2 (z)

(2.49)

356


By direct computations, Z ∂ 2 K (x, x0 , t) 1 2t 2 w(z) dz, (2.50) = K − + 0 ∂t 2 82 (x, x0 , z, t) 83 (x, x0 , z, t) C Z ∂ 2 K (x, x0 , t) z = K 0 2 2 ∂x C tanh(z)8 (x, x 0 , z, t) 2x 2 z 2 + w(z) dz, (2.51) tanh2 (z)83 (x, x0 , z, t) Z ∂ 2 K (x, x0 , t) z2 x 2 x2 = −2K w(z) dz, (2.52) 0 3 ∂ y2 C 8 (x, x 0 , z, t) and ∂ 2 K (x, x0 , t) ∂ 2 K (x, x0 , t) + x2 2 ∂x ∂ y2 Z z = K0 2 C tanh(z)8 (x, x 0 , z, t) 2x 2 z 2 + w(z) dz. sinh2 (z)83 (x, x0 , z, t) Consider the first term of the right-hand side in the last expression. Using the properties of the function w and integrating by parts, we obtain Z Z z w(z) − 2zw0 (z) w(z) dz = dz 2 2 C tanh(z)8 (x, x 0 , z, t) C 8 (x, x 0 , z, t) Z h i w(z) wz = − 2 dz 2 82 (x, x0 , z, t) C 8 (x, x 0 , z, t) Z 0 z +2 w(z) dz 2 C 8 (x, x 0 , z, t) Z Z w(z) w(z) = dz + 2 dz 2 (x, x , z, t) 2 (x, x , z, t) 8 8 0 0 C C Z zw(z)80 (x, x0 , z, t) −4 dz 83 (x, x0 , z, t) Z C w(z) =3 dz 2 (x, x , z, t) 8 0 C Z x2 z z2 w(z) −4 i x0 z − − dz. (2.53) 2 3 2 tanh(z) sinh (z) 8 (x, x0 , z, t) C


357

Finally, 2 ∂ 2 K (x, x0 , t) 2 ∂ K (x, x 0 , t) + x ∂x2 ∂ y2 Z Z w(z) i x0 zw(z) =3 dz − 4 dz 2 (x, x , z, t) 3 (x, x , z, t) 8 8 0 0 C C Z x 2 w(z)z +2 dz tanh(z)83 (x, x0 , z, t) Z Z C w(z) 8(x, x0 , z, t) − i x0 z dz + 4 w(z) dz =− 2 83 (x, x0 , z, t) C 8 (x, x 0 , z, t) C Z x 2 w(z)z +2 dz 3 C tanh(z)8 (x, x 0 , z, t) Z Z w(z) t2 dz + 2 w(z) dz =− 2 3 C 8 (x, x 0 , z, t) C 8 (x, x 0 , z, t) ∂ 2 K (x, x0 , t) = . (2.54) ∂t 2

Observe that w plays the role of the transport term in wave theory. 2.6. Source not at the origin In this subsection we make several remarks concerning the case where the Dirac distribution is given at a point (y, y0 ) 6 = (0, 0) and the observer is fixed at the point (x, x0 ). An analog of Theorem 1 is valid in this case as well. The following is true. 6 The wave kernel K w (x, x0 , t) defined by K (x, x0 , t) = (sin(t L 1/2 )/L 1/2 )δ(y, y0 ), where −L/2 is the Gruˇsin operator, is given by PROPOSITION

K w (y, y0 , x, x0 , t, z) =

K (y, y0 , x, x0 , t, z) + K (y, y0 , x, x0 , t, z) , 2

where K (x, x0 , t) = K 00

Z s 0

1 1 du, u sinh(u) 8(x, x0 , t, u)

(2.55)

(2.56)

and K 00 = 1/(2π)2 , 0 is an appropriate contour, and 8(y, y0 , x, x0 , t, z) is the phase given by the expression 8(y, y0 , x, x0 , t, z) = −

x 2 + y2 xy t2 + + i(y0 − x0 ) + . 2 tanh z sinh z 2z

(2.57)

Set f (y, y0 , x, x0 , t, z) = z8(y, y0 , x, x0 , t, z). The singularities of the integrand are located at the zeros of F(y, y0 , x, x0 , t, z) = 0. More precisely, the singularity is lo-

358


cated at the intersection of Re 8(y, y0 , x, x0 , t, z) = 0 and Im f (y, y0 , x, x0 , t, z) = 0. Proof The proof is similar to the proof of Theorem 1, and using the same arguments, we obtain the following expression of the phase: x 2 + y2 xy t2 + + i(y0 − x0 ) + . (2.58) 2 tanh z sinh z 2z The contour starts in the region where Re 8(y, y0 , x, x0 , t, z) < 0, but it turns out to be more complex. 8(y, y0 , x, x0 , t, z) = −

The analog of Proposition 4 is more complex and will be discussed in a future work. 3. Wave kernels in one dimension In this section we study the wave kernel for two operators. The first operator is the harmonic oscillator ∂x x − x 2 , where we are able to give an integral representation formula, and the second is the Klein Gordon operator ∂x x − a 2 (a is a constant), where we recover well-known results. 3.1. Wave kernel for the harmonic oscillator The purpose of this subsection is to discuss the wave kernel for the equation and initial conditions ∂tt u = ∂x x u − x 2 u, ∂t u(x, 0) = δx , u(x, 0) = 0. We obtain the following result. 7 The wave kernel for the operator ∂x x − x 2 can be expressed in the form Z s i 1 2 2 K (x, t) = et /2z−x coth(z)/2 dz, 4π C 2z sinh(z) PROPOSITION

(3.1)

where C is a contour symmetric with respect to the x-axis going through the origin, obtained by a smooth deformation of the circle C(1/2c, 1/2c). The singularity at z = 0 is essential. For t < |x| the kernel vanishes. The phase φ(x, t, z) = t 2 /(2z) − x 2 coth(z)/2 satisfies the Hamilton-Jacobi equation ∂φ 2 ∂φ 2 ∂φ =− + − x 2. (3.2) 2 ∂z ∂t ∂x


359

Proof Using expression (1.38) for the Dirac operator and separation of variables, we can express the kernel in the form √ 1 sin( 2n + 1t) 2 Hn (0)Hn (x)e−x /2 . K (x, t) = √ √ n n!2 π 2n + 1 n=0 +∞ X

We now use integral formula (2.10) to express the time dependence: √ r Z c+i∞ π 1 sin t 2n + 1 2 e(1/2)(t v−(2n+1)/v) v −3/2 dv. = √ 2 2iπ 2n + 1 c−i∞

(3.3)

(3.4)

Then the kernel assumes the form K w (x, x0 , t) ∞

1 X = √ 2iπ 2 0

Z

c+i∞

e(1/2)(t

2 v−(2n+1)/v)

c−i∞

v −3/2 dv

1 2 Hn (x)e−x /2 Hn (0). n!2n

By Mehler formula (1.36) for Re v > 0, we have e−1/2v

+∞ X e−1/2v −x 2 e−2/v /(1−e−2/v ) Hn (x)Hn (0)e−n/v =√ e . n 2/v 2 n! 1 − e n=0

(3.5)

We can simplify as in (2.14), and we find that in the new variable z = 1/v, √ √ dv/v = −dz/z, and v = 1/ z, the contour x = c is transformed to a circle C = C(1/2c, 1/2c), centered at (1/2c, 0) of radius 1/2c, obtaining finally Z s i 1 2 2 K (x, t) = et /(2z)−x coth(z)/2 dz. (3.6) 4π C 2z sinh(z) The details of the computation are given in Section 2 for the Gruˇsin operator. We can deform C to a closed contour symmetric with respect to the u-axis (u = Re(z)) as follows. The contour starts outside the region defined by Re(t 2 /(2z) − x 2 coth(z)/2) > 0 and Re z ≥ 0 (see Fig. 4 for the curve 2x,t ). At the point where 2x,t meets the imaginary axis for the first time, we continue along this axis until the next point vλ (where λ = x 2 /t 2 ), defined in Section 2.1. Then the contour is continued by being allowed to come back to Re z > 0, avoiding the singularity at a multiple of iπ. This construction is continued each time a singularity has to be avoided. The contour reaches the origin along the imaginary axis. Then the path is symmetrized with respect to the u-axis. Thus the contour avoids the singularities sinh(z) = 0 for z 6 = 0 and stays in the region u ≥ 0.

360


To prove that the operator satisfies finite speed propagation and vanishes when t < x, we may use the residue theorem, and since the singularity at z = 0 is removable, K (x, t) = 0 when t < x. (3.7) We may verify by an elementary computation that the phase φ satisfies HamiltonJacobi equation (3.2). Remark. When t > x, the point zero is an essential singularity and yields the main contribution. Indeed, the integral on the imaginary axis can be expressed as K a (x, t) Z as t2 t2 i 1 x 2 coth(y) x 2 coth(y) = cos − − i sin − dy, 4π −a 2y sin(y) 2y 2 2y 2 which reduces by symmetry of the first term (understood as principal value) to Z as t2 1 x 2 coth(y) 1 K a (x, t) = sin − dy, (3.8) 4π −a 2y sin(y) 2y 2 so that 1 K a (x, t) = 4π

Z

a −a

s

t2 − x2 1 sin dy + O(1). 2y sin(y) 2y

(3.9)

K a depends (to a first approximation) only on t 2 − x 2 , and we have for a close to zero, Z a t2 − x2 1 1 sin dy + O(1). (3.10) K a (x, t) ∼ √ 2y 2 2π 0 y The integral in (3.10) is convergent, as can be seen by using the following change of variable z = 1/y. 3.2. Wave kernel for the Klein-Gordon operator In this paragraph we show how it is possible to recover the well-known result concerning the wave kernel for the translation-invariant Klein-Gordon operator. The result is given in term of the Bessel function J0 . We prove the following. PROPOSITION 8 The wave kernel for the equation

∂2 w = ∂x x w − a 2 w, ∂t 2 w(x, 0) = 0, wt (x, 0) = δ0 (x),

(3.11)


361

is given by p 1 J0 (a t 2 − x 2 )H (t 2 − x 2 ), 2 where H is the Heaviside function and J0 is the Bessel function. K (x, t) =

(3.12)

Proof √ The general solution of (3.11) is given by the family of functions eikx sin( k 2 + a 2 t) √ and eikx cos( k 2 + a 2 t). Integrating the family (recall (1.9)), we obtain √ Z sin( k 2 + a 2 t) 1 cos(kx) √ K (x, t) = dk. (3.13) 2π R k2 + a2 Using the relation √ √ Z π/2 0+ (1/2)(u−(k 2 +a 2 )t 2 /u) −3/2 sin( k 2 + a 2 t) e u du = √ 2iπ −∞ k2 + a2 (see (1.43)), we get √ Z Z 0+ 1 π/2 2 2 2 K (x, t) = e(1/2)(u−(k +a )t /u) u −3/2 du cos(kx) dk. 2π 2iπ R −∞ But Z e R

−k 2 t 2 /2u

√ 2πu −x 2 u/(2t 2 ) cos(kx) dk = e t

for u ∈ C − R− (usual cut). Hence Z 0+ 1 1 2 2 2 2 2 2 K (x, t) = e((1−x /t )/2)(u−t a /(1−x /t )) u −1 du, 2 2iπ −∞

(3.14)

(3.15)

(3.16)

(3.17)

and applying (1.43) again, we see that for t 2 − x 2 > 0, K (x, t) =

p 1 J0 (a t 2 − x 2 ). 2

(3.18)

It is well known that K is zero for t 2 < x 2 , and if H denotes the Heaviside function, we obtain (3.12). 4. The Heisenberg wave kernel via the heat kernel In this section we construct a representation formula for the Heisenberg wave kernel √ (the kernel of cos( −21 H t)) by inverting the so-called transmutation formulas. We also have to deform an integration path in the complex plane. The results are expressed in Theorems 2 and 3 of this section. In the corollaries, analyticity results are given, and the behavior of the kernel near its singular support is described.

362


PROPOSITION 9 Let L be a nonnegative self-adjoint operator, u > 0. Then Z ∞ Z ∞ √ √ 1 1 2 −Lu −t 2 /(4u) cos( L t) dt = √ e e−t /(4u) cos( Lt) dt. e =√ πu 0 4πu −∞ (4.1)

Proposition 9 is a well-known transmutation formula. A proof may be found in [8]. PROPOSITION 10 For every a 6= 0, u > 0, and nonnegative integer n,

4n+1/2 e−a /(2u) = √ u n+1 u 2

Z 0

∞

e−t

2 /(4u)

∂ (n−1/2) 2 δ (t − 2a 2 ) dt. ∂t

(4.2)

Proof We have Z ∞ Z ∞ ∂ 2 2 e−t /(4u) δ (n−1/2) (t 2 −2a 2 ) dt = 1/(4u) e−t /(4u) δ (n−1/2) (t 2 −2a 2 )2t dt. ∂t 0 0 Setting t 2 − 2a 2 = y, we see (recalling the formula for fractional differentiation, cf. [5, pp. 739 – 740]) that the right-hand side is, by definition, equal to 1 4u

e−a /(2u) (y) dy = 4u 2

∞

Z

e

−(2a 2 +y)/(4u) (n−1/2)

δ

0

∞

Z 0

e−y/(4u) ∂ n 1 √ dy 0(1/2) ∂ y n y

e−a /(2u) R∞ = √ 0 1/2 (4u)n+1 0 e−y/(4u) / y dy √ 2 π e−a /u −a 2 /(2u) =√ e , = π (4u)n+1/2 4u n+1/2 2

and the proposition follows. Note that one may derive (4.2) from (4.1) and the well-known expressions for √ cos( −1 t). We set Z ∞ 1 e− f (x,x0 ;τ )/u V (τ ) dτ, (4.3) P(x, x0 ; u) = (2πu)n+1 −∞ where f (x, x0 ; τ ) = ατ coth(2ατ )|x|2 − i x0 τ = f (x, x0 , τ ; α)

(4.4)


363

and

2ατ n . (4.5) sinh(2ατ ) Here α is a positive constant and f (x, x0 , τ ; α) is the function introduced in Section 1.2. V (τ ) =

PROPOSITION

11

Let n

1H =

1h X ∂ ∂ 2 + 2αx2 j 2 ∂ x2 j−1 ∂ x0 j=1

n

+

1 X ∂ ∂ 2 i − 2αx2 j−1 . 2 ∂ x2 j ∂ x0

(4.6)

j=1

Then P(x, x0 ; t) = et1 H (x, x0 ; 0, 0).

(4.7)

A proof may be found in [1]. Observe that the kernel P(x, x0 ; t) is the fundamental solution of the heat equation ∂P = 1H P (4.8) ∂t with the singularity at the origin. 4.1. Deforming the path of integration We deform the path of integration used in formula (4.3) from the real axis to the curve 0 discussed in Section 1.2. This path has been introduced for studying singularities of the Heisenberg wave kernel by Taylor [15, pp. 80 – 86]. We consider the case where x0 ≥ 0; the other case is similar. PROPOSITION 12 If x 6= 0, x0 6= 0, then

P(x, x0 ; u) =

1 (2πu)n+1

Z 0x,x0

e− f (x,x0 ,τ )/u V (τ ) dτ.

(4.9)

Proof Let N = N (x0 /|x|2 ) denote the number of purely imaginary zeros of ∂ f /∂τ , and for R → ∞ consider the closed contour 0 R (see Fig. 5) formed by the interval {−R ≤ τ1 ≤ R, τ2 = 0}, the vertical segments τ1 = ±R joining (−R, 0) and (R, 0) with the unbounded branches of 0x,x0 , and the portion of 0x,x0 between those intersection points.

364


Figure 5. The integration path 0 R

By Cauchy’s integral theorem, Z e− f (x,x0 ;τ )/u V (τ ) dτ = 0. 0R

Recall that V (R + iτ2 ) ∼ 2(R + iτ2 )n as R → ∞. Moreover, f (x, x0 ; R + iτ2 ) = (R + iτ2 ) coth(R + iτ2 ) − i Rx0 + x0 τ2 , so that for every ε > 0, h (|R| + τ )(1 − ε) + x τ i 2 0 2 |e− f (x,x0 ;R+iτ )/u | ≤ exp − u if R is large enough. The length of the vertical interval is proportional (asymptotically) to R (see Prop. 3). Hence the integral on the vertical line τ1 = R, and similarly on τ1 = −R, tends to zero as R → ∞. The proposition follows from (4.3). Remark. Recall that 00,x,x0 denotes the closure of the set of non-purely-imaginary points of 0x,x0 . The analyticity of V (τ ) and of f (x, x0 ; τ ) on (iθ2k , iθ2k+1 ) implies that the two integrals over (iθ2k , iθ2k+1 ), performed in opposite directions, cancel out, and we may rewrite (4.9) as Z 1 P(x, x0 ; u) = e− f (x,x0 ,τ )/u V (τ ) dτ. (4.10) (2πu)n+1 00,x,x0


365

4.2. Computation of the wave kernel We analyze the wave kernel for the isotropic Heisenberg Laplacian, that is, the solution w(x, x0 ; t) of the partial differential equation n

∂ 2w h X ∂ ∂ 2 + 2αx = 2 j ∂ x2 j−1 ∂ x0 ∂t 2 j=1

+

n X ∂ ∂ 2 i − 2αx2 j−1 w(x, x0 , t) ∂ x2 j ∂ x0

(4.11)

j=1

with the initial conditions w(x, x0 ; 0) = δ(x, x0 ; 0, 0) and (∂w/∂t)w(x, x0 ; 0) = 0. √ This is the kernel of cos( −21 H t), and we calculate it by combining Propositions 9, 10, and 12 and by setting L = −21 H . PROPOSITION 13 If x 6= 0, x0 6= 0, then

P(x, x0 ; 2u) =

1

Z

√ 2π n+1 u

0x,x0

V (τ )

∞

Z

e−t

2 /(4u)

0

∂ × δ (n−1/2) t 2 − 2 f (x, x0 ; τ ) dt dτ. ∂t

(4.12)

Proof Recall that f (x, x0 ; τ ) is positive on 0x,x0 . By (4.2) (and by setting f = a 2 ), 1 e− f (x,x0 ,τ )/(2u) (4πu)n+1 Z ∞ 1 ∂ 1 2 = e−t /(4u) δ (n−1/2) t 2 − 2 f (x, x0 ; τ ) dt. √ n+1 ∂t 2π u 0 Substituting in (4.9), we get (4.12). Note also that P(x, x0 ; 2u) =

1

√ 2π n+1 u

Z 00,x,x0

V (τ )

∞

Z

e−t

2 /(4u)

0

∂ × δ (n−1/2) t 2 − 2 f (x, x0 ; τ ) dt dτ. ∂t

(4.13)

366


THEOREM 2 If x 6= 0, x0 6= 0, then

p w(x, x0 ; t) = cos( −21 H t)(x, x0 ) Z 1 ∂ (n−1/2) 2 δ t − 2 f (x, x0 ; τ ) V (τ ) dτ. = n+1/2 π 0x,x0 ∂t

(4.14)

Proof Set L = −21 H , and apply Proposition 9 to (4.12), getting the relation 1 √ 4πu

Z

∞

2 /(4u)

w(x, x0 ; t) dt Z Z ∞ 1 ∂ 2 = V (τ ) e−t /(4u) δ (n−1/2) t 2 − 2 f (x, x0 ; τ ) ds dτ. √ n+1 ∂t 2π u 0x,x0 0 (4.15) e−t

−∞

Recall that on the unbounded branch of 0x,x0 , coth(2τ ) → 1 as τ → ∞, so that f (x, x0 ; τ ) = Re f (x, x0 ; τ ) ∼ |x|2 τ1 + x0 τ2 ≥ δ|τ | for a certain δ = δ(|x|, x0 ) > 0. Proceeding as in the proof of Proposition 10 and setting t 2 − 2 f (x, x0 ; τ ) = y, we see that the inner integral on the right-hand side of R∞ √ √ (4.15) is equal to e− f (x,x0 ;τ )/(2u) /( π u n+1 ) 0 e−y/(4u) / y dy, so that for fixed u, the right-hand side of (4.15) is proportional to Z

V (τ )e

0x,x0

− f (x,x0 ;τ )/(2u)

Z

∞

0

e−y/(4u) dy dτ √ y Z Z ≤C V (τ )e−δ|τ | dτ 0x,x0

∞ 0

e−y/(4u) dy dc. √ y

Hence we may interchange the order of integration in (4.15) to obtain the equation Z

∞

e

−t 2 /(4u)

−∞

=

π

w(x, x0 ; t) dt = 2 Z ∞ hZ 1 −t 2 /(4u) e n+1/2 0

∞

Z

e−t

2 /(4u)

w(x, x0 ; t) dt

0

0x,x0

V (τ )

i ∂ (n−1/2) 2 δ (t − 2 f (x, x0 ; τ )) dτ ds. ∂t (4.16)

Relation (4.16) holds for all u > 0; hence for all 1/u > 0 and by uniqueness for the Laplace transform, we get (4.14). Note. Formula (4.14) remains valid if the integration is extended over 00,x,x0 .


367

COROLLARY 1 If x 6= 0, x0 6= 0 and t 6= d j for all i ≤ j ≤ N , then w(x, x0 ; t) is a real analytic function of all its arguments.

p Note. Recall the formula d j (x, x0 ) = 2 f (x, x0 ; iθ j ) for i ≤ j ≤ N . Note also that w is a solution of the Heisenberg wave equation. Proof Let t0 be different from d j for all 1 ≤ j ≤ N . Note that w(x, x0 ; t) ≡ 0 for t < d1 (finite speed; see [13] or [15]). We may assume that t0 ∈ (d2k+1 , d2k+2 ) for a certain k (or d N < t0 ). The only possible singularities of w may arise from the contributions of the parts of 0x,x0 between iθ2k+1 and iθ2k+2 (or iθ N and ∞). There exists a positive ε such that the interval (t0 − ε, t0 + ε) contains no d j . Recall that ∂ f /∂τ 6= 0 between iθ2k−1 and iθ2k+2 (or between iθ N and ∞). The reality of f on 0x,x0 along with the Cauchy-Riemann equations imply that there exists an analytic function h( f ) (the 2 2 inverse function of f ) such that h(d2k+1 /2) = iθ2k+1 , h(d2k+2 /2) = iθ2k+2 . Let ∞ ψ = C0 (t0 − ε, t0 + ε) be such that ψ(t) ≡ 1 for t in a neighborhood of t0 . To prove Corollary 1, it suffices to prove the regularity of the distribution v(x, x0 ; t) given by Z p v(x, x0 ; t) = δ (n−1/2) t 2 − 2 f (x, x0 ; τ ) V (τ )ψ 2 f (x, x0 ; τ ) dτ. 0x,x0

Introduce a new real variable σ = 2 f (x, x0 , τ ). Then the function h(σ/2) is well √ √ defined in ( t0 − ε, t0 + ε ) ⊃ supp ψ. Hence Z h 0 (σ/2) σ (n−1/2) 2 ψ(σ ) dσ. v(x, x0 ; t) = δ (t − σ )V h 2 2 Set V (h(σ/2))ψ(σ )(h 0 /2)(σ/2) = g(σ ). Thus g(σ ) is real analytic near t02 , and so is Z v(x, x0 ; t) = g(t 2 − σ )δ (n−1/2) (σ ) dσ near t0 . It is possible to use the same change of variables on all of 00,x,x0 in order to obtain a representation of w(x, x0 ; t). Recall that f (x, x0 ; τ ) maps the branch of 0x,x0 which joins iθ2k−1 and iθ2k and for which τ > 0 in a differentially invertible manner 2 onto (d2k−1 /2, dk2 /2). Let h k (σ ) denote the inverse function of f (x, x0 ; τ ) defined on 2 (d2k−1 /2, dk2 /2). Set wk (σ ) = V (h k (σ )) + V (−h k (σ )). Similarly, let h N (σ ) denote the inverse of f (x, x0 ; τ ), defined on (d N2 /2, ∞) and parametrizing the right branch of 0 N joining iθ N and ∞. The note after the proof of Theorem 2 yields the following.

368


THEOREM 3 If x 6= 0, x0 6= 0, then

p w(x, x0 ; t) = cos( −21 H t)(x, x0 ) [N /2] Z 2 h 0 (σ/2) 1 n X dk ∂ (n−1/2) 2 = n+1 δ (t − σ )Wk (σ ) k dσ 2 ∂t 2 π k=1 d2k−1 Z ∞ h0 σ o ∂ (n−1/2) 2 + δ (t − σ )Wk (σ ) k dσ . (4.17) 2 ∂t 2 2 dN One may read off (4.17) the nature of the singularities of w(x, x0 ; t) when t is near d j . An alternative determination of the singularities may be found in Nachman [13, p. 713]. We restrict ourselves to the leading singularity. 2 Let x 6= 0, x0 6= 0, and let all the θ j be distinct. Then there exist C j (x, x0 ) such that COROLLARY

w(x, x0 ; t) ∼ C j (x, x0 )δ (n) (t − d j ) if j is odd, w(x, x0 , t) ∼ C j (x, x0 )(t − d j )−n−1

if j is even,

(4.18)

for t near d j (x, x0 ). Proof Consider first the case where j is odd, j = 2k − 1. By Theorem 3, the only contribution to the singularities of w(x, x0 ; t) for t near d2k−1 or d N arises from in2 tegration near the local minimum of f at iθ2k−1 (or σ = d2k−1 ). By assumption, 0 2 2 2 (∂ f /∂τ )(x, x0 ; iθ2k−1 ) 6 = 0, so that h k (σ/2) ∼ C j /(σ − d2k−1 )1/2 . (We denote different constants by C j .) Hence the main singularity of w(x, x0 ; t) is given by Z dσ ∂ (n−1/2) 2 Cj δ (t − σ ) 2 2 ∂t (σ − d2k−1 )1/2 d2k−1 Z 2 − σ) ∂ δ (n−1/2) (t 2 − d2k−1 = Cj dσ √ ∂t σ Z0 ∂ 1 ∂ (n) (−1/2) 2 dσ 2 = Cj δ (t − d2k−1 − σ)√ ∂t 2 ∂t σ Z0 ∂ 1 ∂ (n) 1 dσ q = Cj √ , 2 0 ∂t 2 ∂t t 2 − d2k−1 −σ σ where the integrations and the differentiations are to be understood in the distribution sense, and only the left endpoint of the σ -interval is indicated. Applying this distribution to a test function ϕ(t), we see that the leading part of w(x, x0 ; t) applied to ϕ(t)


369

is ∂ 1 ∂ (n) 1 q 0 ∂t t ∂t t2 − d2

Z Z Cj

dσ √ ϕ(t) dt σ 2k−1 − σ

2 t 2 −d2k−1

1 dσ h 1 ∂ n ∂ i q ϕ (t) dt √ 2 0 t 2 − d2k−1 − σ σ t ∂t ∂t Z ∞ h 1 ∂ n ∂ i 2 = Cj ϕ (t) dt = C j ϕ (n) (2d2k−1 ) + ··· 2 t ∂t ∂t dk−1 Z Z

= Cj

Ra √ (recall that 0 dσ/ (a − σ ) σ = π if a > 0), yielding the first relation in (4.18). Similarly, if j = 2k is even, then f has a local maximum at iθ2k (σ = dk2 ). Then the main singularity of w(x, x0 ; t) is given by Z Cj

2 d2k

∂ (n−1/2) 2 dσ δ (t − σ ) 2 ∂t (d2k − σ )1/2 Z ∂ (n−1/2) 2 dσ 2 = Cj δ (t − d2k + σ)√ ∂t σ Z0 ∂ 1 ∂ (n) 1 dσ q = Cj √ . 0 ∂t 2 ∂t t2 − d2 + σ σ k

Applying this distribution to a test function ϕ(t), we get the leading term Cj

Z Z 1 ∂ (n) 1 dσ q √ ϕ(t) dt t ∂t 2 0 t 2 − d2k + σ σ Z Z h 1 ∂ n ∂ i dσ q = Cj ϕ (t) dt. t ∂t ∂t 0 (t 2 − dk2 + σ )σ

√ Note that 1/ (a + σ )σ makes sense for a < 0 as well if σ > −a and √ max(−a,0) dσ/ (a + σ )σ ∼a→0 C ln |a|. Hence the leading term is equal to

R1

Z Cj

ln |t

2

h 1 ∂ in ∂ ϕ (t) dt. t ∂t ∂t

− dk2 |

Set t − dk = ρ. Then ln |t 2 − dk2 | = ln |ρ| + ln(2dk + ρ) ∼ ln |ρ| as ρ → 0 and h Z Z ∂ in ∂ ϕ(dk + ρ) 1 ϕ (dk + ρ) dρ ∼ C j dρ, C j ln |ρ| ρ + dk ∂ρ ∂ρ ρ n+1 where the last integral is understood in the distribution sense.

370


Remark. The preceding results continue to hold in the anisotropic case (described at the end of Sec. 1.2) if the condition x 6= 0 is replaced throughout by x 00 6 = 0. The wave kernel may be computed if x = 0. We discuss the isotropic case first. Recall that dk2 (0, x0 ) = kπ x0 /α. THEOREM 4 For every positive integer n and α, x0 > 0, there exist constants a j,k , k = 1, 2, . . . , 1 ≤ j ≤ n − 1, such that

w(0, x0 , t) =

∞ X n−1 X

∂ (n+ j−1/2) 2 δ (t − dk2 ). ∂t

(4.19)

√ X ∞ π ∂ (−1)k+1 k δ (1/2) (t 2 − dk2 ). 2α ∂t

(4.20)

a j,k

k=0 j=0

If n = 1, then w(0, x0 , t) =

k=1

Proof By (4.3), P(0, x0 ; 2u) =

1 (4πu)n+1

Z

∞

ei x0 τ/(2u) V (τ ) dτ.

−∞

There exists a positive such that for every k, n there exists a function Wk,n (τ ) holomorphic in |τ − πki/(2α)| < 2, Wk,n (πki/(2α)) 6= 0, and V (τ ) = Wk,n (τ )/(τ − πki/(2α))n . Using simple estimates (cf. [1]), it follows that ∞ Z X Wk,n (τ ) 1 dτ. ei x0 τ/(2u) (τ − πki/(2α))n (4πu)n+1 |τ −πki/(2α)|= k=0 (4.21) Applying the residue theorem, we see that

P(0, x0 ; 2u) =

P(0, x0 ; 2u) =

1 u n+1

∞ X

e−πkx0 /(4αu)

n−1 X b j,k

k=1

j=0

uj

=

∞ X k=0

e−dk /(4u) 2

n−1 X j=0

b j,k n+1+ j u

.

(4.22) Application of (4.2) with a 2 = dk2 /2 yields P(0, x0 ; 2u) =

∞ X k=1

1 √ πu

n−1 ∞X

Z 0

j=0

a j,k e−t

2 /(4u)

∂ (n+ j−1/2) 2 δ (t − dk2 ) dt, (4.23) ∂t

and (4.19) follows from Proposition 9 (P(0, x0 ; 2u) = e−2u1 H (0, x0 )).


371

If n = 1, then (see [1, p. 654]) P(0, x0 ; 2u) =

∞ 1 X 2 (−1)k+1 ke−dk /(4u) . 16αu 2 k=1

Applying (4.2) once again with

a2

=

dk2 /2,

we get (4.20).

In the anisotropic case, the function V (τ ) has poles at the points πki/(2α j ), k = 1, 2, . . . , 1 ≤ j ≤ n. We leave the formulation and proof of the anisotropic analog of (4.19) to the diligent reader. 5. Wave kernels via the continuation method Recall that L denotes a second-order positive semidefinite self-adjoint operator. Set sin(L 1/2 t) , L 1/2 so that W L (t) is the (operator-valued) solution of the wave equation W L (t) =

∂ 2 WL = −L W L ∂t 2

(5.1)

(5.2)

with the initial conditions W L (0) = 0,

W L0 (0) = I.

(5.3)

We find an explicit representation for the kernel of W L (t) when −L/2 is the Heisenberg Laplacian and when −L is a degenerate elliptic operator of the type studied in [2]. Our method involves analytic continuation of the Green function of L − ∂ 2 /∂ y 2 , and it is applicable whenever L is positive definite or zero is in the continuous spectrum of L (so that L −1/2 is well defined, at least as a closed operator with a dense domain). The Green function (−L + ∂ 2 /∂ y 2 )−1 is defined as that (operator-valued) solution G(y) of the equation (−L + ∂ 2 /∂ y 2 )G(y) = I · δ(y) which tends to zero as |y| → ∞. The main tool is the following proposition, essentially due to Taylor [15]. 14 Let L be a positive semidefinite operator. Then 1 ∂2 1/2 − L + 2 (L −1/2 e−|y|L ) = −I, 2 ∂y 1 −1/2 −|y|L −1/2 ∂ 2 −1 L e =− −L+ 2 , 2 ∂y

PROPOSITION

(i)

∂ 2 −1 W L (t) = lim Im 2 − L + 2 (·, it + ) . →0 ∂y

(5.4) (5.5)

(5.6)

372

(ii)


Let m be a positive integer. Then for every a > 0, m−1 δ(t − a) 1 0 1 ∂ lim Im 2 , = c m t ∂t t (a + (it + )2 )m →0+ m−1 H (t − a) 1 00 1 ∂ , lim Im 2 = c √ m t ∂t (a + (it + )2 )m−1/2 →0+ t 2 − a2

(5.7) (5.8)

0 (R ) of where the limits are understood in the distribution sense (i.e., in Dm + 0 00 t), and cm , cm are negative constants.

Proof Note that ∂ −|y|L 1/2 1/2 e = −L 1/2 sign(y)e−|y|L , ∂y ∂ 2 −|y|L 1/2 1/2 1/2 e = Lsign2 (y)e−|y|L − 2L 1/2 δ(y)e−|y|L , ∂ y2

(5.9) (5.10)

1/2

implying (5.4). The limits of L −(1/2) e−|y|L and of (L − ∂ 2 /∂ y 2 )−1 as |y| → +∞ vanish. The solution of the second-order ordinary differential equation (in y) (−L + ∂ 2 /∂ y 2 )u = −I is uniquely determined by the limits lim y=±∞ u(y), and (5.5) 1/2 follows. The operator-valued function L −(1/2) e−y L is holomorphic in the half-plane Re y > 0, its boundary values at y = it satisfy the wave equation (5.2), its imaginary part is uniformly bounded in compact subsets of the half-plane Re y ≥ 0, and 1/2 the function L −(1/2) sin(L 1/2 t) = − Im(L −(1/2) e−it L ) also satisfies the initial conditions (5.3), proving (5.6). To prove (5.7), set L = −12m+1 . Then the kernel of (−L + ∂ 2 /∂ y 2 )−1 is given by ∂ 2 −1 cm 12m+1 + 2 (x, y; 0, 0) = (cm < 0) (5.11) 2 ∂y (|x| + y 2 )m (for x ∈ R2m+1 ). On the other hand, it is well known that for t > 0, 1 ∂ m−1 δ(t − |x|) W−12m+1 (t)(x, 0) = dm (dm > 0). (5.12) t ∂t t Formula (5.7) follows from (5.11), (5.12), and (5.6). Similarly, to prove (5.8), set L = −12m , and recall the formulas (for x ∈ R2m ) c˜m ∂2 12m + 2 (x, y; 0, 0) = 2 , (5.13) ∂y (x + y 2 )(2m−1)/2 and (for t > 0) W−1m (t)(x, 0) = d˜m

1 ∂ m−1 H (t − |x|) . √ t ∂t t2 − x2

(5.14)


373

Recall (see [3], [2]) that the Green kernels of operators such as the Heisenberg Laplacian, the Heisenberg Laplacian +∂ 2 /∂ y 2 , and certain degenerate elliptic second-order operators are known to be of the form Z V (z) dz, q f R (z) where V is an analytic function of z (only), whereas f is a complex-valued action (of the type introduced and discussed in Sec. 1.2) associated with the symbol of L (or L − ∂ 2 /∂ y 2 ). The integration path may be deformed to a contour on which f is real and (5.7), (5.8) may be applied. 5.1. The Heisenberg Laplacian Here we consider for simplicity only the isotropic case. Thus, given a positive integer n and a positive constant α, the Heisenberg Laplacian 1 H is defined on R2n+1 = {(x1 , . . . , x2n , t)} by 1H =

n

n

j=1

j=1

1 X ∂ ∂ 2 1 X ∂ ∂ 2 + 2αx2 j + − 2αx2 j−1 . 2 ∂ x2 j−1 ∂y 2 ∂ x2 j ∂ x0

(5.15)

R. Beals and P. Greiner [3] computed the Green kernel of 21 H + ∂ 2 /∂ y 2 with a pole at the origin (of R2n+2 ): ∂ 2 −1 − 21 H + 2 (x1 , . . . , x2n , x0 , y) ∂y Z ∞ 2ατ n dτ = cn n+1/2 , (5.16) 2 sinh(2ατ ) −∞ |x| ατ coth(2ατ ) + y 2 /2 − i x0 τ P 2 where |x|2 = 2n j=1 x j and cn is a positive constant. We utilize this computation in order to derive formulas for the wave kernel W H (t)(x, x0 ; 0, 0), where H denotes the operator −21 H . Setting 2ατ n , f (x, x0 ; τ ) = |x|2 ατ coth(2ατ ) − i x0 τ, (5.17) V (τ ) = sinh(2ατ ) we may rewrite (5.16) with a different cn as Z ∞ ∂ 2 −1 V (τ ) − 21 H + 2 (x1 , . . . , xn , x0 , y) = cn . 2 n+1/2 ∂y −∞ [2 f (τ ) + y ]

(5.18)

Note that f (τ ) = f (x, x0 ; τ ) = f (|x|, x0 , τ ; 2α). Hence the properties of the curve 0x,x0 (the set where f is real) are as described in Proposition 3. Remark. If x0 = 0, then 0x,x0 coincides with the real axis.

374


It is well known that one may perform the integrations in (5.16) or (5.18) on the deformed contour 0x,x0 . (Note that Re f > 0 between R and 0x,x0 , so that the fractional power is single valued.) The following is a consequence of Proposition 14. THEOREM 5 If x 6= 0, then for t > 0,

W H (t)(x, x0 ; 0, 0) = cn

Z 0x,x0

1 ∂ n H t − √2 f (x, x0 ; τ ) V (τ ) dτ p . t ∂t t 2 − 2 f (x, x0 ; τ )

(5.19)

Proof Note that V (τ ) and τ coth(2ατ ) are both even functions and are real if τ ∈ R. Hence V (−τ ) = V (τ ) and f (−τ ) = f (τ ) (x0 is real!). Moreover, the map τ → −τ maps 0x,x0 onto itself, with the orientation reversed. Hence Z V (τ ) dτ Im (2 f + (it + )2 )n+1/2 0 Z Z 1 V (τ ) dτ V (τ ) dτ = − 2i 0 (2 f (τ ) + (it + )2 )n+1/2 (2 f (τ ) + (−it + )2 )n+1/2 Z Z0 V (τ ) dτ 1 V (−τ )d(−τ ) = + 2 n+1/2 2i 0 (2 f (τ ) + (it + )2 )n+1/2 0 (2 f (−τ ) + (−it + ) ) (we use the fact that f (τ ) = f (τ ) on 0) Z Z 1 V (τ ) dτ V (τ ) dτ = − 2 n+1/2 2i 0 (2 f (τ ) + (it + )2 )n+1/2 0 (2 f (τ ) + (−it + ) ) Z h i 1 = V (τ ) Im dτ. (5.20) (2 f (τ ) + (it + )2 )n+1/2 0 Application of (5.8) (with m = n + 1) to (5.18) proves (5.19). Remark. As in Section 4, (5.19) may be written as involving only integrals of a real variable. If x = 0, x0 6= 0, then we may assume x0 > 0. The integrals in (5.16) and (5.18) may be computed using the residue theorem: the function V (τ ) has a pole of order n at τ = kπi/(2α), k = ±1, ±2, . . . , and we may deform the path of integration so that only the poles with Im(τ ) > 0 matter. For y 6 = 0 there is no singularity at τ = 0. For every > 0 sufficiently small, we have ∞ Z X ∂ 2 −1 V (τ ) dτ − 21 H + 2 (0, x0 , y) = cn . (5.21) 2 n+1/2 ∂y (−4i x 0τ + y ) |τ −kπi/(2α)|= k=1


375

It is well known that there exist infinitely many geodesics joining (0, x0 ) to the origin with lengths r kπ x0 dk = , k = 1, 2, . . . . (5.22) α 6 If x = 0, x0 > 0, then for t > 0, THEOREM

W H (t)(0, x0 ; 0, 0) =

1 ∂ n+ j H (t − √kπ x /α) 0 p cn,k, j , 2 t ∂t t − kπ x0 /α j=0

∞ X n−1 X k=1

(5.23)

where cn,k, j are functions of x0 . (Note that for fixed t the sum in (5.22) is finite.) Proof By (5.17), we have for each k = 1, 2, . . . that V (τ ) =

Wk (τ ) , (τ − kπi/(2α))n

where Wk (τ ) is regular near τ = kπi/(2α). Hence Z dτ V (τ ) 2 n+1/2 (−2i x τ 0 +y ) |τ −kπi/(2α)|= 2πi ∂ n−1 (Wk (τ ) = (n − 1)! ∂τ n−1 (−2i x0 τ + y 2 )n+1/2 τ =kπi/2 n−1 2πi X n − 1 (n−1− j) = Wk (n − 1)! j j=0

kπi (n + 1/2) · · · (n + 1/2 + j − 1)(2i x ) j 0 · . 2α [kπ/α + y 2 ]n+1/2+ j

(5.24)

Inserting (5.24) in (5.21) and using (5.8), we get (5.23). Remark. Note that for every nonnegative integer m and a positive number c, 1 ∂ m H (t − c) . (5.25) √ 2t ∂t t 2 − c2 √ Moreover, (5.1) implies that (∂/∂t)W H (t) = cos( −21 H t). Hence Theorems 5 and 6 are equivalent to Theorems 2 and 4, respectively. δ m−1 (t 2 − c2 ) =

376


5.2. Degenerate elliptic operators In this subsection we compute W L when L is a degenerate elliptic operator of the type considered in [2]. For simplicity we consider here a subclass, consisting of operators of the form 1 ∂ 2 ∂ 2 ∂2 L=− (5.26) + + (Bx1 , Bx1 ) 2 , 2 ∂ x1 ∂ x2 ∂ x0 where xi ∈ Vi (i = 1, 2), V1 , V2 are real Euclidean vector spaces, and ∂/∂ xi denotes the gradient in Vi (i = 1, 2), ( , ) denotes the inner product in V1 , and B is a positive definite matrix on V1 . It was proved in [2] that −L has a fundamental solution of the form Z ∞ V (τ ) dτ G(x1 , x2 ; x0 , x10 , x20 , x00 ) = −c , (5.27) 0 0 0 ˜ −∞ f (x 1 , x 1 , x 2 − x 2 , x 0 − x 0 , τ )q where Bτ 1/2 , sinh(Bτ ) τ f˜(x1 , x10 , x2 , x0 , τ ) = −i x0 τ + B coth(Bτ )(x1 − x10 ), x1 − x10 2 Bτ |x2 − x20 |2 + τ B tanh x1 , x10 + , 2 2 dim V1 + dim V2 , q= 2 V (τ ) = det

(5.28)

(5.29) (5.30)

and c is a positive constant. We consider here only the case where x10 = 0 (without loss of generality, we may assume x20 = t 0 = 0), leaving the case where x10 6= 0 to the future. Examples. Special cases of (5.26) are the Gruˇsin operator ∂ 2 /∂ x 2 + x 2 (∂ 2 /∂ x02 ) and the Baouendi-Goulaouic operator ∂ 2 /∂ x12 + ∂ 2 /∂ x22 + (x1 (∂/∂ x0 ))2 . Setting V˜2 = V2 × R, we see from (5.27) that a fundamental solution of 2L − ∂ 2 /∂ y 2 with a pole at the origin is given by the following (special case) of (5.27): Z ∞ V (τ ) dτ G(x1 , x2 , x0 , y; 0, 0, 0, 0) = c . (5.31) −∞ ( f˜(x 1 , 0, x 2 , x 0 , τ ) + y 2 /2)q+1/2 Set n = dim V1 , let 0 ≤ a1 ≤ a2 ≤ · · · ≤ an denote the eigenvalues of B (repeated according to their multiplicities), let (x1 ) j denote the component of x1 in the jth eigendirection, and let x100 denote the projection of x onto the eigenspace belonging


377

to the largest eigenvalue an . Then n τ X |x2 |2 f˜(x1 , 0, x2 , x0 , τ ) = −i x0 τ + a j (x1 )2j coth(a j τ ) + . 2 2

(5.32)

j=1

Thus f˜(x1 , 0, x2 , x0 , τ ) is the sum of a positive number and a function of the form (1.26). Note that G is arbitrarily small if |(x1 , x2 , x0 , y)| is sufficiently large. Hence the right-hand side of (5.31) represents the kernel of (−L + ∂ 2 /∂ y 2 )−1 . Observe that Proposition 3 holds if x100 6 = 0, x0 ≥ 0 (and L is the (degenerate) Laplace-Beltrami operator associated with the distance d on V1 × V2 ). Hence ∂ 2 −1 − −L+ 2 (x1 , x2 , x0 , y; 0, 0, 0, 0) ∂y Z V (τ ) dτ =c . (5.33) 0x1 ,t ( f˜(x 1 , 0, x 2 , x 0 , τ ) + y 2 /2)q+1/2 The next theorem follows from (5.33) and Proposition 14 in the same manner as in the proof of Theorem 5. Note, however, the distinction between the case where dim V1 + dim V2 is even and where q it is odd. For fixed x1 , x2 , and x0 , let h k (σ ) denote the inverse of the function (of τ ) 2 f˜(x1 , x2 , x0 , τ ) restricted to that part of 0x1 ,x0 (0 does not depend on x2 ) lying between i22k−1 and i22k and in the right half-plane. THEOREM 7 Let x100 6 = 0.

(i)

If dim V1 + dim V2 = 2 p, where p is a positive integer, then W L (t)(x1 , x2 , x0 ; 0, 0, 0) q 1 ∂ p H t − 2 f˜(x1 , x2 , 0, x0 , τ ) q = c0 V (τ ) dτ. (5.34) 0x,x0 t ∂t t 2 − 2 f˜(x1 , x2 , 0, x0 , τ ) Z

(ii)

If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then W L (t)(x1 , x2 , x0 ; 0, 0, 0) = c00 (Here d2k−1 < t ≤ d2k or d N < t.)

Proof

V (h k (t)) 1 1 ∂ p − Re . t t ∂t h 0k (t)

(5.35)

378


We apply Proposition 14 to

−L+

∂ 2 −1 (x1 , x2 , x0 , y; 0, 0, 0, 0) ∂ y2 Z V (τ ) dτ = −c˜ ˜ 0x,x0 (2 f (x 1 , 0, x 2 , x 0 , τ ) + y 2 )q+1/2

(c˜ > 0). (5.36)

If dim V1 + dim V2 = 2 p, where p is a positive integer, then q = p and (5.8) is applicable, with m = p + 1, and (5.34) follows as in the proof of Theorem 5. If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then q = p + 1/2, so that q + 1/2 = p + 1 and (5.7) is applicable to the integrand, with m = p + 1. Hence

W L (t)(x1 , x2 , x0 ; 0, 0, 0) = cˆ

Z 0x,x0

1 ∂ p δ t − V (τ ) t ∂t

q

2 f˜(·, τ ) t

dτ.

(5.37)

Now let t ∈ (d2k−1 , d2k ) or t > d N . Then h 0k (σ ) is nonzero in an open interval (d2k−1 , d2k ) (or d N , ∞), and we may change the variable of integration from τ to σ , integrating over an interval rather than over an arc of 0x,x0 . Recall that we also have to take into account the branch of 0x,x0 lying in the left half-plane, where τ = −h k (σ ) and the orientation is reversed. Hence Z V (h k (σ )) 1 ∂ p δ(t − σ ) dσ W L (t)(x1 , x2 , x0 ; 0, 0, 0) = cˆ h 0k (σ ) t ∂t t Z V (h k (σ )) 1 ∂ p δ(t − σ ) + cˆ dσ, (5.38) t ∂t t hˆ 0k (σ ) the integration is performed over an interval containing t (if t < d1 , then W L (t)(x1 , x2 , x0 ; 0, 0, 0) = 0), and (5.35) follows. Remark. For both the Gruˇsin operator and the Baouendi-Goulaouic operator, n = 1 and x100 = x1 . More generally, if B is a scalar operator, then the same equality x100 = x1 holds. The analysis of the case where x1 6 = 0, x100 = 0 is very complicated (cf. [1]) and is not attempted here. The case where x1 = 0, x0 = 0 is relatively simple, as the following theorem shows. 8 Let x1 = 0, x0 = 0. If dim V1 + dim V2 = 2 p, where p is a positive integer, then THEOREM

W L (t)(0, x2 , 0; 0, 0, 0) = c

1 ∂ p H (t − |x |) 2 q . t ∂t t2 − x2 2

(5.39)


379

If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then W L (t)(0, x2 , 0; 0, 0, 0) = c

1 ∂ p δ(t − |x |) 2 . t ∂t t

(5.40)

380


Proof It follows from (5.29) and (5.33) that Z ∞ ∂ 2 −1 V (τ ) dτ − −L+ 2 (0, x2 , 0, y; 0, 0, 0, 0) = c 2 2 q+1/2 ∂y −∞ ((x 2 + y )/2) c1 = 2 . (5.41) (x2 + y 2 )q+1/2 If dim V1 + dim V2 = 2 p, where p is a positive integer, then q = p and we may apply (5.8) to (5.41) (with m = p + 1), obtaining (5.39). If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then q = p + 1/2 and we may apply (5.7) to (5.41) (with m = p + 1), obtaining (5.40). The case where x1 = 0, x0 6= 0 is considerably more complicated. The results depend on the parity of the multiplicities of the eigenvalues of B. We treat here only the case where B is scalar, and we distinguish between even and odd n. The general case, where each eigenvalue has either even or odd multiplicity, is essentially a combination of the scalar cases. 9 Let x1 = 0, x0 > 0. Assume that B = a I and dim V1 = n = 2n 0 , where n 0 is a positive integer. If dim V1 + dim V2 = 2 p, where p is a positive integer, then THEOREM

W L (t)(0, x2 , x0 ; 0, 0, 0) 0

=

∞ nX −1 X k=1 j=0

ck, j

q 1 ∂ p+ j H t − 2kπ x0 /a + x22 q · . (5.42) t ∂t t 2 − 2kπ x0 /a − x22

If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then q 0 −1 ∞ nX 1 ∂ p+ j δ t − 2kπ x0 /a + x22 X 0 . W L (t)(0, x2 , x0 ; 0, 0, 0) = ck, j · t ∂t t k=1 j=0

(5.43) The constants ck, j and

0 ck, j

in (5.42) and (5.43) also depend on a and on n.

Note that as in Theorem 6, the sums in (5.42) and in (5.43) are finite for t fixed. Proof Let x0 > 0 (the case where x0 < 0 is treated similarly). Then f˜(0, 0, x2 , x0 , τ ) = −i x0 τ + x22 /2 is real for τ in the imaginary axis and is positive if Im τ > 0. The


381

function V (τ ) defined in (5.28) is given by aτ n 0 V (τ ) = , sinh(aτ ) and it has poles of order n 0 at τ = kπi/a, k = ±1, ±2, . . . . As in the discussion leading to (5.18), we get from (5.36) the representation (valid for y 6 = 0 and > 0 sufficiently small) ∂ 2 −1 − −L+ 2 (0, x2 , x0 , y; 0, 0, 0, 0) ∂y ∞ Z X V (τ ) dτ =c 2 2 q+1/2 [−2i x τ 0 + x2 + y ] k=1 |τ −kπi/a|= ∞ Z X Vk (τ ) dτ =c , n 0 [−2i x τ + x 2 + y 2 ]q+1/2 (τ − kπi/a) |τ −kπi/a|= 0 2 k=1 (5.44) where Vk is regular near τ = kπi/a. But Z |τ −kπi/a|= (n 0 − j−1)

· Vk

n 0 −1 Vk (τ ) 2πi X n 0 − 1 dτ = 0 0 (n − 1)! j (τ − kπi/a)n [−2i x0 τ + x22 + y 2 ]q+1/2 j=1

kπi (q + 1/2)(q + 1/2 + 1) · · · (q + 1/2 + j − 1)(2i x ) j 0 . (5.45) q+1/2+ j 2 a 2 |2kπ x0 /a + x + y | 2

If dim V1 +dim V2 = 2 p, where p is a positive integer, then q +1/2+ j = p+ j +1/2, and we may apply (5.8) (m = p+ j +1) and substitute (5.45) in (5.44) to obtain (5.42). If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then q + 1/2 + j = p + j + 1; we may apply (5.7) (m = p + j + 1), and, substituting (5.45) in (5.44), we get (5.43). If n is odd, then V (τ ) is no longer single-valued. First we have to determine the boundary values of V (τ ) on the imaginary axis. 15 Let n be an odd integer, n = 2n 0 + 1, and let a denote a positive number. Then there exist temperate (one-dimensional) distributions E + , E − such that a(is + ) n/2 E + (s) = lim , →0+ sinh(a(is + )) a(is − ) n/2 E − (s) = lim . (5.46) →0+ sinh(a(is − )) PROPOSITION

382


Moreover, singsupp(E + ) = singsupp(E − ) = {kπ/a, k = ±1, ±2, . . .}. Outside of the singular support, the functions E + (s) and E − (s) coincide with each other and are real if |s| < π/a or 2k(π/a) < |s| < (2k + 1)(π/a), k = ±1, ±2, . . . , and are purely imaginary with E + (s) = −E − (s) if (2k − 1)(π/a) < |s| < 2k(π/a), k = ±1, ±2, . . . , and in this case, |as| n/2 . (5.47) E + (s) = (−1)k+1 i | sin(as)| Both E + and E − are n 0 th derivatives of a locally integrable function whose singularities are of the form 1/(s − k(π/a))1/2 . Proof The function sinh(z)/z vanishes if and only if z = kπi, k = ±1, ±2, . . . . Moreover, sinh(z)/z is positive if z is real. Hence the function (aiτ /sinh(aiτ ))n/2 is well defined in the half-planes Im(τ ) > 0 and Im(τ ) < 0 with (aiτ /sinh(aiτ ))n/2 positive if τ is real (except at the poles). The zeros of sinh(i z) at z = kπ/a are simple, so that

1 ai z n/2 =O )n/2 sinh(ai z) |z − kπ/a|

(5.48)

for Im(z) 6 = o, z near kπ/a. Hence the limits in (5.46) exist as distributions (see [7, p. 63]). If s is real, then the function sin(as)/(as) is positive for |s| < π/a or 2k(π/a) < |s| < (2k + 1)(π/a), k = ±1, ±2, . . . , and negative elsewhere. Hence the functions (ai z/sinh(ai z))1/2 and (ai z/sinh(ai z))n/2 have branch points at z = kπ /a, k = ±1, ±2, . . . , and cuts in the intervals where sin(as)/(as) is negative. The positive choice of the square root near z = 0 (in particular, for z = is with |s| < π/a) implies that E + (s) and E − (s) must be negative for 2π < s < 3π and then positive for 3π < s < 5π, and so on. Hence the alternating signs for the imaginary case (5.47). Recall that kπ z = (−1)k + h k (z), (5.49) sin(z) z − kπ where h k (z) is holomorphic near kπ. Hence the function (having a branch point at kπ) aiτ n/2 aiτ n 0 aiτ 1/2 = · sinh(aiτ ) sinh(aiτ ) sinh(aiτ ) can be expanded in an algebraic Laurent series in powers of ((τ − kπ)/a)1/2 , starting with −n = −(2n 0 + 1). This expansion may be integrated n 0 times. Set now E = E + − E − . Then E(s) = 0 if |s| < π/a or 2k(π/a) < |s| < (2k + 1)(π/a), k = ±1, ±2, . . . , and E(s) is purely imaginary if (2k − 1)(π/a) < |s|
0. Assume that B = a I and dim V1 = n = 2n 0 + 1, where n 0 is a nonnegative integer. If dim V1 + dim V2 = 2 p, where p is a positive integer, then q Z ∞ 1 ∂ p H t − 2sx0 + x22 q W L (t)(0, x2 , x0 ; 0, 0, 0) = c E(s) ds. (5.51) t ∂t 0 t 2 − 2sx + x 2 0

2

If dim V1 + dim V2 = 2 p + 1, where p is a nonnegative integer, then q Z ∞ 1 ∂ p δ t − 2sx0 + x22 ds, W L (t)(0, x2 , x0 ; 0, 0, 0) = c E(s) t ∂t t 0

(5.52)

and there exist real constants ck such that the leading singularity of W L (t)(0, x2 , x0 ; 0, 0, 0) is given by √ (2k + 1)π /(ax0 ) − π/(2ax0 ) n/2+ p √ t − (2k + 1)π /(ax0 ) + √ ∞ X c2k H t − (2k)π /(ax0 ) − π/(2ax0 ) . (5.53) + n/2+ p √ t − (2k)π/(ax0 ) − k=1

∞ X c2k+1 H t − k=0

n/2+ p

n/2+ p

(The distributions 1/s+ and 1/s− are defined, e.g., in [7, pp. 68 – 71].) Note that the integrals in (5.51) and in (5.52) are actually performed over compact intervals, and note that the sum in (5.53) is finite if t is fixed. Proof Let y 6 = 0. Recall that by (5.33), Z ∂ 2 −1 (aτ /sinh(aτ ))n/2 dτ − −L+ 2 (x1 , x2 , x0 , y; 0, 0, 0, 0) = c , 2 2 q+1/2 ∂y 0x1 ,t (−2i x 0 τ + x 2 + y ) (5.54) where c is a positive constant, V (τ ) = (aτ /sinh(aτ ))n/2 , and q = p + (1/2). An estimate similar to the one used in the proof of Proposition 12 implies that for every > 0, R > 0, the integral in (5.54) may be extended over the contour composed of {−∞ < τ1 ≤ −, τ2 = R}, {τ1 = −, 0 ≤ τ2 ≤ R}, {− ≤ τ ≤ , τ2 = 0}, {τ1 = , 0 ≤ τ2 ≤ R}, and { ≤ τ1 < ∞, τ2 = R}. For every positive integer j, set

384


R −∞ |V (i R j + σ )| dσ , − |V (i R j −q−1/2 y 2 )−q−1/2 = O(R j ) for Im τ =

R j = ( j + 1/2)π. Then while (−2i x0 τ + x22 + −(−L+

R∞

n/2

+ σ )| dσ are O(R j ) R j , j → ∞. Hence

∂ 2 −1 ) (x1 , x2 , x0 , y; 0, 0, 0, 0) ∂ y2 Z 0 V (− + is) d(is) = c lim j→∞ 2 2 q+1/2 R j [−2i x 0 (− + is) + x 2 + y ] Z V (− + is) d(is) + 2 2 q+1/2 − [−2i x 0 (− + is) + x 2 + y ] Z Rj V ( + is) d(is) + [−2i x0 ( + is) + x22 + y 2 ]q+1/2 0 Z 0 V (− + is) d(is) =c 2 2 q+1/2 ∞ [−2i x 0 (− + is) + x 2 + y ] Z V (− + is) d(is) + 2 2 q+1/2 − [−2i x 0 (− + is) + x 2 + y ] Z ∞ V ( + is) d(is) + . [−2i x0 ( + is) + x22 + y 2 ]q+1/2 0

(As a side remark, note that the integrals do not converge absolutely if dim V2 ≤ 1. The following argument is valid without absolute convergence; one could also introduce additional artificial x2 -variables and apply a method of descent.) The regularity of the integrands at τ = 0 (note that y 6 = 0) implies that we may let tend to zero and obtain the representation Z ∞ ∂ 2 −1 E(s) ds . (5.55) − −L+ 2 (x1 , x2 , x0 , y; 0, 0, 0, 0) = ci 2 ∂y [y + 2x0 s + x22 ]q+1/2 0 Recall that E(s) is purely imaginary. Hence Im

−L+

∂ 2 −1 (x1 , x2 , x0 , y; 0, 0, 0, 0) ∂ y2 Z ∞ = ci E(s) Im 0

1 ds. (5.56) [y 2 + 2x0 s + x22 ]q+1/2

Putting y = it + , where t > 0, and letting tend to zero from the right, we get (at least formally) (5.51) and (5.52) from (5.56), (5.7), and (5.8). The intersection of the wave front set W F(E(s)) with the set W F 0 (K )R1 , where K is one of the right-hand sides of (5.7) or (5.8), is empty. By [7, Th. 8.2.13], the “integrals” in (5.51) and (5.52) make sense and are the limits of the integrals in (5.56).


385

Formula (5.53) follows from the observation that near the branch points kπ/a, the leading singularity of E + (E − ) is given by (1/(s − kπ/a − i0)n/2 ) (1/(s − kπ/a + i0)n/2 ). For the Gruˇsin operator n = 1, dim V2 = 0, so that p = 0, and we get from (5.52) the simple expression √ Z 2kπ ∞ X |s| 1/2 δ t − 2x0 s 0 k W L (t)(0, x0 ; 0, 0) = c (−1) ds t (2k−1)π | sin(s)| k=1

= (−1) j c0

1/2 t 2 /(2x0 ) , − sin(t 2 /(2x0 ))

(5.57)

where j is a positive integer determined by (2 j − 1)π < t 2 /(2x0 ) < 2 jπ , and W L (t) = 0 if no such j exists. 6. Directions for further studies We suggest here a certain number of open problems connected to this paper. The first question is how to extend the methods to find the wave kernel on a Riemannian P ∗ manifold (Mn , g) for a degenerate operator L = m 1 Łi Łi , where Łi are m vector fields such that their Lie brackets generate the tangent space at each point of the manifold. In particular, we are interested in the role played by the geometry. Another possible extension is to consider the wave kernel for ∂2 w = 1w − V (x)w, ∂t 2 w(P, 0) = 0, wt (P, 0) = δ P ,

(6.1)

where V is a double well potential. A related question in dimension 2, is to study ∂2 w = ∂x x w + V (x)∂x0 x0 w, ∂t 2 w(x, x0 , 0) = 0, wt (x, x0 , 0) = δ P .

(6.2)

We can choose, for example, V (x) = x 2 (1 − x)2 . The point P can be (0, 0) or (1, 0). What is the picture of interferences? Finally, we suggest a problem in the direction of the nonlinear wave equation. In particular, we are interested in the global existence or possible blow up in finite time

386


for the following wave equation: ∂2 w = 1 H w + |w| p−1 w, ∂t 2 w(P, 0) = g, wt (P, 0) = f,

(6.3)

where p ≤ pc = 1 + 2/n is the critical exponent in the Stein-Sobolev inequality on the Heisenberg group and where 1 H is the Heisenberg Laplacian. The initial data are smooth enough. We expected some anisotropic phenomena due to the interaction between the nonlinearities and the propagation along the bicharacteristics. Acknowledgment. We are very much indebted to the referee for a careful reading of the manuscripts and for comments that greatly improved the exposition. References [1]

[2]

[3] [4] [5]

[6]

[7]

[8]

[9] [10]

R. BEALS, B. GAVEAU, and P. C. GREINER, Hamilton-Jacobi theory and the heat

kernel on Heisenberg groups, J. Math. Pures Appl. (9) 79 (2000), 633 – 689. MR 2001g:35047 331, 334, 336, 337, 338, 340, 363, 370, 371, 378 R. BEALS, B. GAVEAU, P. GREINER, and Y. KANNAI, Exact fundamental solutions for a class of degenerate elliptic operators, Comm. Partial Differential Equations 24 (1999), 719 – 742. MR 2000c:35079 333, 371, 373, 376 R. BEALS and P. GREINER, Calculus on Heisenberg Manifolds, Ann. of Math. Stud. 119, Princeton Univ. Press, Princeton, 1988. MR 89m:35223 333, 373 A. BELLAÏCHE and J.-J. RISLER, eds., Sub-Riemannian Geometry, Progr. Math. 144, Birkhäuser, Basel, 1996. MR 97f:53002 331, 333, 334, 336 R. COURANT and D. HILBERT, Methods of Mathematical Physics, Vol. II: Partial Differential Equations, Wiley Classics Lib., Wiley, New York, 1989. MR 90k:35001 330, 362 B. GAVEAU, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95 – 153. MR 57:1574 331, 335 ¨ L. HORMANDER , The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis, Grundlehren Math. Wiss. 256, Springer, Berlin, 1983. MR 85g:35002a 382, 383, 384 Y. KANNAI, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. Partial Differential Equations 2 (1977), 781 – 830. MR 58:29247 332, 362 p , The method of ascent and cos A2 + B 2 , Bull. Sci. Math. 124 (2000), 573 – 597. MR 2001h:35025 333 Y. KANNAI and S. KIRO, The initial value problem for a degenerate wave equation, Proc. Amer. Math. Soc. 104 (1988), 125 – 130. MR 90d:35193 331


[11]

[12]

[13]

[14]

[15]

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W. MAGNUS, F. OBERHETTINGER, and R. P. SONI, Formulas and Theorems for the

Special Functions of Mathematical Physics, 3d ed., Grundlehren Math. Wiss. 52, Springer, New York, 1966. MR 38:1291 342, 343, 345 R. MELROSE, “Propagation for the wave group of a positive subelliptic second-order differential operator” in Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), Academic Press, Boston, 1986, 181 – 192. MR 89h:35177 331 A. I. NACHMAN, The wave equation on the Heisenberg group, Comm. Partial Differential Equations 7 (1982), 675 – 714. MR 84e:58074 331, 332, 335, 336, 351, 354, 367, 368 R. S. STRICHARTZ, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221 – 263, MR 88b:53055; Corrections, J. Differential Geom. 30 (1989), 595 – 596, MR 90f:53081 331, 334 M. E. TAYLOR, Noncommutative Harmonic Analysis, Math. Surveys Monogr. 22, Amer. Math. Soc., Providence, 1986. MR 88a:22021 331, 333, 338, 340, 363, 367, 371

Greiner Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada; [email protected] Holcman Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel; [email protected]; current: Department of Physiology, University of California at San Francisco, Keck Center, 513 Parnassus Ave., San Francisco, California 94143-0444, USA Kannai Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel; [email protected]


RIEMANNIAN MANIFOLDS WITH MAXIMAL EIGENFUNCTION GROWTH CHRISTOPHER D. SOGGE and STEVE ZELDITCH

Abstract On any compact Riemannian manifold (M, g) of dimension n, the L 2 -normalized eigenfunctions {φλ } satisfy ||φλ ||∞ ≤ Cλ(n−1)/2 , where −1φλ = λ2 φλ . The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S n . But, of course, it is not sharp for many Riemannian manifolds, for example, for flat tori Rn / 0. We say that S n , but not Rn / 0, is a Riemannian manifold with maximal eigenfunction growth. The problem that motivates this paper is to determine the (M, g) with maximal eigenfunction growth. Our main result is that such an (M, g) must have a point x where the set Lx of geodesic loops at x has positive measure in Sx∗ M. We show that if (M, g) is real analytic, this puts topological restrictions on M; for example, only M = S 2 or M = RP 2 (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any M fail to have maximal eigenfunction growth. In addition, we construct an example of (M, g) for which Lx has positive measure for an open set of x but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result. 1. Introduction The problem that motivates this paper is to characterize compact Riemannian manifolds (M, g) with maximal eigenfunction growth. Before stating our results, let us describe the problem precisely. We first recall that the associated Laplace-Beltrami operator 1 = 1g has eigenvalues {−λ2ν }, where 0 ≤ λ20 ≤ λ21 ≤ λ22 ≤ · · · are counted with multiplicity. Let {φν (x)} be an associated orthonormal basis of L 2 -normalized eigenfunctions. If λ2 is in the spectrum of −1, let Vλ = {φ : 1φ = −λ2 φ} denote the corresponding DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 3, Received 17 May 2001. Revision received 4 September 2001. 2000 Mathematics Subject Classification. Primary 35P20; Secondary 58J05, 58J50. Sogge’s research partially supported by National Science Foundation grant number DMS-0099742. Zelditch’s research partially supported by National Science Foundation grant number DMS-0071358.

387

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eigenspace. We then measure the eigenfunction growth rate in terms of L ∞ (λ, g) =

sup

φ∈Vλ :||φ|| L 2 =1

||φ|| L ∞ .

If eλ ( f ) is the projection of f ∈ L 2 onto Vλ , let E λ f = associated partial sum operators. Note that X E λ (x, y) = φν (x)φν (y)

(1) λν ≤λ eλν ( f )

P

be the

(2)

λν ≤λ

is the kernel, where {φν } are as above. If r X p(x, ξ ) = g jk (x)ξ j ξk

(3)

√ is the principal symbol of

−1, then by the local Weyl law (see [Av], [L], [Ho1]), Z X E λ (x, x) = |φν (x)|2 = (2π)−n dξ + R(λ, x) (4) p(x,ξ )≤λ

λν ≤λ

with uniform remainder bounds |R(λ, x)| ≤ Cλn−1 ,

x ∈ M.

Since the integral in (4) is a continuous function of λ and since the spectrum of P the Laplacian is discrete, this immediately gives λν =λ |φν (x)|2 ≤ 2Cλn−1 (see Lem. 3.4), which in turn yields L ∞ (λ, g) = O(λ(n−1)/2 )

(5)

on any compact Riemannian manifold. The bound (5) cannot be improved in the case of the standard round sphere (S n , can) (by the zonal spherical harmonics) and on any rotationally invariant metric on S 2 . It is, however, not attained by metrics of revolution on the 2-torus T 2 , and the problem arises whether there is any metric g on a surface of positive genus for which it is attained. More generally, we pose the following. PROBLEM

Determine the (M, g) for which L ∞ (λ, g) = (λ(n−1)/2 ). Here, we are using the notation that (λ(n−1)/2 ) means O(λ(n−1)/2 ) but not o(λ(n−1)/2 ).

MANIFOLDS WITH MAXIMAL EIGENFUNCTION GROWTH

389

Our main result, Theorem 1.1, implies a necessary condition on a compact Riemannian manifold (M, g) with maximal eigenfunction growth: there must exist a point x ∈ M for which the set Lx = {ξ ∈ Sx∗ M : ∃T : expx T ξ = x}

(6)

of directions of geodesic loops at x has positive surface measure. Here, exp is the exponential map, and the measure || of a set is the one induced by the metric gx on Tx∗ M. For instance, the poles x N , x S of a surface of revolution (S 2 , g) satisfy |Lx | = 2π. THEOREM 1.1 Suppose that |Lx | = 0. Then, given ε > 0, there exist a neighborhood N = N (ε) of x and a positive number 3 = 3(ε), so that

sup

φ∈Vλ

kφk L ∞ (N ) ≤ ελ(n−1)/2 , kφk L 2 (M)

λ ∈ spec

√ −1 ≥ 3.

(7)

If one has |Lx | = 0 for every x ∈ M, then sup

φ∈Vλ

2(n + 1) kφk L p (M) = o(λn(1/2−1/ p)−1/2 ) if p > . kφk L 2 (M) n−1

(8)

The L p -bonds improve on some estimates in [So2], which say that sup

φ∈Vλ

kφk p = O(λδ( p) ), kφk2

2 ≤ p ≤ ∞,

(9)

where δ( p) =

( n

1 1 1 2 − p − 2, n−1 1 1 2 2 − p ,

2(n+1) n−1

≤ p ≤ ∞,

2≤ p≤

2(n+1) n−1 .

(10)

Also, in §8 we see that the last part of Theorem 1.1 is false for the endpoint case where p = 2(n + 1)/(n − 1). Recently, M. Taylor [T] proved bounds like (7) under more restrictive hypotheses involving the caustics of the wave group, and in [T] he also discusses applications of these sorts of bounds to problems involving the convergence of eigenfunction expansions. Also, P. Sarnak [Sa] has recently obtained limit formulae for the L 4 -norm of certain eigenfunctions on arithmetic hyperbolic quotients. Our eigenfunction bounds are based on the following estimates for the local Weyl remainder.

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THEOREM 1.2 As in (4), let R(λ, x) denote the remainder for the local Weyl law at x. Then

R(λ, x) = o(λn−1 ) if |Lx | = 0.

(11)

Additionally, if |Lx | = 0, then, given ε > 0, there exist a neighborhood N of x and a 3 < ∞, both depending on ε, so that |R(λ, y)| ≤ ελn−1 ,

y ∈ N , λ ≥ 3.

(12)

As we were informed by V. Guillemin, A. Laptev, and D. Robert after completing this paper, these estimates overlap previous results of Yu. Safarov in the paper [S] (see also [SV, §1.8] and [Iv2]). Because our proof is somewhat different from that of Safarov and our exposition contains complete details, we have decided to retain our original proof. Theorem 1.2 is a kind of local analogue of the classical Duistermaat-Guillemin (see [DG]; see also V. Ivri˘ı [Iv1]) result that N (λ) = (2π)−n Vol(M)λn + o(λn−1 ) if the set of initial directions (x, ξ ) of closed geodesics has measure zero in S ∗ M. Here, N (λ) denotes the number of eigenvalues less than or equal to λ (counted with multiplicity). Their assumption is the natural one for this theorem since only closed geodesics contribute singularities to the trace of the wave group U (t), t 6= 0; however, for pointwise bounds at x one needs to assume zero measure of closed loops through x. We should point out that (11) by itself is not strong enough to give the above sup-norm estimates for eigenfunctions. They also require the more delicate second statement, which says that x → lim supλ→∞ λ−(n−1) R(λ, x) is continuous at points where |Lx | = 0. We now give some concrete applications and examples of these general results. Our first application is to the real analytic setting, where the result has strong implications on the topology of M. 1.3 Suppose that (M, g) is real analytic and that L ∞ (λ, g) = (λ(n−1)/2 ). Then (M, g) is a Y`m -manifold, that is, a pointed Riemannian manifold (M, m, g) such that all geodesics issuing from the point m return to m at time `. In particular, if dim M = 2, then M is topologically a 2-sphere S 2 or a real projective plane RP 2 . THEOREM

For the definition and properties of Y`m -manifolds, we refer to [Bes, Chap. 7]. By a theorem due to L. Bérard-Bergery (see [BB], [Bes, Th. 7.37]), Y`m -manifolds M satisfy the conditions that π1 (M) is finite and that H ∗ (M, Q) is a truncated polynomial ring in one generator. This, of course, implies that M = S 2 or RP 2


391

(topologically) when n = 2. We remark that the loops are not assumed to close up smoothly. An interesting example to keep in mind here is the triaxial ellipsoid E a1 ,a2 ,a3 = {(x1 , x2 , x3 ) ∈ R2 : x12 /a12 +x22 /a22 +x32 /a32 = 1} with a1 < a2 < a3 . This is a metric on S 2 with four umbilic points. At each umbilic point, all the geodesics leaving the point return to it at the same time. Only two such geodesics close up smoothly, namely, the middle closed geodesic {x2 = 0} traversed in the two possible directions. For more on the geometry of the ellipsoid, we refer to [A], [K], and [CVV]. Our second application is to generic metrics. THEOREM 1.4 L ∞ (λ, g) = o(λ(n−1)/2 )

for a generic Riemannian metric on any manifold.

The proof is just to show that the condition |Lx | = 0, ∀x ∈ M, holds for a residual set of metrics with respect to the Whitney C ∞ -topology. It appears that this is a new geometric result. To our knowledge, the most general class of metrics for which an improvement of the general sup-norm bound O(λ(n−1)/2 ) has previously been proved is that of manifolds without conjugate points satisfying exponential bounds on the geodesic flow (see [Be]), where L ∞ (λ, g) = O(λ(n−1)/2 /log λ). The result is not stated in [Be] but follows from its logarithmic improvement on the remainder estimate in the Weyl law. Estimates of the error term in the Weyl law also appear in [V], but it appears that the methods of that paper require integration in x and hence do not imply L ∞ bounds; in particular, the assumptions are on closed geodesics rather than loops. It is natural to ask whether the converse of these results is also true. Regarding the first remainder estimate in Theorem 1.2, we prove the conditional converse. 1.5 If the loopsets of (M, g) are clean, or even almost clean, then the following implication holds: R(λ, x) = o(λn−1 ) =⇒ |Lx | = 0. THEOREM

Clean and almost-clean loopsets are generalizations of the same notions for fixed point sets and are defined in §7. Almost-clean (M, g) include manifolds containing round annuli, that is, annuli isometric to equatorial annuli of the round sphere. Although it has not been proven in detail, one can surely obtain such manifolds by generically deforming the round metric on S n in small balls, say, by adding handles. The geodesic flow in such cases is not clean in the sense of J. Duistermaat and Guillemin [DG] but is “usually” clean outside a hypersurface (the boundaries of

392

SOGGE and ZELDITCH

the balls). The proof of Theorem 1.5 is similar to the proof of the converse of the Duistermaat-Guillemin theorem in [Z2], and the assumption of almost-cleanness is taken from there. However, the naive converse of our sup-norm result is simply false. 1.6 There exist C ∞ -Riemannian tori of revolution (T 2 , g) such that THEOREM

∃x : |Lx | > 0,

R(λ, x) = (λ(n−1)/2 ),

but

||φλ ||∞ = o(λ(n−1)/2 ). ||φλ ||2

Thus, remainder blow-up is not sufficient for maximal eigenfunction growth. We note that it is very likely that the ellipsoid is also an example of such behavior: we have R(λ, x) = (λ(n−1)/2 ) at each umbilic point, but it is conjectured that ||φλ ||∞ /||φλ ||2 = o(λ(n−1)/2 ) (J. Toth). Another interesting result illustrating the fact that remainder blow-up need not imply eigenfunction growth was pointed out to us by Sarnak. In his (unpublished) dissertation, A. Karnaukh [Ka] proved the lower bound R(λ, x) = (λ1/2 ) for any x ∈ M on surfaces of constant negative curvature. Yet the conjectured rate (Sarnak) for sup-norms of eigenfunctions on such surfaces is O(λε ). In §8, we discuss some interesting open problems on maximal eigenfunction growth. This paper is organized as follows. In §2, we review some basic results on the wave group and geodesic flow and on several Tauberian lemmas. This background is sufficient for the proofs of Theorems 1.1 and 1.2 in §3. The proof of Theorem 1.2 is in part a variant of Ivri˘ı [Iv1] (see also [Ho3] for a proof of the Duistermaat-Guillemin theorem in [DG] that the remainder term in Weyl’s law is o(λ(n−1)/2 ) if the set of closed geodesics has measure zero). In particular, we need to analyze the small-time behavior of the restriction to the diagonal in M of the kernel of CU (t)B if C and B are zero-order pseudodifferential operators. The case where C = Id is the identity operator was analyzed by Ivri˘ı [Iv1] (see also [Ho3]); however, since we cannot take traces in our proof, we need to study this slightly more technical case. Fortunately, we are able to reduce the analysis to the special case discussed in §2. Additionally, Theorem 1.2 uses some continuity properties of |Lx |. The proof of Theorem 1.1 for the L ∞ -norms is then a simple matter, and the other cases follow by interpolating the L p -bounds in [So1]. We then apply the results to some concrete classes of metrics. As preparation, we discuss the geometry of loops in §4. In §§5 and 6, we give the applications of Theorem 1.1 to real analytic manifolds and generic Riemannian manifolds, respectively. In §7, we study converses to Theorems 1.1 and 1.2 and construct an example where R(λ, x) = (λ(n−1)/2 ) but L ∞ (λ, g) = o(λ(n−1)/2 ). Finally, in §8 we consider related open problems. Regarding notation, C denotes a constant that is not necessarily the same at each occurrence. Also, integrals like the one in (4) should be interpreted as C ∞ -densities


393

that are homogeneous in λ and defined so that the scalar product with ψ ∈ C ∞ (M) is the integral of the product by ψ with respect to the symplectic measure d x dξ on T ∗ M. 2. Wave group and geodesic flow In this section, we collect the relevant background on the wave group and geodesic flow for the proofs of Theorems 1.1 and 1.2. Essentially, they are based on the shorttime asymptotics of the wave equation, longtime wave front relations, and Tauberian lemmas. We need substantially more geometry of loopsets to prove Theorems 1.5 and 1.6, but we postpone the discussion until §4. Throughout this paper, (M, g) denotes a compact Riemannian manifold of dimension n ≥ 2, T ∗ M denotes its cotangent bundle, and S ∗ M denotes its unit-sphere bundle with respect to g. We denote by exp t H p the geodesic qP flow of g, defined as the flow of the Hamiltonian vector field H p of p(x, ξ ) = g jk (x)ξ j ξk , the prin√ cipal symbol for −1. By definition, exp t H p is homogeneous, that is, commutes with the natural R+ -action, r × (x, ξ ) = (x, r ξ ) on T ∗ M\0. We also define the exponential map at x by expx ξ = π ◦ exp t H p (x, ξ ). These definitions are standard in microlocal analysis but differ from the usual geometer’s definitions, which take P p 2 = i, j g i j (x)ξi ξ j as the Hamiltonian generating the geodesic flow. The geometer’s geodesic flow is not homogeneous. 2.1. Wave group and spectral projections Let d E λ be the spectral measure associated with the partial summation operators E λ √ for −1. Then we prove our estimates for the local Weyl law remainder term R(λ, x) by studying the Fourier transform Z √ U (t) = e−itλ d E λ = e−it −1 . This, of course, is the wave group, that is, the unitary group generated by solves the Cauchy problem 1 ∂ √ + −1 U (t) = 0, i ∂t

√ −1. It

U (0) = Id .

It is well known that the kernel U (t, x, y) is a Fourier integral operator in the class I −1/4 (R × M × M, C ), where C is the Lagrangean (t, x, y; τ, ξ, η) : τ + p(x, ξ ) = 0 and (x, ξ ) = exp(t H p (y, η)) . The last condition means that the orbit of H p which is at (y, η) at time zero is at (x, ξ ) at time t. In particular, x and y are joined by a geodesic of length |t|.

394

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We are mainly concerned with the restriction of the wave kernel to the diagonal in M × M. The restriction is defined by U (t, x, x) = 1∗ U (t, x, y), where 1 : R × M → R × M × M is 1(t, x) = (t, x, x). Although the notation conflicts with that for the Laplacian, it is standard and should cause no confusion. By general wave front set considerations, one always has W F U (t, x, x) ⊂ C1 , where C1 = (t, τ, x, ξ − η) : τ = −|ξ |, exp t H p (x, η) = (x, ξ ) .

(13)

Indeed, the canonical relation underlying 1∗ is given by (see [DG, (1.20)]) W F 0 (1) = (t, τ, x, (ξ + η)); (t, τ, x, ξ, x, η)) ⊂ T ∗ (R × M) × (R × M × M) . The intersection defining C1 is essentially that between C and the second component D := (t, τ, x, ξ, x, η) ⊂ T ∗ (R × M × M) of W F 0 (1). If C ∩ D is a clean intersection (cf. [DG], [Ho3, §29.1]), C1 is a Lagrangian submanifold of T ∗ (R × M)\0 and one has the much stronger statement that U (t, x, x) ∈ I 0 (R × M, C1 ), where I 0 (R×M, C1 ) is the class of Fourier integral operators of order zero associated to the canonical relation C1 . However, we do not need this hypothesis in the proofs of Theorems 1.1 and 1.2. We note that (x, η) is the initial (co-)tangent vector and that (x, ξ ) is the final (co-)tangent vector of a geodesic loop of length t at x; loops need not be smoothly closed. If there are no geodesic loops of length |t0 | through x0 ∈ M, it follows that (t, x) → U (t, x, x) is free of singularities at (t0 , x0 ). Since the length of a geodesic loop is at least the injectivity radius δ of (M, g), it follows that U (t, x, x) is smooth for |t| < δ. 2.2. Small-time asymptotics for microlocal wave operators In the proof of Theorem 1.2, we need to calculate asymptotics for the small-time behavior of the diagonal restriction [U (t)Q](x, x), {(x, x) : x ∈ M} of the oneparameter family of operators U (t)Q : C ∞ (M) → C ∞ (M),


395

where Q(t, x, Dx ) is a polyhomogeneous zero-order pseudodifferential operator. In the case where Q is independent of t, this calculation was carried out by L. Hörmander [Ho3], following Ivri˘ı [Iv1], and, as we see, the slightly more general case mentioned above follows in a straightforward way from this special case. To set up the notation, write Q(t, x, Dx ) = B(x, Dx ) + t R(t, x, Dx ),

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where, of course, B(x, Dx ) = Q(0, x, Dx ). We then let b and bs denote the principal and subprincipal symbols R(0, x, Dx ). qP of B, while r denotes the principal symbol of √ jk We recall that p = g (x)ξ j ξk denotes the principal symbol of −1 and that p s = 0 for this operator. The calculation we require then is the following. PROPOSITION 2.1 For any compact Riemannian manifold (M, g), the restriction K of the kernel of U (t)Q to R × {(x, x) : x ∈ M} is conormal with respect to {0} × {(x, x) : x ∈ M} in a neighborhood of this submanifold. Moreover, there is a δ > 0, so that when |t| < δ, Z ∞ ∂ A(x, λ) −iλt K (t, x) = e dλ, (15) ∂λ −∞

where A ∈ S n , A(x, 0) = 0, and Z Z i −n s −n ∂ A(x, λ) − (2π) (b + b ) dξ + (2π) {b, p} dξ ∂λ 2 p(x,ξ )0 0x , the second statement follows from the first and from the definition of cleanliness. The first statement follows from Proposition 4.6 together with [M, Th. 16.2] on the finite-dimensional approximation to x . There it is proved that a certain compact manifold B a of broken geodesics is a faithful model for the infinite-dimensional loop space ax of loops with energy E ∈ [0, a]. In particular, the energy function E is Bott-Morse on x if and only if it is Bott-Morse on B a and has the same critical points. Since Bott-Morse functions have isolated critical

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components, there are only finitely many critical components of energy less than any given L 2 (or length less than L) in x , and hence the set of critical values Lspx is finite in each compact subset of R. In the general case (without cleanliness assumptions), it follows from this finitedimensional approximation and Sard’s theorem that Lspx is a set of measure zero in R. The following proposition plays a role in the proof of Theorem 1.5. PROPOSITION 4.8 Let (M, g) be a compact Riemannian manifold. Then we have the following. (i) Suppose that for some T > 0 the loopset S0xT is clean and |S0xT | > 0. Then Sx∗ M = S0xT . (ii) If (M, g) has clean loopsets at x and if |Lx | > 0, then there exists T > 0 such that Sx∗ M = S0xT .

Proof (i) The hypothesis implies that S0xT is a submanifold of Sx∗ M of full dimension. S (ii) Since Lspx is discrete by Corollary 4.7, 0x = T >0 T 0xT . Since Lx is the countable union of their radial projections to Sx∗ M and has positive measure, at least one component T 0xT must be of dimension n − 1. Since T 0xT ⊂ T Sx∗ M by Proposition 4.2, again equality must hold. Examples We now present a few simple examples to illustrate the loopset notions and their connection to eigenfunctions, mainly with a view of clarifying the notation and terminology. It would take us too far afield to provide even a sketchy discussion of the wide variety of possible phenomena that occur; we limit ourselves to some rather standard cases. 4.1. Flat tori Let Rn / 0 be a flat torus. A geodesic loop at x is a helix that returns to x. Loops on flat tori are always closed geodesics, and they correspond to lattice points in 0. Thus, o n γ :γ ∈0 , Lspx = |γ |, γ ∈ 0 . 0x = 0, Lx = |γ | We observe that 0x is a countable discrete subset of Tx∗ Rn / 0, that Lx is a countable dense subset of Sx∗ M, and that 0g is a countable set of embedded Lagrangean comS ponents each diffeomorphic to Rn / 0. Also, C1 = γ ∈0 {(|γ |, τ, x, 0) : x ∈ M}.


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This example is nongeneric because (among other things) every loop is a closed geodesic. Flat tori are examples of metrics without conjugate points and with completely integrable geodesic flows. Their eigenfunctions (generically, exponentials) have remarkably small sup-norms (see [TZ1], [TZ2] for rigidity results on eigenfunctions with low sup-norms). We now generalize the discussion. 4.2. Manifolds without conjugate points Let (M, g) denote a manifold without conjugate points, that is, a Riemannian manifold such that each exponential map expx is a covering map. Manifolds of nonpositive curvature are examples, so this class of metrics is open on any M (though it may be empty for some M). By definition, there are no Jacobi fields along any loop satisfying Y (0) = Y (1) = 0, so the Jacobi operator is nondegenerate. Equivalently, each loop functional L x is a Morse function on x . Thus, for all x, 0x is a countable discrete set of points, Lx is countable, and 0xT is a finite set. Unlike the case of flat tori, loops are not generally closed geodesics. Not much is known about the growth of eigenfunctions on such manifolds beyond the bound already mentioned in [Be], except in the case of some special arithmetic hyperbolic manifolds (see [Sa]). 4.3. Surfaces of revolution Surfaces of revolution provide a simple (but nontrivial) class of examples for which the loopsets may be explicitly determined. Topologically, the surface must be either the 2-sphere S 2 or the 2-torus T2 . It would take us too far afield to discuss the geometry of loops and eigenfunctions on general surfaces of revolution, so we give only a few indications of how to determine the loops explicitly and refer the reader to the papers [CV], [Bl], [KMS], and [Z3] for further background. A 2-sphere of revolution is a Riemannian sphere (S 2 , g) with an action of S 1 by isometries. We may write g = dr 2 +a(r )2 dθ 2 in geodesic polar coordinates at one of the poles (fixed points) of the S 1 -action. The poles are always self-conjugate points. They are distinguished in the analysis of eigenfunctions because they are the unique points at which the eigenfunctions attain the maximal bounds. Tori of revolution generalize flat tori, but nonflat cases must have conjugate points (Hopf’s theorem). The metric on a torus of revolution may be written in standard angle coordinates as g = d x 2 + a(x)2 dθ 2 . Tori of revolution have no poles, and as we see in Theorem 1.6, eigenfunctions need not attain the maximal bounds. In fact, we conjecture that tori of revolution never have maximal eigenfunction growth. Otherwise, by our main result, there would have to exist a point at which a positive measure of closed geodesics recur. As already mentioned, this cannot occur in the analytic case for any metric on T2 . We conjecture that it also cannot occur for any

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C ∞ -metric of revolution on T2 due to the complete integrability of such geodesic flows on T2 . Such integrability constrains the behavior of loops, as we now recall. The complete integrability of geodesic flows on surfaces of revolution is due to the existence of the Clairaut integral I (x, ξ ) = ξ(∂/∂θ). Here, ∂/∂θ is the vector field generating the rotations. The Clairaut integral is constant along geodesics, or, otherwise put, unit speed geodesics must lie on level sets of I within Sg∗ M. Compact regular level sets are necessarily tori and are known as the invariant tori. We denote by TI0 ⊂ Sg∗ M a torus with Clairaut integral I = I0 . For notational simplicity, we ignore the fact that the level sets of I may have several components. Near any regular torus TI , one can always find a local action-angle such that the geodesics may be expressed as winding lines on the invariant tori (φ1 + tω1 , φ2 + tω2 ). Here, ω j are the frequencies of motion (which generally vary with I ). Now consider the geodesic loops at a point x ∈ S 2 . If x is a pole (a fixed point of the rotation), then all the geodesics passing through x are meridians, and the projection from the torus of meridians {I = 0} to S 2 under the standard projection π : T ∗ S 2 → S 2 collapses the initial tangent vectors at x to the meridians to the point x. Thus, 0x2π = Tx∗ M, Lx = Sx∗ S 2 , and Lspx = {n L , n ∈ Z}, where L is the length of a closed meridian geodesic. All of these loops are closed geodesics. The blow-down singularity of π : {I = 0} → S 2 is the feature responsible for the fact that the eigenfunctions associated to {I = 0} (the zonal eigenfunctions) have maximal blow-up at the poles. Now let M = S 2 or M = T2 , and assume that x is not a pole, as automatically occurs in the case of tori of revolution. Under π : T ∗ M → M, the tori TI project to annuli on M, and with a finite number of exceptions, the projection map π I : TI → M, restricted to a torus, is a two-to-one cover with fold singularities (the caustic sets). It follows that there are either zero, one, or two geodesics of TI passing through x, as x accordingly does not lie in π I (TI ), lies on its boundary, or lies in its interior. Using action-angle variables, we can determine the tori TI that contain a geodesic loop x. Such a loop is a winding line on TI which passes through the set π I−1 (x) at least twice. If it passes through the same point of π I−1 (x) twice, then it is a closed geodesic; otherwise, it is a nonsmooth loop. The set of tori that contain periodic geodesics are known as periodic tori and are independent of the point x. Under a generic twist condition (see [Bl], [Z3]), the set of periodic tori in Sg∗ M is countable. It follows that for generic surfaces of revolution, the set of closed geodesics through any point is countable. More complicated are loops that are not closed geodesics, that is, winding lines that pass through distinct points of π I−1 (x). These do depend on x. One can write down explicit equations for the values of I such that TI contains a winding line that loops at a given point x. It is clear from the equations that, for fixed nonpolar x, a generic surface of revolution has only countably many loops at x. As mentioned


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above, we suspect that the loops always form a set of measure zero, but it would take us too far afield to settle the question here. For further information on geodesics on tori of revolution, we refer to [KMS]. 4.4. Triaxial ellipsoids We recall that E a1 ,a2 ,a3 = {(x1 , x2 , x3 ) ∈ R3 : x12 /a12 + x22 /a22 + x32 /a32 = 1} with 0 < a1 < a2 < a3 . Jacobi proved in 1838 that the geodesic flow of E a1 ,a2 ,a3 (for any (a1 , a2 , a3 )) is completely integrable. The two integrals of the motion are the length H (x, ξ ) = |ξ |x and the so-called Joachimsthal integral J . More recent discussions of the geodesics of the ellipsoid can be found in [A], [K], and [CVV, §3]. There are four distinguished umbilic points ±P, ±Q which occur on the middle closed geodesic {x2 = 0}. All geodesics leaving P arrive at −P at the same time, then leave −P and return to P at the same time (see [K, Th. 3.5.16] or [A]). Thus, the triaxial ellipsoid is an example of a Y`m -metric that is not a Zoll metric. At all other points x ∈ E a1 ,a2 ,a3 , the set of initial directions of geodesics which return to x is countable (see [K]). The moment map (H, J ) : T ∗ E a1 ,a2 ,a3 → R2 is regular away from the middle closed geodesic, and its only singular level in S ∗ E a1 ,a2 ,a3 is that (H = 1, J = 0) of {x2 = 0}. For a description of the singular level, we refer to [CVV]. The middle geodesic is homoclinic; that is, the trajectories on the level are forward and backward asymptotic to it. Other level sets of the moment map are regular Lagrangean tori, and the discussion of periodic orbits and loops is similar to the case of surfaces of revolution. 4.5. Zoll surfaces Now let us consider the extreme case of Zoll metrics on S 2 , that is, metrics all of whose geodesics are closed (see [Bes]). Among such metrics, there is an infinitedimensional family of surfaces of revolution. There is an even larger class with no isometries. We may suppose, with no loss of generality, that the least common period of the S ∗ 2 geodesics equals 2π. Then 0x2π = Sx∗ S 2 for every x, and 0x = ∞ n=1 2πnSx S . As x S∞ varies, we obviously get 0 = n=1 2πnS ∗ S 2 . We also note that C1 is parameterized by R × T ∗ M → T ∗ (R × M × M), (t, x, ξ ) → t, −|ξ |x , x, ξ, exp −t H p (x, ξ ) . Since all geodesics are periodic, we obtain C1 = {(t, τ, x, 0)} ≡ Z × R × M,

clearly, a homogeneous Lagrangean submanifold of T ∗ (R × M)\0.

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As we discuss in §8, although geodesics are recurrent at every x, we do not expect eigenfunctions of 1 to have maximal growth everywhere, or even anywhere in general. Rather, it is the eigenfunctions of a closely related pseudodifferential operator A which have maximal growth. This operator (see [CV] for its definition) has integral eigenvalues of high multiplicity, so its spectral projections have the same supnorm growth as do zonal spherical harmonics on the standard sphere. However, these spectral projections are not eigenfunctions of the actual Laplacian in general. 5. Real analytic Riemannian manifolds Our first application is to characterize Riemannian manifolds in the analytic setting with maximal eigenfunction growth. THEOREM 5.1 Suppose that (M, g) is a compact real analytic Riemannian manifold with L 2 normalized eigenfunctions {φλ } satisfying ||φλ ||∞ = (λn−1 ). Then (M, g) is a Y`m -manifold for some `, m. In particular, M = S 2 , RP 2 (topologically) if n = 2.

We recall (cf. [Bes, Def. 7.8, p. 182]) that a compact manifold M with a distinguished point m is called a Y`m -manifold if all geodesics issuing from m come back to m at time `. We further recall that a compact Y`m -manifold has the properties that π1 (M) is finite and H ∗ (M, Q) has exactly one generator. Proof By assumption, there exist a constant C > 0 and a subsequence of L 2 -normalized (n−1)/2 eigenfunctions {φλ j } such that ||φλ j ||∞ ≥ Cλ j . This contradicts the last statement of Theorem 2.1; hence, there exists a point m such that |Lm | > 0. Since g is real analytic, exp : Tm∗ M → M is a real analytic map; hence, 0m is an analytic set. In any local coordinate patch U ⊂ M containing m, 0m is the zero set of a pair of real-valued real analytic functions, that is, has the form ( f 1 (ξ ), . . . , f n (ξ )) = (m 1 , . . . , m n ). The solutions are the same as for ( f 1 (ξ )−m 1 )2 +· · ·+( f n (ξ )−m n )2 = 0, so 0m is the zero set of a real analytic function. By a theorem due to F. Bruhat and H. Whitney, the zero set of a real analytic function is locally a finite union of embedded real analytic submanifolds Yiki of dimensions 1 ≤ ki ≤ n − 1. Thus, for each ξ there exists a ball Bδ (ξ ) such that 0m ∩ Bδ (ξ ) =

d [

Yiki .

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i=1

For a more recent discussion, we refer to [N, Props. 17, 18] (see also [H]). We claim that for some (ξ, δ) there exists a component Yin−1 of dimension n − 1.


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If not, 0m is of Hausdorff dimension less than or equal to n − 2. But then its radial projection ρ : 0m → Sm∗ M would also have Hausdorff dimension less than or equal to n − 2. In fact, each ray through the origin in Tm∗ M intersects 0m in at most countably many points. So the radial projection preserves the dimension. But this contradicts the fact that 3m = ρ(0m ) has positive Lebesgue measure. Now let 4 ⊂ 0m be an open embedded real analytic hypersurface of T ∗ M. Consider the rays {tξ : 0 ≤ t ≤ 1} ⊂ Tm∗ M and the union [ C= {tξ : 0 ≤ t ≤ 1}. ξ ∈4

Thus, each ray in C exponentiates to a geodesic loop that returns at t = 1. As proved in Proposition 4.2, it follows by the first variation formula that the length |ξ | of each loop must be a constant independent of ξ ∈ 4. We conclude that |ξ | = ` for some ` ∈ R+ and ξ ∈ Y . But this equation is real analytic and hence must hold on all of 4; hence, 4 ⊂ `Sm∗ M. Again, by real analyticity, `Sm∗ M ⊂ 0m . This is the same as saying that (M, g) is a Y`m -manifold. In §7 we prove that the converse to Theorem 1.2 holds automatically in the analytic setting. 6. Generic metrics: Proof of Theorem 1.4 We now prove that there exists a residual subset of the space G of C ∞ -metrics with the Whitney C ∞ -topology for which ||φ||λ = o(λ(n−1)/2 ). In view of Theorem 1.1, it suffices to prove the following. LEMMA 6.1 g There exists a residual set R ⊂ G of metrics such that |Lx | = 0 for every x ∈ M when g ∈ R .

Proof R g Recall that |Lx | = 0 if and only if S ∗ M 1/L ∗g (x, ξ ) dξ = 0 if L ∗g (x, ξ ) is the loopx length function defined before. To make use of this, choose a coordinate patch ⊂ M with coordinates y = κ(x) ranging over an open subset of Rn . We then fix K ⊂ κ() to be compact, and we let Z dσ , F(g) = sup ∗ (y, ξ ) n−1 L y∈K S g using the induced coordinates {y, ξ } for T ∗ ⊂ T ∗ M. Here also, dσ is the standard surface measure on S n−1 , and we are abusing notation a bit by letting L ∗g (y, ξ ) denote

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the pushforward of L ∗g using κ. It then suffices to show that the set of metrics for which n 1o G N = g : F(g) < N is open and dense. Density just follows from the fact that F(g) = 0 for any non-Zoll real analytic metric. Such metrics are dense in G . The main step in proving that these sets are also open is to show that the function R f (g, y) = S n−1 dσ/L ∗g (y, ξ ) is upper semicontinuous on G × K . This holds since 1/L ∗g (y, ξ ) is a positive, (locally) bounded upper semicontinuous function on G ×× S n−1 if we equip G with the C 3 -topology. Therefore, if (g j , y j ) → (g, y), y j ∈ K , we have Z Z dσ dσ ≤ sup ∗ sup ∗ S n−1 j L g j (y j , ξ ) S n−1 L g j (y j , ξ ) j Z Z Z dσ dσ dσ =⇒ lim sup ≤ lim sup ≤ , ∗ (y , ξ ) ∗ (y , ξ ) ∗ (y, ξ ) n−1 n−1 n−1 L L L j j S S S j j gj gj g using the dominated convergence theorem and upper semicontinuity. Now let us prove that the sets G N are open. Let g ∈ G N . By the definition of F, f (g, y) < 1/N for each y ∈ K . Since f is upper semicontinuous, the set { f < 1/N } is open, so there exist δ(y) such that Bδ(y) (y) × Bδ(y) (g) ⊂ { f < 1/N } if Bδ (y) and Bδ (g) denote the δ-Euclidean and C 3 -balls of y and g, respectively. Here, g is fixed, so we do not indicate the dependence of the δ’s on it. As y varies over K , the balls Bδ(y) (y) give an open cover of K , and by compactness there exists a finite subcover {Bδ(y j ) (y j ), j = 1, . . . , N }. Let δ = min j δ(y j ), so that Bδ (y j ) × Bδ (g) ⊂ { f < 1/N } for j = 1, . . . , N . If g 0 ∈ Bδ (g), then f (y, g 0 ) < 1/N for all y ∈ K . Hence, F(g 0 ) < 1/N , and so G N is open. 7. Converse results: Proofs of Theorems 1.5 and 1.6 The purpose of this section is to prove Theorems 1.5 and 1.6. We need to go into somewhat more detail about loopsets. We first consider the simplest converse in which the loopsets are assumed to be clean. 7.1. Proof of Theorem 1.5 for manifolds with clean loopsets THEOREM 7.1 Suppose that (M, g) is a compact Riemannian manifold. Then we have the following. (i) If 0xT 6= ∅ is clean and if {T } is isolated in Lspx , then R(λ, x) = o(λn−1 ) =⇒ |S0xT | = 0.


(ii)

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If (M, g) has clean loopsets, then R(λ, x) = o(λn−1 ) =⇒ |Lx | = 0.

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We begin the proof with some considerations that are valid for all compact Riemannian manifolds. For each (T, x) we introduce the sequence of probability measures X 1 |φ j (x)|2 δei T λ j (θ) E λ (x, x)

µT,λ,x (θ) =

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λν ≤λ

on S 1 . Here, δei T λ j (θ ) denotes the point mass at the indicated point of S 1 . LEMMA 7.2 Let (M, g) be any compact Riemannian manifold. Suppose that R(λ, x) = o(λn−1 ). Then µT,x,λ → dθ for all T 6= 0.

Proof The kth Fourier coefficient of µT,λ,x (θ) equals µˆ T,λ,x (θ) =

X 1 |φ j (x)|2 ei T kλ j . E λ (x, x)

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λν ≤λ

Hence, R(λ, x) = o(λn−1 ) implies E λ (x, x)µˆ T,λ,x (θ) =

λ

Z 0

eikT λ d E λ (x, x)

λ

Z

e

=

ikT λ

n

λ

Z

dλ +

0

eikT λ d R(λ, x)

0 λ

Z =n

eikT λ λn−1 dλ + eikT λ R(λ, x)

0 λ

Z − ikT

eikT λ R(λ, x) dλ

0

= o(λn ) for k 6= 0.

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Since dθ is the unique probability measure µ on S 1 satisfying µ(k) ˆ = δk0 , we have Z eikθ dµT,λ,x (θ) → 0 (∀k, T 6 = 0) ⇐⇒ dµT,λ,x (θ) → dθ . (49) S1

Hence, Lemma 7.2 follows from (48) and (49). In view of Lemma 7.2, to complete the proof it suffices to show the following.

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LEMMA 7.3 We have the following. (i) Suppose that {T } is an isolated point in Lspx and that 0xT is clean. Then µT,λ,x (θ) → dθ implies that S0xT is a submanifold of Sx∗ M of dimension less than or equal to n − 2. (ii) We have µT,λ,x (θ) → dθ (∀T 6 = 0) =⇒ |Lx | = 0.

Proof (i) By (49), (i) asserts that if 0xT is clean, then Z eikθ dµT,λ,x (θ) → 0 (∀k, T 6 = 0) =⇒ |S0xT | = 0.

(50)

S1

We prove this by studying the asymptotics of [U (T )E λ ](x, x) as λ → ∞, which are dual to the singularity of U (T +s)(x, x) at s = 0. These asymptotics were studied (in greater generality) in [Z2, §2], and we follow that discussion in the calculation to follow. Following [Z2], we write Z U (T + s, x, x) = U (T + s, y, y 0 )δx (y)δx (y 0 ) dy dy 0 . M×M

Viewing x and T as fixed, we obtain a distribution in s with singularities at times in the loop-length spectrum Lspx (referred to as “sojourn times” in the more general context of [Z2]). If {T } is isolated in Lspx , then there exists an interval around s = 0 such that U (T + s, x, x) is singular only at s = 0. If 0xT is a clean intersection, it then follows from the clean composition calculus that U (T + s, x, x) is a Lagrangean distribution in s in this interval (see [Z2, §2] for the specific case at hand or [DG], [Ho3] for the general theory). This implies that U (T + s, x, x) has an isolated singularity at s = 0 with a singularity expansion U (T + s, x, x) = ar (T, x)(s + i0)−r + ar −1 (T, x)(s + i0)−r +1 + · · · , where · · · denotes smoother terms and where the order r = dim S0xT . Equivalently, we may write Z ∞ ∂ A T (x, λ) −isλ U (T + s)(x, x) = e dλ, (51) ∂λ −∞ where A T (x, λ) is a symbol of order r − 1 in λ. By Lemma 2.5 or by [Z2, Lem. 3.1], it follows that U (T )E λ (x, x) − A T (x, λ) = O(λn−1 ). (52) Therefore, Lemma 7.3(i) is equivalent to order of

A T (x, λ) ≤ n − 1 ⇐⇒ |0xT | = 0.

(53)


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The principal symbol of U (T + s, x, x) at s = 0 (for fixed x) was calculated in [Z2, Prop. 1.10, Lem. 3.6, Th. 3.15]. The formula depends on dim S0xT . When dim S0xT = n − 1, then the order of A T equals n and the symbol is given by σn A T (x, ·) = |S0xT |. (54) Now suppose that dµT,x,λ → dθ. Then the first moment of dµT,x,λ tends to zero. If dim S0xT = n − 1, it must equal Sx∗ M. But then the order n symbol of A T cannot equal zero. This contradiction completes the proof of (i). (ii) This case follows from (i) and is easier since the hypothesis is much stronger. By Proposition 4.2, if |Lx | > 0, then there exists T > 0 such that 0xT = T Sx∗ M. As just proved, this implies that the symbol of order n of A T is positive. But R(x, λ) = o(λ(n−1)/2 ) implies that it equals zero. 7.2. Real analytic metrics We also have a converse theorem for real analytic metrics, which is a simple corollary of Theorem 7.1. 7.4 Let (M, g) be real analytic. Then R(x, λ) = o(λn−1 ) =⇒ |Lx | = 0. COROLLARY

Proof The set of vectors {ξ ∈ Sx∗ : expx ξ = x} is an analytic set. Hence, it has only countably many components. Since L ∗ is constant on smooth components, Lspx is countable for each x. Suppose now that |Lx | > 0 for some x. Then, as discussed in the proof of Theorem 5.1, there must exist T > 0 such that S0xT = T Sx∗ M. The time {T } is isolated in Lspx . Moreover, 0xT is necessarily a clean intersection since by homogeneity of exp T H p we have (exp T H p )Tx∗ M = Tx∗ M. Hence, the estimate follows from Theorem 7.1. 7.3. Almost clean We now generalize the result to almost-clean loopsets. In the argument above, we assumed that, for (T, x), U (T + s)(x, x) is a Lagrangean distribution in s, but much less is sufficient since we used only the principal symbol and we required only o(λn ) from the remainder estimate. Definition 7.5 We say that 0xT is an almost-clean loopset if 0xT ⊂ Tx∗ M is a (homogeneous) submanifold with boundary and if the intersection exp T H p Tx∗ M ∩ Tx∗ M is a clean in-

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tersection in the interior of 0xT . We say that (M, g) has almost-clean loopsets if this property holds for all T . Simple examples of metrics with an almost-clean loopset 0xT are provided by generic bumps on standard spheres. In other words, let Bx (r ) denote the ball of radius r around a point x ∈ (S n , g0 ) (the standard sphere), and let e2u g0 denote a conformal perturbation with supp u = Bx (r ). For points x in the exterior of the bump Bx (r ), 0x2π contains any direction that avoids the bump. For generic u it does not contain any other loops at time T = 2π. Hence, S0x2π is a submanifold with boundary of Sx∗ M with boundary the set of initial tangent vectors to geodesics which intersect ∂ Bx (r ) tangentially. More generally, we could put several bumps on the sphere and obtain a number of components. Instead of bumps, one could put handles and thus obtain any topology. This example is important in the proof of Theorem 1.6. Since 0xT could have several components, there is a possibility that cancellation between components can cause slow growth in U (T )E λ (x, x) even when |Lx | > 0. We adopt a reasonable hypothesis, easy to check in our main examples, which rules this out. 7.6 Suppose that {T } is an isolated point in Lspx of (M, g), that 0xT is an almostclean loopset, and that the (common) Morse indices of loops in all components of 0xT of dimension n − 1 are the same. Then R(λ, x) = o(λn−1 ) =⇒ |S0xT | = 0. Suppose that (M, g) has almost-clean loopsets and that, for all T , the (common) Morse indices of loops in all components of 0xT of dimension n − 1 are the same. Then R(λ, x) = o(λn−1 ) =⇒ |Lx | = 0. (55)

THEOREM

(i)

(ii)

Proof We prove only (i) since, as in the clean case, (ii) is an immediate consequence. Nothing changes from the proof in the clean case until (51). Since the clean composition hypothesis is not assumed to hold, U (T + s, x, x) is not known to be Lagrangean, although the singularity at s = 0 is still assumed to be isolated. We therefore cut the wave kernel up into pieces that can be controlled easily. The following argument is somewhat similar to the proof of [Z2, Th. 3.20]. We fix (T, x) and choose a cover S ∗ M adapted to S0xT as follows. By assumption, S0xT is a submanifold with boundary S ∗ M for all T . It therefore has only finitely many


423

connected components. We write S0xT =

nT [

Tj

Kx ∪

j=1

rT [

Tj

Nx ,

n=1

Tj

where {K x } are the components (possibly empty) of dimension n − 1 and where Tj {N x } are the components of dimensions less than or equal to n − 2. The Morse Tj indices of the loops in each component K x are the same; we are assuming the Morse Tj index is the same constant for all K x , as is the case in many natural examples. Tj Tj Tj Tj We then form the cover {int K x , Tε (∂ K x ), Tε (N x ), O }, where Tε (∂ K x ) is Tj the ε-tube (tubular neighborhood of radius ε) around the boundary of K x , where Tj Tj Tε (N x ) is the ε-tube around N x , and where O covers the rest of S ∗ M. We then define a partition of unity subordinate to the cover as follows: we dej Tj fine functions bε ∈ C0∞ (int K x ) which are invariant under exp T H p and idenTj Tj tically equal to 1 in points of int K x which lie outside of Tε (∂ K x ). We furj T j j T j ther define cε ∈ C0∞ (Tε (∂ K x )), dε ∈ C0∞ (Tε (N x )), and rε ∈ C0∞ (O ) so that Pn T j Pn T j PrT j j=1 bε + j=1 cε + j=1 dε + rε ≡ 1. We note that b, c, d, r also depend on T ; since it is fixed throughout the calculation, we do not indicate this in the notation. In a well-known way (see [Ho3]), we may quantize the symbols to obtain a pseudodifferential partition of unity: nT X

Bεj +

j=1

nT X

Cεj +

j=1

rT X

Dεj + Rε ∼ I.

j=1

We then have U (T )E λ (x, x) =

nT hX

Bεj U (T )E λ

j=1 rT hX

+

nT i hX i (x, x) + Cεj U (T )E λ (x, x) j=1

i

Dεj U (T )E λ (x, x) + Rε U (T )E λ (x, x).

j=1 j

Since Bε is microsupported in the set where 0xT is a clean intersection, it follows j that Bε U (T + s)](x, x) is a Lagrangean distribution for s near zero and its principal R j symbol of order n is given by (i m j S0 T bε dνx ), as in the clean case. Here, m j is the Tj

x

Morse index of loops in K x (which we assume to be independent of j). It follows by the standard trace asymptotics (see Lem. 2.5) that Z j Bε U (T )E λ (x, x) = i m j bεj dνx λn + O(λn−1 ). S0xT

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Also, Rε U (T + s, x, x) is smooth in s ∈ R since no loops at x occur in the microsupport of Rε . We conclude that U (T )E λ (x, x) =

nT Z X

bεj

nT hX i n dνx λ + Cεj U (T )E λ (x, x)

T j=1 S0x j=1 rT i hX Dεj U (T )E λ (x, x) + O(λn−1 ). + j=1

(56)

We estimate the two middle terms in the same way; the harder one is the C-term, so we leave the details of the D-terms to the reader. We have j C U (T )E λ (x, x) ε X = eiλν t φν (x)Cεj φν (x) λν ≤λ

≤

X λν ≤λ

∼ λn =O

|φν (x)||Cεj φν (x)|

sZ

q

≤

sX

|φν

(x)|2

λν ≤λ

sX

j

|Cε φν (x)|2

λν ≤λ

j

Sx∗ M

|cε |2 dνx

√ Tj Vol(Tε (∂ K x )) λn = O( ελn ).

(57)

In the above we used the fact that, for any zeroth order pseudodifferential operator, Z X |Cε φν (x)|2 = C E λ C ∗ (x, x) = |c|2 dνx λn + O(λn−1 ). Sx∗ M

λν ≤λ Tj

Since Tε (∂ K x ) is a tube around a hypersurface in S ∗ M, the integral is O(ε). Thus, by the same Fourier coefficient calculation as in the clean case, R(λ, x) = o(λn−1 ) implies that U (T )E λ (x, x) = o(λn ) for all T 6= 0. It follows for any ε > 0 that nT X j=1

i

mj

Z S0xT

√ bεj dνx = O( ε).

By the assumption that m j ≡ m for all j, it follows then that |S0xT | = 0. This completes the proof of (i). To prove (ii), it suffices to observe that Lspx is countable, and we have just seen that each S0xT has measure zero. Thus, Theorem 1.5 is proved.


425

7.4. Counterexamples: Proof of Theorem 1.6 We now present an example of a C ∞ - (but not analytic) surface (M, g) satisfying ∃x : |Lx | > 0,

|R(λ, x)| = (λ(n−1)/2 )

but

L ∞ (λ, g) = o(λ(n−1)/2 ).

Thus, neither |Lx | > 0 nor |R(λ, x)| = (λ(n−1)/2 ) is sufficient to imply maximal eigenfunction growth. Our example is of a surface of revolution, that is, a surface (M, g) whose metric is invariant under an action of S 1 by isometries. The two possible M are M = S 2 and M = T 2 (the two-torus). On M = S 2 we can construct metrics such that ∃x : |Lx | > 0, but no sequence of eigenfunctions has maximal growth at x; but the invariant eigenfunctions do have maximal growth at the poles. To obtain an example of a surface that does not have eigenfunctions of maximal growth, we instead choose M = T 2 . We often write M = S 1 × S 1 with the angle coordinates (x, θ ). We refer to the circle over which x varies as the base circle, and to the orbits as the fiber circle. Our torus of revolution is of the following kind. Definition 7.7 (T 2 , g) is called a torus of revolution with an equatorial band if in the √ standard coordinates (x, y) on T 2 we have ga = d x 2 + a(x)2 dθ 2 , where a = 1 − x 2 for x ∈ (−ε, ε). Such a surface may be obtained by revolving the graph y = eu(x) , 0 ≤ x ≤ 2π, of a 2π-periodic strictly positive function eu(x) around the x-axis, and then by joining the ends x = −1, x = 1 together. The equatorial band is the part Bε ∼ (ε, ε) × S 1 lying over angular interval (−ε, ε) (for some 0 < ε < 1/2), and it is isometric to an equatorial band lying over the same interval of the z-axis of the standard S 2 . Clearly, |Lx | = 2π for the open annulus of x ∈ Bε . We now construct u such that the eigenfunctions of (T 2 , gu ) fail to have maximal growth. We break up the base circle x ∈ S 1 into four parts: a flat part on [−1, −2ε] ∪ [2ε, 1], a spherical part [−ε, ε], and a “bridge” over (−2ε, −ε) ∪ (ε, 2ε) between them. We construct a metric that is invariant under the involution x → −x (i.e., ei x → e−i x ) as follows. Introduce C ∞ cutoff functions on S 1 satisfying ( 1, x ∈ [−ε, ε], χ1 (x) = 0, x ∈ [−1, −2ε] ∪ [2ε, 1], ( 0, x ∈ [−ε, ε], χ2 (x) = 1, x ∈ [−1, −2ε] ∪ [2ε, 1].

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We now define a class of profile curves of the following form: a f (x) =

p

1 − x 2 χ1 (x) +

1p 1 − ε2 χ2 (x) + f (x) 1 − χ1 (x) − χ2 (x) , 2 f (x) = f (−x), f ∈ C ∞ ([ε, 2ε]).

For concreteness, we construct f so that a f is √ monotonically decreasing from the boundary of√the round region (where a f (ε) = 1 − ε2 ) to the flat region (where a f ≡ (1/2) 1 − ε2 ). We begin with one such function f 0 and then consider small perturbations f = f 0 + εu. ˙ We denote the associated surface of revolution by 2 (T , g f ), and its Laplacian by 1 f . It is clear that only the values of f on ±([ε, 2ε]) matter. We equip the space of u˙ ∈ C ∞ ([ε, 2ε]) with the Whitney C ∞ -topology. Lmult(λ) Our first result involves the splitting of eigenspaces E λ = n=1 Vλ,n of 1 f0 into irreducibles for S 1 . Here, Vλ,n denotes the joint eigenspace of 1, ∂/∂θ. Because the eigenvalues of Sturm-Liouville operators on S 1 have multiplicity at most two, and dim Vλ,n = 2, the multiplicity of each Vλ in E λ is at most one. Therefore, mult(λ) counts the total number of distinct irreducibles in E λ . The following lemma shows that, by a generic small perturbation of f 0 , we can arrange that all of the eigenspaces are irreducible. We note that the same proof shows more generally that all eigenspaces are irreducible for generic surfaces of revolution. This result is analogous to the earlier result of S. Zelditch [Z1] that all eigenspaces are irreducible for generic G-invariant metrics on Riemannian d-manifolds when G is a finite group and d ≥ dG for some minimal dimension dG depending on G. We refer to [Z1] for background, references, and arguments that are valid when G = S 1 . LEMMA 7.8 Let Mirr denote the subspace of C ∞ ([ε, 2ε]) of functions f such that all eigenspaces of 1 f are irreducible for the S 1 -action. Then Mirr is residual in C ∞ ([ε, 2ε]). In particular, its intersection with any closed ball Bδ ( f 0 ) around f 0 in C ∞ ([ε, 2ε]) is residual in the ball.

Proof We need to prove that Mirr contains a countable intersection of open dense sets. As in [Z1], we denote by Mk the metrics for which all of the first k-eigenspaces are irreducible. It suffices to prove that each Mk is open dense. Since eigenvalues and eigenfunctions move continuously with perturbations, it is a simple fact based on the minimax characterization of eigenvalues that Mk is open in C ∞ ([ε, 2ε]) and hence that its intersection with any closed ball is a relatively open subset of the ball. We refer to [Z1] for the standard argument. The novel point to prove is the density of each Mk .


427

We now separate variables. For each n ∈ Z, we denote the joint eigenvalues and normalized eigenfunctions of 1, ∂/∂θ by {λ2n, j , 8n, j }. Thus, 8n, j (x, θ ) = einθ 8n, j (x, 0). We refer to such joint eigenfunctions as equivariant eigenfunctions. √ With no loss of generality, we may conjugate 1 by a to put it into Liouville normal form (i.e., transform it to the half-density Laplacian). The functions 8n, j (x, 0) are then complex eigenfunctions of the associated Sturm-Liouville operator Ln =

d2 n2 + + V, dx2 a(r )2

V = a −1/2

d −1 d −1/2 a a . dx dx

When n 6 = 0, we have 8n,λ (x, θ) = einθ u n,λ (x) + ivn,λ (x) , where u n,λ , vn,λ are the real eigenfunctions of L n . When n = 0, there is only a onedimensional space of invariant eigenfunctions. To prove that Mk is dense, it suffices to prove that, for each of the first keigenspaces, it is possible to find a perturbation that splits the eigenspace into irreducibles. Since only a finite number of irreducibles are involved, it suffices to do so for one eigenspace at a time and for one pair of irreducibles in its decomposition. The case where n = 0 for one of the eigenspaces is somewhat different, so we assume n, m 6 = 0 at first. We argue by contradiction. Suppose that, for a metric g0 , the eigenspace E λ contains at least two complex normalized eigenfunctions 8n,λ , 9m,λ with m, n 6= 0. We would like to find a curve of metrics which splits up their eigenvalues. It is clear that m 6 = n since the eigenvalues of Sturm-Liouville operators are at most double. We recall that the variation of the eigenvalue under a deformation e2u ε g0 is given by

˙ n,λ , 8n,λ i = (u1 λ˙ = h18 ˙ + 1u)8 ˙ n,λ , 8λ = 2λhu8 ˙ n,λ , 8n,λ i, ˙ is the variation of 1. The same calculation holds for 9m,λ . If the eigenvalue where 1 does not split under any deformation, then we have hu8 ˙ n,λ , 8n,λ i = hu9 ˙ m,λ , 9m,λ i,

∀u˙ ∈ C ∞ (ε, 2ε).

Since T

Z

|8n,λ (x, 0)|2 − |9m,λ (x, 0)|2 u˙ d x = 0,

∀u˙ ∈ C ∞ (ε, 2ε),

0

it follows that |8n,λ (x, θ)| = |9m,λ (x, θ)| in (ε, 2ε) × S 1 . It follows that on this region, 8n,λ (x, θ) = ei(n−m)θ eiα(x) 9m,λ (x, θ) for some real-valued function α. To analyze the situation, we write 8(x, θ) = einθ eρ(x)+iτ (x) .

428

SOGGE and ZELDITCH

Then 1einθ eρ(x)+iτ (x) = λeinθ eρ(x)+iτ (x) gives a0 n2 ρ 00 + iτ 00 + (ρ 0 )2 + 2iτ 0 ρ 0 − (τ 0 )2 + (ρ 0 + iτ 0 ) = λ + 2 . a a Again, separating into real and imaginary parts, this becomes ( a0 0 n2 2 00 0 2 0 2 < : a(x) 2 − λ = ρ + (ρ ) − (τ ) + a ρ , =:

0 = 2ρ 0 τ 0 + τ 00 (x) +

a 0 (x) 0 a(x) τ (x).

(58)

From the imaginary-part equation, we obtain τ 0 = eτ

0 (0)

(a −1 e−2ρ ) =⇒ τ = τ (0) + eτ

0 (0)

x

Z

(a −1 e−2ρ ) dy.

(59)

0

Thus, the phase τ is determined up to two constants τ (0), τ 0 (0) by the modulus eρ of 8n,λ . We further observe, by combining this with the real-part equation, that ρ solves ρ 00 + (ρ 0 )2 +

a0 0 n2 0 ρ − e2τ (0) (a −2 e−4ρ ) = − λ2 . a a(x)2

Suppose now that ρ1 = |8|, ρ2 = |9|. Subtracting their two equations gives d2 a0 d 0 0 d (ρ − ρ ) = (ρ + ρ ) (ρ − ρ ) + (ρ1 − ρ2 ) − a −2 e2τ (0) (e−4ρ1 − e−4ρ2 ). 1 2 1 2 1 2 dx a dx dx2 Here, we assumed that τ1 (0) = τ2 (0), which is no loss of generality since we may multiply 9 by a ei(τ1 (0)−τ2 (0)) to obtain a scalar multiple with this property. If we put f = ρ1 − ρ2 , then we obtain the inequality d2 a0 0 0 0 f − a −2 e2τ (0) f G(ρ1 , ρ2 ) f = ρ + ρ + 1 2 a dx2 =⇒ | f 00 (x)| ≤ A| f 0 (x)| + B| f (x)| for some constants A, B, namely, the supremums of (ρ10 + ρ20 + a 0 /a) and G, respectively. It follows then by Aronszajn’s unique continuation theorem that f = 0 on an interval implies f ≡ 0. Hence, we have |8(x)| = |9(x)| globally. Of course, this is no contradiction. There do exist metrics on T 2 possessing pairs of equivariant eigenfunctions with the same eigenvalue and same pointwise norm. Indeed, for the standard R2 /Z2 , one may take 8n,λ (x, θ) = ei(kx+nθ) , 9m,λ (x, θ ) = ei(`x+mθ) , where k 2 + n 2 = m 2 + `2 . We now assert that such pairs exist only for flat metrics of revolution on T 2 .


429

Indeed, suppose that 8 = F9. We also note that F(x, θ) = ei(n−m)θ F(x, 0). Write F(x, 0) = f (x), 9(x, 0) = ψ(x). Since | f (x)| = 1 also, there exist smooth functions ρ, τ, α such that ψ(x) = eρ(x)+iτ (x) ,

9(x, θ ) = eimθ ψ(x),

f (x) = eiα(x) ,

−λ2 F9 = 1(F9) = 91F + 2∇9 · ∇ F + F19 ⇐⇒ 91F + 2∇9 · ∇ F = 0 ⇐⇒ ∇(9 2 ∇ F) = 0. Simple calculations give (ψ(x)2 )0 0 a0 0 m 2 − n2 00 f (x) = 2 f (x) + f (x) + f a a(x)2 ψ(x)2 0 m 2 − n2 a 0 (x) 0 0 0 0 2 00 =⇒ = 2 ρ (x) + iτ (x) iα (x) − α (x) + iα (x) + i α (x). a(x) a(x)2

∇(9 2 ∇ F) = 0 =⇒

Breaking up into real and imaginary parts gives ( 2 2 −n 2 0 0 0 < : ma(x) 2 = −2τ (x)α (x) − α (x) , =:

0 = 2ρ 0 (x)α 0 (x) + α 00 (x) −

a 0 (x) 0 a(x) α (x).

The imaginary-part equation has the unique solution α 0 (x) = (n − m)a −1 (x)e−2ρ(x) , consistent with the given initial conditions. Substituting into the real-part equation gives m 2 − n 2 = −2(n − m)τ 0 (x)a −1 (x)e−2ρ(x) − (n − m)2 e−4ρ(x) .

(60)

Substituting (59) into (60) gives m 2 − n 2 = Ce−4ρ =⇒ ρ(x) = Cm,n

(61)

for some constant Cm,n . Substituting this into the real-part equation of (58) and substituting the value of τ 0 in (59) gives (C 2 − n 2 )a −2 = λ,

(62)

which implies that a(x) is constant. To complete the proof, we need to estimate eigenfunctions on our torus of revolution T with metric 2 ga = d x 2 + a(x) dθ 2 , −1 ≤ x ≤ 1, −π ≤ θ ≤ π, (63)

430

SOGGE and ZELDITCH

where a is a positive periodic function on R/Z. We show that for L 2 -normalized eigenfunctions of the form 8 = 8n,λ = einθ 8(x)

(64)

with eigenvalue λ, we have the bounds k8k L ∞ (T 2 ) ≤ Cλ1/2−σ

(65)

for some σ > 0 to be specified later. These bounds of course imply that for metric (63) we have L ∞ (λ, g) = o(λ1/2 ); however, as we sketch after the elementary proof, the bounds we obtain from this simple argument are far from optimal. LEMMA 7.9 For any bridge function f , the joint L 2 -normalized eigenfunctions {8n,λ } of 1 and of the S 1 -action on (T 2 , ga ) satisfy ||8n,λ ||∞ = o(λ1/2 ).

Proof To prove (65), fix an even function ρ ∈ C0∞ (R) with integral one and having the property that ρ(t) = 0 if |t| > 1. Then, if ρˆ is the Fourier transform of ρ, it follows that for a given δ > 0, √ 8 = χλδ 8 = ρˆ δ(λ − −1) 8. Note that χλδ

1 = 2π δ

Z

e−it (λ−

√ −1)

ρ(δ −1 t) dt.

(66)

(67)

Let χλδ (w, z), w, z ∈ T 2 , be the kernel of χλδ . Then by finite propagation speed √ for cos(t −1), it follows that χλδ (w, z) = 0

if dist(w, z) > 2δ

with dist(w, z) being the distance between w and z measured by the metric ga . Because of this, we can explain the role of δ in our estimates. First, if δ > 0 is small, the kernel is supported on small scales, and so after rescaling to the unit scale in local coordinates, it behaves increasingly like ones arising from a flat metric as δ → 0. This makes calculations in the proof easier; however, as δ → 0 one sees from (66) that χλδ looks less and less like an eigenspace projection operator, which reflects negatively on (65). Ideally, one would like to take δ to be, say, half the injectivity radius, but then the calculations that arise in the proof seem formidable.


431

To proceed, let us note that if δ is larger than λ1−α0 , where α0 > 0 is fixed, and if δ is smaller than half the injectivity radius, then we can write X ± χλ (w, z) = λ1/2 δ −3/2 (w, z)e±iλ dist(w,z) , aλ,δ ±

where the amplitudes satisfy the bounds γ

± |Dw,z aλ,δ (w, z)| ≤ Cγ δ −|γ |

with Cγ independent of δ. This follows from routine stationary phase calculations using (67). (The case δ ≈ 1 was carried out in [So3, pp. 138 – 140].) Also, the support properties of χλδ (w, z) imply that ± aλ,δ (w, z) = 0

if dist(w, z) > 2δ.

To proceed, let us fix w ∈ T 2 and use the natural coordinate (x, θ) corresponding to (63). In proving bounds for φ at w, in view of (64), we may assume that w has coordinates (x0 , 0). If (x, θ) are then the coordinates of z, it follows from (64) that XZ ± φ(x0 , 0) = K λ,n,δ (x0 , x)φ(x) d x, (68) ± ± where K λ,n,δ is given by the oscillatory integral Z ± ± 1/2 −3/2 K λ,n,δ (x0 , x) = λ δ e±iλ dist((x0 ,0), (x,θ )) einθ aλ,δ (x0 , 0, x, θ) dθ.

By applying the Schwarz inequality, we of course get 1/2 Z 1/2 XZ ± |8(x0 , 0)| ≤ |K λ,n,δ (x0 , x)|2 d x |8(x)|2 d x ±

≤C

XZ

± |K λ,n,δ (x0 , x)|2 d x

1/2

,

±

using in the last step the fact that d Vol = a(x) d x dθ with a(x) > 0 and the fact that 8(x, θ ) has L 2 -norm one. ± Because of this, our proof of (65) boils down to estimating the L 2 -norms of K λ,n,δ and then optimizing in δ. To do this, note that the integrand of the oscillatory integral ± defining K λ,n,δ vanishes unless |θ| + |x0 − x| ≤ Cδ. Therefore, it is natural to make the change of variables θ → δθ, x → x0 + δy, which results in λ 1/2 Z ± ± K λ,n,δ (x0 , x0 + δy) = e±iλδ9δ (x0 ;y,θ ) aλ,δ (x0 , 0; x0 + δy, δθ) dθ, (69) δ

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SOGGE and ZELDITCH

where 9δ = δ −1 dist (x0 , 0), (x0 + δy, δθ) + nδθ. If the warping factor a(x) for the metric (63) were constant in a 2δ-neighborhood of x0 , then one would have (∂/∂θ )2 9δ = −(a(x0 )y)2 /|y 2 + (a(x0 )θ)2 |3/2 , which results in a lower bound of (a(x0 )y)2 for the Hessian of the phase function in this case. For nonconstant a one has q 2 δ −1 dist (x0 , 0), (x0 + δy, δθ) = y 2 + a(x0 )θ + rδ (x0 , y, θ) on the support of the oscillatory integral, where for a given j = 0, 1, 2, . . . , ∂ j rδ ≤ Cδ. ∂θ Consequently, there are uniform constants c0 > 0 and C0 < ∞ so that ∂ 2 9δ ≥ c0 y 2 ∂θ

if y 2 ≥ C0 δ.

Since the amplitude in (69) has uniformly bounded derivatives, we conclude from the stationary phase that λ,n,δ (x 0 , x 0

± K

λ 1/2 + δy) ≤ C (y 2 δλ)−1/2 = Cδ −1 |y| if |y| ≥ C0 δ −1/2 . δ

± One also has the trivial bound |K λ,n,δ | ≤ C(λ/δ)1/2 . Using these two bounds, we can ± compute the L 2 -norm of the kernel K λ,n with respect to the original coordinates:

Z 1/2 2 1/2 1/2 dx K ± (x0 , x0 + δy) 2 dy (x , x) = δ 0 λ,n,δ λ,n,δ

Z K ±

≤ Cδ

−1/2

≤ Cδ

−3/4

Z

+∞ δ 1/2

y

−2

dy

1/2

1/2

+ Cλ

δ 1/2

Z

dy

1/2

0

+ Cλ1/2 δ 1/4 .

One optimizes this by choosing δ = λ−1/2 , which implies that one has (65) with σ = 1/8. Completion of the proof of Theorem 1.6 Two of the three properties follow from Corollary 7.4 and from the fact that |Lx | > 0 within the band. For the third property, we need to prove the existence of a point x0 such that R(λ, x0 ) = (λ(n−1)/2 ). For this we appeal to Theorem 7.6(i). Let x be a point on the equator of the round part. Then 0x2π has an almost-clean contribution coming from the great circles in the equatorial band. We claim that by choosing the


433

flat part to be long enough there, 0x2π contains no other loops. Indeed, any other loop would have to enter the flat part. It cannot reverse direction, so it must travel around the flat part before re-entering the spherical part. As long as the flat part is at least 2π units in length, any such geodesic loop at x does not belong to 0x2π . We believe that the correct bound should be λ1/4 . We discuss this further below. 8. Further problems and conjectures We conclude with some related open problems, questions, and speculations. 1 What is a sufficient condition for maximal eigenfunction growth of (M, g)? PROBLEM

As mentioned above, |Lx | > 0 is not sufficient to imply the existence of a sequence of eigenfunctions blowing up at x at the maximal rate. In fact, to our knowledge, the only Riemannian manifolds known to exhibit maximal eigenfunction growth are surfaces of revolution and compact rank one symmetric spaces. In the case of surfaces of revolution, invariant eigenfunctions must blow up at the poles because all other eigenfunctions vanish there and yet the local Weyl law (4) must hold. In the case of compact rank one symmetric spaces, the exceptionally high multiplicity of eigenspaces allows for the construction of eigenfunctions of maximal sup-norm growth. In each case, Lx has full measure, but the mechanisms producing maximal eigenfunction growth involve something more. All of these examples are completely integrable, and eigenfunctions with maximal sup-norms may be explicitly constructed by the Wentzel-Kramers-Brillouin (WKB) method. Let us describe the symplectic geometry underlying these examples because it is very similar to the situation we found in the real analytic case. The eigenfunctions with maximal sup-norms in these cases are actually semiclassical (equal to nonhomogeneous) Lagrangian distributions (equal to oscillatory integrals) associated to certain (geodesic flow) invariant Lagrangian submanifolds diffeomorphic to S 1 × Sx∗ M ∼ S 1 × S n−1 . They are the images of S 1 × Sm∗ M under the Lagrange immersion ι : S 1 × Sm∗ M → T ∗ M\0,

ι(t, x, ξ ) = G t (x, ξ ),

where G t : T ∗ M\0 → T ∗ M\0 is the geodesic flow. Under the natural projection π : S ∗ M → M and its restriction π : ι(S 1 × Sm∗ M) → M to the Lagrangian, the sphere Sm∗ M is “blown down” to m. As discussed in [TZ1] and [TZ2] (among other places), singularities of projections of Lagrangean submanifolds cause sup-norm blow-up of the associated quasimodes; blow-down singularities cause maximal sup-norm blowup.

434

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In the real analytic case with maximal eigenfunction growth, there also exists a point m such that all geodesics leaving m return to m at a fixed time `. If they are smoothly closed curves at m, then as above these geodesics fill out a closed, embedded Lagrangean submanifold of the form ι(S 1 × Sm∗ M) above. A rather naive question is whether the sequence {φ j } of eigenfunctions which “blows up” at the maximal rate at m is a semiclassical (i.e., nonhomogeneous) Lagrangian distribution (or oscillatory integral) with wave front on the ι(S 1 × Sm∗ M). It seems quite wild to conjecture that maximally growing eigenfunctions should be Lagrangian distributions, but maximal growth seems so unusual a condition that the two situations may coincide. A related question is whether there at least exist high-order quasimodes associated to ι(S 1 × Sm∗ M). (High-order means solving the eigenvalue equation to high order in λ−1 j .) Such quasimodes would have to blow up at the maximal rate at m, and since the eigenfunctions also do, one might expect a strong relation between √ them. As a preliminary issue, we ask what kind of sequences of eigenvalues λ j of 1 correspond to eigenfunctions with maximal growth. Here, we considered only the case where maximal eigenfunction growth occurs at m and where the geodesics at m close up smoothly. There may also exist cases where maximal eigenfunction growth occurs at a point m but where loops are not smoothly closed, that is, where the “first return map” on unit tangent vectors is not the identity. In this case, we have an immersed Lagrangean manifold with boundary ι([0, L] × Sm∗ M). The umbilic points of the triaxial ellipsoid are an example of the latter, but we conjecture (with John Toth) that maximal eigenfunction growth does not occur at these umbilic points. Are there any examples where maximal eigenfunction growth occurs but where geodesic loops are not smoothly closed? Without the assumption of maximal eigenfunction growth, we do not expect such high-order quasimodes to exist. Nor do we expect maximal (Laplace) eigenfunction growth in all situations where Lx has full measure, even at all points x. For instance, we do not expect maximal eigenfunction growth on all Zoll manifolds (manifolds all of whose geodesics are closed), even though the converse estimate R(λ, x) = (λ(n−1)/2 ) holds and even though embedded Lagrangian manifolds of the type ι(S 1 × Sm∗ M) exist. Associated to these Lagrangians are quasimodes of order zero for the Laplacian. They are actual eigenfunctions for a related positive elliptic pseudodifferential operator A (we refer again to [CV] for its definition) and have maximal growth (as eigenfunctions of A). Yet they are far from actual Laplace eigefunctions, and we doubt that maximal eigenfunction growth is common among (nonrotational) Zoll surfaces. To explain this speculation, we recall the result (see [V]) that on the p standard S 2 , almost every orthonormal basis of eigenfunctions satisfies ||φλ || = O( log λ ), even though special eigenfunctions (zonal spherical harmonics) have maximal eigen-


435

function growth. It is possible that eigenfunctions of typical Zoll surfaces resemble such typical bases of spherical harmonics rather than the special ones with extremal eigenfunction growth. We may imagine analytically deforming the standard S 2 into a nonstandard (nonrotational) Zoll S 2 . There is an orthonormal basis of eigenfunctions spherical harmonics which moves analytically with the deformation. p Is it a basis with maximal eigenfunction growth or a more generic basis with O( log λ ) growth in general? 2 Characterize (M, g) with maximal L p -norms of eigenfunctions. PROBLEM

We have left this problem open for p ≤ 2(n + 1)/(n − 1), and we expect the condition on (M, g) to change at the critical Lebesgue exponent p = 2(n + 1)/(n − 1). Indeed, as C. Sogge proved in [So2], the “geometry” of extremal eigenfunctions changes at this exponent. Specifically (cf. [So3, pp. 142 – 144], [So1]), for p > 2(n + 1)/(n − 1), eigenfunctions concentrated near a point tend to have extreme L p -norms, while for 2 < p < 2(n + 1)/(n − 1), ones concentrated along stable closed geodesics tend to have this property; if p = 2(n + 1)/(n − 1), at least in the case of the round sphere, both types give rise to (λδ( p) )-bounds. J. Bourgain [B] has constructed a metric on a 2-torus of revolution for which the maximal L 6 -bound is attained, although |Lx | = 0 for all x. The eigenfunctions are similar to the highest-weight spherical harmonics on S 2 which concentrate on the equator. Note that this is the Lebesgue exponent where one expects the behavior of eigenfunctions that maximize this quotient to change. Thus, we do not expect “|Lx | > 0” to be a relevant mechanism in producing large L p -norms below the critical exponent. Possibly the existence of stable elliptic closed geodesics is involved. We plan to study the problem elsewhere. PROBLEM 3 At the opposite extreme, characterize compact Riemannian manifolds (M, g) with minimal eigenfunction growth, such as occurs on a flat torus.

In [TZ1] it is proved that in the (quantum) integrable case, the only (M, g) with bounded multiplicities and uniformly bounded eigenfunctions are flat tori and their quotients. In fact, in [TZ2] a certain minimal growth rate on sup-norms is given in nonflat cases. It would be desirable to remove the condition that the geodesic flow is completely integrable. We conjecture (with John Toth) that KAM systems have a certain minimal growth rate of eigenfunctions. But the result should be true much more generally.

436

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Acknowledgments. It is a pleasure to thank R. Hardt, W. Minicozzi, and B. Shiffman for very helpful conversations on analytic sets and for directing us to [N]. We are particularly grateful to V. Guillemin, A. Laptev, and D. Robert for the reference to Yu. Safarov’s work [S]. We also thank J. Toth for bringing the ellipsoid example to our attention. References [A]

V. I. ARNOLD, Mathematical Methods of Classical Mechanics, 2d ed., Grad. Texts in

[Av]

¨ V. G. AVAKUMOVIC´ , Uber die Eigenfunktionen auf geschlossenen Riemannschen

[Be]

´ P. H. BERARD , On the wave equation on a compact Riemannian manifold without

[BB]

´ L. BERARD-BERGERY , Quelques exemples de variétés riemanniennes où toutes les

Math. 60, Springer, New York, 1998. MR 96c:70001 391, 415 Mannigfaltigkeiten, Math. Z. 65 (1956), 327 – 344. MR 18:316a 388, 398 conjugate points, Math. Z. 155 (1977), 249 – 276. MR 56:13295 391, 413

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[Bl] [B] [CV]

[CVV]

[D] [DG]

[H]

[Ho1] [Ho2]

géodésiques issues d’un point sont fermées et de même longueur, suivis de quelques résultats sur leur topologie, Ann. Inst. Fourier (Grenoble) 27 (1977), 231 – 249. MR 55:9124 390 A. L. BESSE, Manifolds All of Whose Geodesics Are Closed, with appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger, and J. L. Kazdan, Ergeb. Math. Grenzgeb. 93, Springer, Berlin, 1978. MR 80c:53044 390, 415, 416 P. M. BLEHER, Distribution of energy levels of a quantum free particle on a surface of revolution, Duke Math J. 74 (1994), 45 – 93. MR 96f:58169 413, 414 J. BOURGAIN, Eigenfunction bounds for compact manifolds with integrable geodesic ´ flow, preprint, Institut des Hautes Etudes Scientifiques, 1993. 435 ` Y. COLIN DE VERDIERE, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent, II: Le cas intégrable, Math. Z. 171 (1980), 51 – 73. MR 81i:58046 413, 416, 434 ` Y. COLIN DE VERDIERE and S. VU˜ NGO.C, Singular Bohr-Sommerfeld rules for 2D integrable systems, preprint, 2000, http://www-fourier.ujf-grenoble.fr/PREP/html/a508/a508.html 391, 415 J. J. DUISTERMAAT, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687 – 706. MR 82d:58029 J. J. DUISTERMAAT and V. W. GUILLEMIN, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39 – 79. MR 53:9307 390, 391, 392, 394, 396, 398, 407, 420 R. M. HARDT, “Stratification via corank one projections” in Singularities (Arcata, Calif., 1981), Part 1, Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, 1983, 559 – 566. MR 85e:32016 416 ¨ L. HORMANDER , The spectral function of an elliptic operator, Acta Math. 121 (1968), 193 – 218. MR 58:29418 388, 398, 407 , The Analysis of Linear Partial Differential Operators, III: Pseudodifferential


[Ho3]

[Iv1]

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[Ka] [K] [KMS]

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[M] [N] [S]

[SV]

[Sa] [So1]

437

Operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985. MR 87d:35002a 397 , The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren Math. Wiss. 275, Springer, Berlin, 1985. MR 87d:35002b 392, 394, 395, 396, 398, 399, 400, 401, 420, 423 V. JA. IVRI˘I, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary (in Russian), Funktsional. Anal. i Prilozhen. 14, no. 2 (1980), 25 – 34. MR 82m:58057 390, 392, 395, 396, 397, 401, 407 , “Semiclassical spectral asymptotics and multiparticle quantum theory” in Partial Differential Operators and Mathematical Physics (Holzhau, Germany, 1994), Oper. Theory Adv. Appl. 78, Birkhäuser, Basel, 1995, 199 – 212. MR 97e:81177 390, 400 A. KARNAUKH, Spectral count on compact negatively curved surfaces, Ph.D. dissertation, Princeton University, Princeton, 1995. 392 W. KLINGENBERG, Riemannian Geometry, de Gruyter Stud. Math. 1, de Gruyter, Berlin, 1982. MR 84j:53001 391, 415 D. V. KOSYGIN, A. A. MINASOV, and YA. G. SINA˘I, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces (in Russian), Uspekhi Mat. Nauk 48, no. 4 (292) (1993), 3 – 130; English translation in Russian Math. Surveys 48, no. 4 (1993), 1 – 142. MR 95e:58174 413, 415 B. M. LEVITAN, On the asymptoptic behavior of the spectral function of a self-adjoint differential equation of the second order, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325 – 352. MR 15:315e 388, 398 J. MILNOR, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud. 51, Princeton Univ. Press, Princeton, 1963. MR 29:634 411 R. NARASIMHAN, Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25, Springer, Berlin, 1966. MR 36:428 416, 436 YU. G. SAFAROV, Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition (in Russian), Funktsional. Anal. i Prilozhen 22, no. 3 (1988), 53 – 65; English translation in Funct. Anal. Appl. 22 (1988), 213 – 223. MR 89i:35108 390, 436 YU. SAFAROV and D. VASSILIEV, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Transl. Math. Monogr. 155, Amer. Math. Soc., Providence, 1997. MR 97h:58163 390 P. SARNAK, in preparation. 389, 413 C. D. SOGGE, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43 – 65. MR 87g:42026 392, 407, 435

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, Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123 – 138. MR 89d:35131 389, 435 , Fourier Integrals in Classical Analysis, Cambridge Tracts in Math. 105, Cambridge Univ. Press, Cambridge, 1993. MR 94c:35178 398, 407, 431, 435 M. E. TAYLOR, The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions, Comm. Partial Differential Equations 27 (2002), 565 – 605. 389 J. A. TOTH AND S. ZELDITCH, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111 (2002), 97 – 132. CMP 1 876 442 413, 433, 435 , L p -norms of eigenfunctions in the completely integrable case, preprint, 2001. 413, 433, 435 J. M. VANDERKAM, L ∞ norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices 1997, 329 – 347, MR 99d:58175; Correction, Internat. Math. Res. Notices 1998, 65. MR 99d:58176 391, 434 A. V. VOLOVOY, Improved two-term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold, Comm. Partial Differential Equations 15 (1990), 1509 – 1563. MR 91m:58158 S. ZELDITCH, On the generic spectrum of a Riemannian cover, Ann. Inst. Fourier (Grenoble) 40 (1990), 407 – 442. MR 91g:58294 426 , Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992), 221 – 260. MR 93e:58184 391, 398, 420, 421, 422 , The inverse spectral problem for surfaces of revolution, J. Differential Geom. 49 (1998), 207 – 264. MR 99k:58188 413, 414

Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218, USA; [email protected]; [email protected]


KRONECKER-WEBER PLUS EPSILON GREG W. ANDERSON

Abstract We say that a group is almost abelian if every commutator is central and squares to the identity. Now let G be the Galois group of the algebraic closure of the field Q of rational numbers in the field C of complex numbers. Let G ab+ be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups, and let Qab+ /Q be the corresponding Galois extension. We prove that Qab+ is generated by the roots of unity, the fourth roots of the rational primes, and the square roots of certain algebraic sine-monomials. The inspiration for the paper came from recent studies of algebraic 0-monomials by P. Das and by S. Seo. 1. Introduction We say that a group is almost abelian if every commutator is central and squares to the identity. Let G be the Galois group of the algebraic closure of the field of rational numbers Q in the field C of complex numbers. Let G ab+ be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups. Let G be the kernel of the natural map of G ab+ to the abelianization G ab of G. By construction, the group G is central in G ab+ and killed by 2. Let Qab (resp., Qab+ ) be the Galois extension of Q in C with Galois group G ab (resp., G ab+ ). The Kronecker-Weber theorem determines the structure of the group G ab and provides an explicit description of the field Qab . The theory of [F] in principle determines the structure of the group G ab+ but does not provide an explicit description of the field Qab+ . Kummer theory identifies the Pontryagin dual of G with H 0 (G ab , Qab× /Qab×2 ). Our purpose in this paper is to exhibit for the latter group an explicit (Z/2Z)-basis, thereby obtaining a description of the field Qab+ as explicit as that provided for the field Qab by the Kronecker-Weber theorem. Our method is more or less elementary and independent of the theory of [F]. The inspiration for our work came from the recent studies [D] and [S] of algebraic 0-monomials. Our main results are as follows. Let A be the free abelian group on symbols of DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 3, Received 14 March 2001. Revision received 7 September 2001. 2000 Mathematics Subject Classification. Primary 11R20; Secondary 11R32, 11R34, 11R37.

439

440

GREG W. ANDERSON

the form (a ∈ Q),

[a] modulo the identifications

[a] = [b] ⇔ a − b ∈ Z. For all prime numbers p < q, if 2 < p, put a pq :=

( p−1)/2 X h i=1

(q−1)/2 (q−1)/2 X h i X h j i ( p−1)/2 X h j ii ki ì − + − − + , p pq q q pq p k=0

`=0

j=1

for example, a3·5 =

h1i 3

+

h 2 i h 4 i h1i − − , 15 15 5

and if 2 = p, put

a pq :=

h i 1 4

−

(q−1)/2 X h k=0

−

1 ki + 4q q

(q−1)/2 X h j=1

ji h 1 ji h j i h 1 j i + − + − − − + , q 2q q 2q 4q 2q

for example, a2·3 :=

h1i 4

−

h 5 i h1i − . 12 3

Let sin : A → Qab× be the unique homomorphism such that ( 2 sin πa (= |1 − e2πia |) if 0 < a < 1, sin[a] = 1 if a = 0

a ∈ Q ∩ [0, 1) .

We prove that the family of real numbers √ ` `:prime ∪ {sin a pq } p,q:prime p 1, Pkν is the kth Gegenbauer polynomial of order ν, and λk,d =

0(k + (n + d)/2) , 0(k + (n − d)/2)

k ≥ 0.

(n−1)/2

(12)

Remark. For n = 1 the quantity (n − 1)−1 Pk (ξ · η) should be replaced by k −1 cos(kφ), where φ is the angle between ξ and η. This is because Pkν (cos φ) ∼ 2νk −1 cos(kφ) as ν → 0.

486

CARLO MORPURGO

For 0 < d < n, (11) shows that Ad has eigenfunctions the spherical harmonics and eigenvalues λk,d , with multiplicity (2k + n − 1)(k + n − 2)!/((n − 1)!k!) if n > 1, and 2 if n = 1. This is also true for higher d > 0 by analytic continuation. (Note that if d − n = 2m with m = 0, 1, 2 . . . , then λk,d = 0 when k = 0, 1, 2, . . . , m.) Thus, (10) follows since the eigenvalues of B are {k + (n − 1)/2}k≥0 . As particular cases, we note that A1 = B and A2 = 1 S n + (n/2)(n/2 − 1), the conformal Laplacian of S n in the round metric. Extension of Ad to the conformal class of the round metric of S n Just as in the case of Rn , we can now define conformally covariant pseudodifferential operators of the sphere, which naturally extend the operator Ad . To each metric of the form W 2/d g S n with 0 < W ∈ C ∞ (S n ) we associate the operators Ad (W ) := W −(n+d)/(2d) Ad W (n−d)/(2d)

acting on L 2 (S n , W n/d d x).

In this notation, Ad (1) = Ad . The Ad (w) are nonnegative (positive if 0 < d < n), elliptic, self-adjoint, pseudodifferential operators of order d, with the same leading d/2 symbol as that of w−d/n 1 S n (Section 4). They are natural in the sense that their definition is compatible with the action of the conformal group of S n (intertwining property). By the general results of [GJMS], when d is an even integer, one can in fact extend Ad to differential operators Ad (g) defined universally in terms of any given smooth metric on the sphere. A special case is, once again, the conformal Laplacian on (S n , g), for d = 2 (for more general d, see [P]). Spectrum of intertwiners and Birman-Schwinger operators By known theorems the operators Ad (W ) defined on the smooth compact Riemannian manifold (S n , W 2/d g S n ) have a discrete set of eigenvalues λ0 (W ) ≤ λ1 (W ) ≤ · · · ↑ ∞. These eigenvalues are of course the inverses of the eigenvalues of the compact −1 n operator A−1 d (W ). On R , the operators 1d (w) are in general not compact, except under certain integrability conditions on w. At this point we notice that, as far as spectral properties are concerned, one might as well have considered the two families of operators, W −1/2 Ad W −1/2 acting on the fixed space L 2 (S n , dξ ) and w−1/2 1d/2 w−1/2 acting on L 2 (Rn , d x), which indeed have the same eigenvalues as the operators Ad (W ) and 1d (w) acting on the corresponding weighted L 2 -spaces. This is in fact the point of view we adopt for the most part in this paper, except when it is perhaps more enlightening to interpret our results in terms of extremal properties satisfied by metrics in a given conformal class. The operator w1/2 1−d/2 w1/2 for d = 2 is known by mathematical physicists as the Birman-Schwinger operator and is a key tool in the problem of estimating

SHARP INEQUALITIES, FUNCTIONAL INTEGRALS, TRACES

487

moments of eigenvalues and the number of bound states of Schrödinger operators (see [S]). One of the features of the eigenvalues of Ad (W ), 1d (w) is their invariance under conformal action. On S n , for example, given a conformal transformation τ ∈ CON(S n ), we have τ ∗ (W 2/d g S n ) = (Wτ )2/d g S n , where Wτ = (W ◦ τ )|Jτ |d/n is the conformal action. The map W → Wτ simply translates the metric W 2/d g S n into an isometric metric, and hence it is natural to expect that it preserves the spectrum. Similar arguments of course hold for Rn . PROPOSITION 2.3 Let 0 < d < n. For 0 ≤ w ∈ L n/d (Rn , d x) the operator w1/2 1−d/2 w1/2 is compact, self-adjoint, and positive on L 2 (Rn , d x); denote its eigenfunctions by {φkw } and its eigenvalues by {µk (w)}, with µ0 (w) ≥ µ1 (w) ≥ · · · ↓ 0. If wh = (w ◦ h)|Jh |d/n with h ∈ CON(Rn ), then

µk (w) = µk (wh ),

φkwh = (φkw ◦ h)|Jh |1/2 .

(13)

If in addition w > 0, then the operator 1d (w)−1 , acting on L 2 (Rn , wn/d d x), is also compact, self-adjoint, and positive, and has eigenvalues {µk (w)}. 1/2 is compact, self-adjoint, For 0 ≤ W ∈ L n/d (S n , dξ ) the operator W 1/2 A−1 d W 2 n and positive on L (S , dξ ); denote its eigenfunctions by {8kW } and its eigenvalues by {νk (W )}, with ν0 (W ) ≥ ν1 (W ) ≥ · · · ↓ 0. If Wτ = (W ◦ τ )|Jτ |d/n with τ ∈ CON(S n ), then νk (W ) = νk (Wτ ),

8kWτ = (8kW ◦ τ )|Jτ |1/2 .

(14)

If in addition W > 0, then the operator Ad (W )−1 , acting in L 2 (S n , W n/d dξ ), is also compact, self-adjoint, and positive, and has eigenvalues {νk (W )}. If w = Wπ = (W ◦ π)|Jπ |d/n |, then µk (w) = νk (W ),

φkw = (8kW ◦ π)|Jπ |1/2 .

(15)

Proof The compactness part follows from results of M. Cwikel [Cw], who showed that 0 operators with kernels f (x − y)g(y) with g ∈ L p and f ∈ Weak L p are compact and bounded in L 2 , where p > 2. Taking f (x) = Nd/2 (x) and g = w1/2 with p = 2n/d gives that the operator 1−d/4 w1/2 is compact and therefore so is its adjoint and w1/2 1−d/2 w1/2 . The remaining statements follow in a straightforward way from the intertwining property and the definition of Ad . Details are left to the reader.

488

CARLO MORPURGO

Proposition 2.3, in particular (15), is repeatedly used to transfer conformally invariant objects from Rn to S n , and vice versa. Zeta functions and 9-functions We recall a few facts about traces. Let X be a locally compact Hausdorff space with a Baire measure µ. If A is a positive, compact, and self-adjoint operator on L 2 (X, dµ), P∞ P then it is trace class if Tr[A] = ∞ k=0 (Aφk , φk ) = k=0 µk (A) < ∞, where the µk (A) are the eigenvalues of A and {φk } is any orthonormal basis of L 2 . Moreover, if R A is positive and has a continuous kernel K (x, y), then Tr[A] = X K (x, x) dµ(x) < ∞. The zeta function of the operator Ad (W ) is defined for 0 < d < n and 0 ≤ W ∈ n/d L (S n ) as ∞ X 1/2 s Z s, Ad (W ) = Tr (W 1/2 A−1 W ) λ j (W )−s , = d

Re s >

j=0

n , d

where λ j (W ) are the (positive) eigenvalues of Ad (W ). If W is smooth, the zeta function is finite for Re s > n/d and can be extended to a meromorphic function on the complex plane (see Section 5 for more details on this). For general W the finiteness for Re s > n/d follows from Theorem 4.1. More generally, for 9 : R+ → C, consider the transformation Z ∞ 9(t) dt, (16) L9 (u) = e−t/u t 0 provided the above integral is finite, and define the 9-function of Ad (W ) as ∞ X 1/2 Z 9 Ad (W ) = Tr L9 (W 1/2 A−1 W ) = L 9 d j=0

1 . λ j (W )

Note that if 9(t) = t s / 0(s), then L9 (u) = u s , and Z 9 (Ad (W )) = Z (s, Ad (W )). The definition above makes sense, provided that L9 (λ0 (W )−1 ) < ∞. This is automatic for the zeta function, but for general 9 it must be assumed. However, from the sharp zeta function inequality in Corollary 4.2 we deduce that λ0 (1) = λ0,d ≤ λ0 (W ), so that it is enough to assume L9 (λ−1 0,d ) < ∞; this condition is always implicitly assumed throughout the rest of the paper. The following Lemma gives a simple condition for the finiteness of Z 9 (Ad ). 2.4 If 9 ≥ 0, then Z 9 (Ad ) < ∞ if and only if Z 1 9(t) t −n/d dt < ∞. t 0 LEMMA


489

Proof The eigenvalues of Ad , counted with their multiplicities, are asymptotic to Ck −d/n , as k → ∞, for some constant C (see, e.g., [Gru, (4.5.6) and references]; it can also be verified directly using (12) and the formula for the multiplicity). P∞ d/n ) < ∞ for some conThus, Tr L9 (A−1 1 L9 (ck d ) < ∞ if and only if stant c > 0, which holds if and only if Z ∞ Z ∞ 9(t) d/n dt n/d e−t x d x < ∞. t 0 λ0,d It is straightforward to check that this implies the statement. Symmetric stable processes, functional integrals, and trace formulas We end this section by giving the main formulas for the zeta and 9-functions in terms of explicit integrals. In the case d ≤ 2 these are essentially versions of the so-called Lieb formula and require the use of path integration. We now recall some basic terminology and results from the theory of Markov processes; the reader is referred to [EK, Chaps. 1, 4] or [DvC, App. C] for more details. For a given locally compact and separable metric space M, we let M∂ = M ∪∂ be its one-point compactification; if M is compact, ∂ is just an isolated point; the point at infinity ∂ is sometimes called the cemetery point. Denote the sample path space, or Skorohod space, by D , that is, the space of functions B : [0, ∞) → M∂ , which are cadlag, that is, right-continuous and with left limits, and which are also such that if B(t) = ∂ or B(t− ) = ∂, then B(s) = ∂ for s ≥ t. Next, denote by C(M∂ ) the Banach space of continuous functions with the sup norm on M and by C0 (M∂ ) the space of continuous functions vanishing at ∂ (or vanishing at infinity). This space is identified with C0 (M), the space of continuous functions on M, so that for all > 0 there is a compact set K ⊆ M with | f (x)| ≤ for all x ∈ / K . Suppose that T (t) is a semigroup on C0 (M), which for simplicity we assume to be given by a positive transition function K (t, x, y): T (t) f = R M K (t, x, y) f (y) dy. T (t) is called a Feller semigroup if it is aRstrongly continuous, positivity-preserving, contraction semigroup on C0 (M), with K (t, x, y) dy ≤ 1. By a known result, given a Borel probability measure ν on M and a Feller semigroup T (t) with transition function K (t, x, y), there exists a Markov process B with transition function K (t, x, y), initial distribution ν, and sample paths in D . A posteriori, by Kolmogorov’s theorem, there exists a probability measure Pν on the measure space (D , F0 ), where F0 is the sigma algebra generated by the coordinate functions on D , so that Pν has the same finite-dimensional distributions as B. In what follows we are only interested in the case ν = δx , the Dirac measure at x ∈ M, and we denote by Px the associated probability measure.

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CARLO MORPURGO

One also defines the conditional (or pinned) measure on D for processes starting at x at time s and ending at y at time t. To do this (see [Ok] for details), one defines the time-inhomogeneous transition function K y,t (s, u, x, z) =

1 K (t − u, z, y)K (u − s, x, z) K (t − s, x, y)

(17)

and then shows that to this transition density there corresponds a probability measure Px,y,t,s on the measure space (D , Fts ), where Fts is the sigma algebra generated by the coordinate functions Bv , v ∈ [s, t] (see [Ok] for a careful construction). Such a measure satisfies Px,y,t,s (Bt = y) = 1. In the following we are considering only the measure Px,y,t := Px,y,t,0 , corresponding to the process which starts at x at time zero and ends at y at time t. Note that the cemetery point ∂ plays no role for the pinned measure. We also let Ft ⊇ Ft0 be a filtration that is complete with respect to the pinned measure (see [Ok, proof of Prop. 4.2]) and right continuous. This is needed in order to guarantee the measurability of first exit times and penetration times used in Section 3. In the particular case M = Rn , the operator 1d/2 is a generator of a Feller semigroup when 0 < d ≤ 2, and the corresponding process is called the d-symmetric d/2 stable Lévy process. The kernel of such a semigroup is e−t1 (x, y) = E d (t, |x − y|), where Z d E d (t, |x|) = e−2πi x·z e−t|z| dz = t −n/d E d (1, |x|t −1/d ). (18) Rn

The main point is that E d (t, |x|) is a positive, radially symmetric, and decreasing function when 0 < d ≤ 2, by the subordination principle (see the proof of Proposition 2.5). The case d = 2 corresponds to the standard Brownian motion on Rn , with continuous paths, and associated Wiener measure Px . On the sphere we have a parallel situation for the operator Ad , 0 < d ≤ 2, which is the generator of the semigroup e−t Ad with kernel e−t Ad (ξ, η) = K d (t, ξ, η), where ∞

K d (t, ξ, η) =

X 1 (n−1)/2 (2k + n − 1)e−tλk,d Pk (ξ · η). n (n − 1)|S |

(19)

k=0

Below we show that this semigroup is positivity preserving; this fact is not so obvious since the intertwining property, which relates Ad and 1d/2 in an easy way, does not carry through for the corresponding heat semigroups. Note that in this case R K d (t, ξ, η) dη < 1, so that the resulting process has finite lifetime. PROPOSITION 2.5 The semigroup generated by Ad is positivity preserving when 0 < d ≤ 2. Hence, Ad generates a Feller semigroup.


491

Proof The semigroup generated by the operator B 2 = 1 S n + ((n − 1)/2)2 is clearly positivity preserving. Using (10), it is enough to show that the function f (x) = e−H (x) is completely monotonic in [0, ∞), where √ 0( x + 1/2 + d/2) H (x) = √ . 0( x + 1/2 − d/2) (Note that as x gets large, H (x) ∼ x d/2 .) If this is true, then by Bernstein’s theoR∞ 2 rem we can write e−t Ad = 0 e−ut B dρ(u) for some bounded and nondecreasing function ρ(u), and this implies the statement. Recall that complete monotonicity means that (−1)n f (n) (x) ≥ 0. To prove this, it is enough to show that H 0 (x) is completely monotonic. This is clear for d = 1. Next, for b > a > 0 we have Z ∞ 0(x + a) 1 e−at = e−t x dt. 0(x + b) 0(b − a) 0 (1 − e−t )1+a−b Thus, rewriting a = (1 + d)/2, b = (1 − d)/2, we can write √ Z ∞ √ e−(1/2+d/2)t 1 0( x + 1/2 + d/2) = e−t x dt, √ 0(−d) 0 (1 − e−t )1+d 0( x + 1/2 − d/2) which, however, holds when −1 < d < 0. To get to the values we need, observe that the x-derivative of the integrand produces an extra t that makes the integral converge for 0 < d < 1; also, using 0(−d)0(d + 1) = −π/ sin πd, we get √ Z sin πd ∞ e−t x t e−(1/2+d/2)t 0 H (x) = 0(d + 1) dt, 0 < d < 1, √ π 2 x (1 − e−t )1+d 0 which we rewrite as 0(d + 1) H (x) = sin πd π2d 0

∞

Z 0

√

e−2t √ x

x

t (sinh t)−d−1 dt.

(20)

√ √ Now this takes care of the case 0 < d < 1 since the function e−c x / x is completely monotonic when c > 0 (product of two completely monotonic functions). To deal with the remaining case 1 < d < 2, we regularize by writing Z ∞ −2t √x 0(d + 1) e 0 t (sinh t)−d−1 − t −d−1 dt H (x) = sin πd √ π2d x 0 Z ∞ −2t √x 0(d + 1) e + sin πd t −d dt √ π2d−1 x 0 Z ∞ −2t √x 0(d + 1) e d = sin πd t (sinh t)−d−1 − t −d−1 dt + x d/2−1 , √ d π2 2 x 0

492

CARLO MORPURGO

and this formula holds for 0 < d < 3. Assuming now 1 < d < 2, it is clear that the whole quantity is completely monotonic. Indeed, the second term is completely monotonic; also, the function inside brackets is negative, and so is sin πd, so that the first term is completely monotonic. What we have shown, in essence, is that the function H (x) − x d/2 has a completely monotonic derivative when 1 < d < 2. (When 0 < d < 1 instead, H (x)−x d/2 is completely monotonic itself, but this does not imply that H 0 is; in this case we need to use (20).) Having proved that when 0 < d ≤ 2 both 1d/2 and Ad generate Feller semigroups, we can define the conditional probability measures for the bridge processes Px,y,t , Pξ,η,t on Rn and S n , respectively. The nonnormalized Markov bridge measures, also known as conditional Wiener measures or pinned measures, are defined as µx,y,t = E d (t, |x − y|)Px,y,t , µξ,η,t = K d (t, ξ, η)Pξ,η,t . The pinned measures are useful in the representations of the kernels for the d/2 Schrödinger semigroups e−t (1 +w) , e−t (Ad +W ) as semigroups on L 2 , where w and W are regular perturbations of the corresponding operators. In particular, for w ∈ L ∞ (Rn ) we have Z Z t −t (1d/2 +w) e (x, y) = exp − w(Bv ) dv dµx,y,t (B), (21) D

0

which is a consequence of the famous Feynman-Kac formula (see [Ok, Props. 4.2, 4.3]). This kernel is also jointly continuous in x, y, as shown, for example, in [DvC, Prop. 3.3]. On S n a similar formula holds for the kernel of e−t (Ad +W ) if W ∈ L ∞ (S n ). For 0 < d ≤ 2 and measurable 9 : R+ → R and w : Rn → R+ , we define Z ∞Z Z Z t dt I9 [w] = 9 w(Bu ) du dµx,x,t (B) d x , (22) n t 0 R D 0 and similarly, for W : S n → R+ , we define Z ∞Z Z Z t dt J9 [W ] = 9 W (Bu ) du dµξ,ξ,t (B) dξ . n t 0 S D 0

(23)

When 9(u) = u s / 0(s), we let Is [w] = I9 [w], Js [W ] = J9 [W ], and for 0 < d < n and s an integer, we define Is [w] =

Z

s Y (Rn )s k=1

Nd (xk − xk+1 )

s Y k=1

w(xk ) d x1 · · · d xs

(24)


and Js [W ] =

Z

s Y (S n )s k=1

Nd (ξk − ξk+1 )

s Y

493

W (ξk ) dξ1 · · · dξs .

(25)

k=1

The consistency of the notation in the case s an integer and d ≤ 2 follows from the proposition below, but it can also be checked directly using the computations of Proposition 5.5. PROPOSITION 2.6 Let 0 < d < n, 0 ≤ W ∈ L n/d (S n ), 0 ≤ w ∈ L n/d (Rn ), and w = (W ◦ w)|Jπ |d/n . If s > n/d with s an integer if d > 2, then Z s, Ad (W ) = Js [W ] = Tr (w1/2 1−d/2 w1/2 )s = Is [w]. (26)

If 0 < d ≤ 2 and 9 is a nonnegative lower semicontinuous function on [0, ∞), with 9(0) = 0, then (allowing both sides to be infinite) Z 9 Ad (W ) = J9 [W ] = Tr L9 (w1/2 1−d/2 w1/2 ) = I9 [w] (27) Proof The middle equalities are a consequence of Proposition 2.3. In the case when s is an integer, the trace formulas follow in a straightforward way 1/2 )s and (w 1/2 1−d/2 w 1/2 )s and integrating by writing the kernels of (W 1/2 A−1 d W along the diagonal (modulo some minor technical fuss to justify that this actually gives the traces). Let now 0 < d ≤ 2. In Rn , formula (27) was derived by Lieb in the case d = 2, but it can be easily extended to the case 0 < d < 2 (and even to generators of the form f (−i∇); see [Dau]). The main ingredient is the Feynman-Kac formula as given in (21), with λ ∈ R and w ∈ L ∞ (Rn ): Z Z t d/2 e−t (1 +λw) (x, y) = exp −λ w(Bv ) dv dµx,y,t (B). (28) D

0

When λ ≥ 0 and w ≥ 0 are smooth with compact support, one can integrate (28) in t, set x = y, and integrate in w(x) d x, obtaining Lieb’s formula for the function 9(u) = ue−λu , and L9 (u) = u/(1 + λu). Then one “builds up” the formula for more general 9 and w by using the fact that linear combinations of functions of type φλ (u) = e−λu are dense in the uniform norm in the set of functions continuous in R+ and vanishing at infinity (see [S, Th. 8.2, p. 90]). In S n the proof is completely similar and is based on the Feynman-Kac formula for the kernel of exp[−t (Ad + λW )], similar to (28).

494

CARLO MORPURGO

Later we use the following finite-dimensional versions of Lieb’s formulas, in terms of classical integrals. LEMMA 2.7 Let 0 < d < n, 0 < d ≤ 2, t > 0, w ∈ L ∞ (R), and Jw = [ess inf w, ess sup w]. Let 9 : E → R be measurable, with E ⊇ t Jw , and 9 continuous on t Jw . Assume Rt further that 9( 0 w(Bv ) dv) is in L 1 (Rn × D , d xdµx,x,t ) or that 9 has constant sign on t Jw . Then, with E d (t, |x|) given as in (18),

Z

Z dx Rn

D

9

Z 0

Z = lim

t

w(Bv ) dv dµx,x,t (B) N Y

N →∞ Rn N j=1

N t X E d (t/N , |x j − x j+1 |)9 w(x j ) d x1 . . . d x N N

(29)

j=1

with the notational convention that x N +1 = x1 . Both sides may be infinite. Note that we do not assume that 9 is nonnegative. Also, the result is probably true if one assumes only lower semicontinuity of 9. Proof The proof of this result for 9(u) = e−λu is a routine argument, which shows that the kernel of the Feynman-Kac formula is pointwise the same as the one of the Trotter d/2 product formula, for the L 2 -semigroup e−t (1 +λw) . For general 9, first observe that it is enough to assume that 9 is continuous with compact support and nonnegative. Then one proceeds, modulo small technicalities, via the same density argument used by Lieb and sketched in the proof of Proposition 2.6. It is not hard to prove that if 9(0) = 0 and w ≥ 0 is bounded with compact support, then both sides of (29) are finite.

3. Symmetrization of functional integrals and related results Rearrangement inequalities for discrete, multiple, and path integrals In the present section we would like to show that the zeta and 9-functions are increased when one replaces the function w ∈ L n/d (Rn ) with its symmetric decreasing rearrangement w∗ , with a precise statement for the case of equality. In this section we prove a stronger result, together with new rearrangement theorems for classical integrals. Most of our results apply in both the Rn and S n settings (and also in the hyperbolic case); however, it is the Rn -versions that we really use. We deal exclusively


495

with the Rn -case, and at the end of this section we make an additional remark about the S n -case. Let us recall that, given a measurable function w : Rn → [−∞, ∞), its symmetric decreasing rearrangement is the function w∗ (unique up to a set of measure zero) which is radially symmetric, nonincreasing, and equimeasurable to w: {x : w∗ (x) > t} = {x : w(x) > t} , ∀t ∈ R. To make sense out of this definition, one should require some decay at infinity. This condition is expressed by assuming {x : w(x) > t} < ∞ for all t > ess inf w. (30) When w ≥ 0, such a condition is certainly satisfied when w ∈ L p (Rn ). With this definition of decreasing symmetric rearrangement, if w(x) = c a.e., then w∗ (x) = c a.e. Also, if A is a measurable set, then (χ A )∗ = χ A∗ , where A∗ is the ball centered at the origin and with the same volume as A. Later on we use the following well-known properties of rearrangement: (a) kw∗ − g ∗ k p ≤ kw − gk p for 1 ≤ p ≤ ∞, so that if wk → w a.e. and in L p , then, up to a subsequence, w∗j → w∗ a.e. and in L p ; (b) if w j ↑ w a.e., then also w∗j ↑ w∗ a.e. The main goal of Section 3 is to prove the following general theorem. THEOREM 3.1 Let t > 0, 0 < d < n, and 0 < d ≤ 2. Let w : Rn → [−∞, +∞) be measurable, let it satisfy (30), and let = x ∈ Rn : w(x) > −∞ , L(t, B) = v ∈ [0, t] : Bv ∈ . (31) R Assume that L(t,B) |w(Bv )| dv < ∞ a.e. in Rn × D with respect to d x dµx,x,t . Let Jw = [ess inf w, ess sup w]\{+∞}, and let 9 : t Jw → R be convex and continuous, with 9(t · ess inf w) = 0. Define Z Z Z t I9 [w, t] = 9 w(Bv ) dv d x dµx,x,t (B). (32) Rn

D

0

Assume either that the integral above is finite or that 9 has constant sign. Then, I9 [w, t] ≤ I9 [w ∗ , t],

(33)

where both sides may be infinite. If in addition is open and connected (up to a set of zero measure), 9 is strictly convex, I9 [w, t] is finite, and the integrand in (32) is not zero a.e., then equality occurs if and only if w(x) = w∗ (x − a) a.e. x ∈ Rn , for some a ∈ Rn .

496

CARLO MORPURGO

COROLLARY 3.2 With the same hypothesis as Theorem 3.1, assume further that 0 ≤ w ∈ L n/d (Rn ) and that I9 [w] = Tr[L9 (w1/2 1−d/2 w1/2 )] < ∞. Then

I9 [w] ≤ I9 [w ∗ ]

(34)

with equality if and only if w(x) = w∗ (x − a) a.e. x ∈ Rn , for some a ∈ Rn . The proof of this corollary is immediate from (33) since I9 [w] = R∞ 0 I9 [w, t] dt/t; for the same reason, equality in (34) implies equality in (33) for a.e. t. Notice that in this case, = Rn up to a set of zero measure. For the purpose of proving 9-function inequalities this corollary is what one really needs; however, Theorem 3.1 also covers the case when w has a singular barrier 0 = Rn \ , where it takes the value −∞. To better understand the functional I9 [w, t], we give several preliminary remarks. Remarks. (1) Observe that 9(t · ess inf w) > 0 (< 0) =⇒ I9 [w, t] = +∞ (−∞).

(35)

This follows, for example, from Lemma 3.12, as explained in the remarks following the lemma. (2) Changing w on a set of zero measure does not affect the value of I9 [w, t]; this can also be seen using Lemma 3.12. (3) With a little extra work the equality statement of Theorem 3.1 can be proved when is a general open set; most likely this is true when is any Borel set. Rt (4) The integrability assumption for w(Bv ) ensures that the integral 0 w(Bv ) dv is well defined and is either equal to −∞ if the path B spends positive time outside , or is equal to a finite quantity if the path is a.e. inside . Note also that the set of paths for which the condition is satisfied is Ft0 -measurable. This follows easily from the fact that the map (v, B) → Bv is progressively measurable, that is, jointly measurable in ([0, t] × D , m × Px , B [0, t] × Ft0 ) (m, B [0, t]) being the Borel measure space in [0, t] , and from Fubini’s theorem. (5) The integrability condition is certainly verified if w ∈ L p () for any p ∈ [1, ∞]. This can be seen as follows. First, let w e = w in , and let w e = 0 outside , R Rt p p R t p so that L(t,B) |w(Bv )| dv = 0 |e w(Bv )| dv ≤ 0 |e w(Bv )| dv. Integrating in R d x d Px,x,t using formula (105) (applied in Rn ) gives precisely |w(x)| p d x. (6) For any Borel set , the first penetration time in c is defined as Z t n o S (B) = inf t > 0 : χ (Bv ) dv < t , (36) 0


497

that is, the first time when the process B is not inside for a.e. time v ∈ [0, t] (see, Rt e.g., [St] or [DvC] for more on this notion). The quantity 0 χ (Bv ) dv is the time spent by the process inside , up to time t. It is known, under our hypothesis on filtrations Ft , that S is a stopping time and that for t > 0, Px (S = t) = 0 (see [DvC, p. 88]). Hence, with probability 1, B : Bv ∈ , a.e. v ∈ [0, t] = {S ≥ t} = {S > t}. Now we claim that the integral (32) is really over the set × {S > t}. For this we recall the notion of S-regular points for c , that is, points of Rn satisfying the Rt condition Px (S = 0) = 1; note that {S = 0} = 0 χc (Bv ) dv > 0, ∀t > 0 ; at regular points the process must spend positive time outside , with probability 1. 3.3 If is any Borel set in Rn and t > 0, a.e. x ∈ / is S-regular for c . Hence, for a.e. n x∈ / and a.e. y ∈ R , we have Px,y,t (S > t) = 0. LEMMA

This means that for a.e. x ∈ 0 and with probability 1, bridges starting at x and ending at x at time t must cross for a positive time the region where the integrand of (32) vanishes. D. Stroock and Z. Ciesielski [St] proved the first part of this lemma for Brownian motions, and their proof extends easily to d-symmetric stable processes. The result is also true for general Feller processes (see [DvC, App. D, Prop. D.7, Rem. 2, pp. 403 – 404]). The second part follows easily from the definition of conditional pinned measure. COROLLARY 3.4 Under the hypothesis of Theorem 3.1, we have Z Z Z t I9 [w, t] = 9 w(Bv ) dv d x dµx,x,t (B). ×{S >t}

(37)

0

Inspired by (29), we prove Theorem 3.1 by approximation with finite-dimensional integrals. The result we need (at least for the inequality statement) is contained in the following theorem, which is a generalization of the standard Riesz rearrangement inequality. Before stating such a result, we introduce a preliminary definition. If I N = {1, 2, . . . , N } and R ⊆ {(i, j) ∈ I N × I N : i < j}, we define [ R= {i, j}. (38) (i, j)∈R

We say that two given ∈ R are connected in R if there is a “chain” of pairs (k` , k`+1 ) ∈ R , ` = 1, . . . , m, with k0 = k, km+1 = k 0 . The set R is connected if k, k 0

498

CARLO MORPURGO

any two points of R are connected. Any set R as above can be split into connected Sp components R = 1 R` . THEOREM 3.5 Let K i j : [0, ∞) → [0, ∞) with i < j, i, j = 1, 2 . . . , N , be measurable and decreasing; let f 1 , . . . , f N : Rn → [−∞, +∞) be measurable and satisfy (30). Assume that none of the K i j vanishes a.e. and that none of the f j = −∞ a.e. Let P P J = j, j ess inf f P j ess sup f j \ {+∞}, and let 9 : J → R be convex and continuous with 9 j ess inf f j = 0. Define

Q 9 [ f1, . . . , f N ] =

N Y

Z

(Rn ) N i< j

K i j (|xi − x j |) 9

N X

f j (x j ) d x1 · · · d x N ,

(39)

j=1

and assume that either the integrand is in L 1 (Rn N ) or 9 has constant sign. Then Q 9 [ f 1 , . . . , f N ] ≤ Q 9 [ f 1∗ , . . . , f N∗ ],

(40)

and if the left-hand side is infinite, so is the right-hand side. Suppose in addition that (i) 9 is strictly convex, (ii) there is a connected set R1 of pairs (i, j) such that K i j is strictly decreasing, (iii) Q 9 is finite. Under these hypotheses, if equality in (40) occurs, then f j (x) = f j∗ (x − a) a.e. for some a ∈ Rn and for all j ∈ R1 , or f j is constant for |R1 | − 1 values of j (with R1 defined as in (38)). Conversely, if f j (x) = f j∗ (x − a) a.e., j = 1, 2, . . . , N for some a ∈ Rn , or f j is constant for at least N − 1 values of j, then equality holds in (40).

Remarks P (a) Observe that if 9 j ess inf f j > 0 (resp., < 0), then (39) is +∞ (resp., −∞). (b) The integral in (39) can be taken over the set 1 × 2 × · · · × N , where j = {x : f j (x) > −∞}. An immediate consequence of Theorem 3.5 is the strict version of the classical multiple Riesz rearrangement inequality. Given measurable K i j : [0, ∞) → [0, ∞) with i < j, i, j = 1, 2 . . . , N , and f 1 , . . . , f N : Rn → [0, ∞), we set Q[ f 1 , . . . , f N ] =

Z

N Y

(Rn ) N i< j

K i j (|xi − x j |)

N Y j=1

f j (x j ) d x1 · · · d x N .

(41)


499

THEOREM 3.6 (Strict Riesz rearrangement inequality) If Q is defined as in (41) and all K i j are decreasing, then

Q[ f 1 , . . . , f N ] ≤ Q[ f 1∗ , . . . , f N∗ ]

(42)

(both sides could be infinite). Let now N ≥ 2, assume that none of the f j vanishes a.e., and assume that Q[ f 1 , . . . f N ] is finite. If equality holds in (42) and there is a connected set R1 such that K i j is strictly decreasing for (i, j) ∈ R1 , then f j (x) = f j∗ (x − a) a.e. for some a ∈ Rn and for all j ∈ R1 . Conversely, if f j (x) = f j∗ (x − a) a.e., j = 1, 2, . . . , N , for some a ∈ Rn , then equality holds in (42). The inequality is a special case of a more general rearrangement inequality due to H. Brascamp, Lieb, and J. Luttinger [BLL]. For a proof of the equality case when N = 2, see [LL, Th. 3.9]. See also the recent paper by A. Burchard and M. Schmuckenschläger [BS], who also derive the equality statement for any N . The above theorem is proved by noticing that if f 1 , . . . , f N are nonnegative measurable functions, then f˜j := log f j are measurable and take values in [−∞, +∞), Q PN ∗ ˜ ˜ ∗ and 1N f j = exp 1 f j . Since log x is increasing, then ( f j ) = log f j and the result follows. To prove Theorem 3.5 we employ an extension of the so-called two-point symmetrization method, which has been used more notably by A. Baernstein and B. Taylor [BT], Beckner [Bec2], and R. Friedberg and Luttinger [FL]. The basic idea is the following. Let H be the set of hyperplanes H on Rn with 0 ∈ / H , and denote by H + , H − the two connected components of Rn \ H with H + 3 0, H − 6 3 0. Let ρ H : Rn → Rn be the reflection about H . For a measurable f : Rn → [−∞, +∞) defined a.e., let the hyperplane rearrangement of f be defined a.e. as ( max( f (x), f (ρ H x)) if x ∈ H + , f H (x) = min( f (x), f (ρ H x)) if x ∈ H − . The first step consists of showing that Q 9 [ f 1 , . . . , f N ] is increased upon replacing each f j by its hyperplane rearrangement. LEMMA 3.7 Under the same hypothesis as that of Theorem 3.5, for any hyperplane H ∈ H ,

Q 9 [ f 1 , . . . , f N ] ≤ Q 9 [ f 1H , . . . , f NH ],

(43)

and if the left-hand side is infinite, then so is the right-hand side. If the additional conditions (i) – (iii) of Theorem 3.5 are also satisfied and equality in (43) occurs, then (a) f j = f jH a.e. in H + (and hence in Rn ), ∀ j ∈ R1 , or

500

CARLO MORPURGO

(b) (c)

f j = f jH ◦ ρ H a.e. in H + , ∀ j ∈ R1 , or f j = f j ◦ ρ H a.e. in H + for at least |R1 | − 1 values of j ∈ R1 . Conversely, if (a), (b), or (c) is true for R1 = {1, 2, . . . , N }, then equality holds in (43).

The key step for the proof of this lemma is to rewrite the integral defining Q 9 as follows: Q 9 [ f1, . . . , f N ] Z X =

Y

(H + ) N ` ,...,` =0,1 i< j N 1

`

ì K i j (|ρ H xi − ρ Hj x j |)9

N X

`k f k (ρ H xk ) d x1 · · · d x N ,

k=1

(44) ` = ρ if ` = 1, and ρ ` = id if ` = 0. This formula can be found by where ρ H H H R R R writing each of the N integrals over Rn in (39) as Rn = H + + H − , carrying each H − into H + by a reflection, and then collecting terms. Inequality (43) is proved if we can show that the integrand in (44) is increased upon replacing each f j with f jH (up to a set of measure zero). This is the content of the following discretized (or combinatorial) version of (40). ˆ . . . , Nˆ }. Consider functions f k : {k, k} ˆ → Let I N = {1, . . . , N }, b I N = {1, b [−∞, +∞). In the set I N ∪ I N , define a “distance” d(P1 , P2 ) = d(P2 , P1 ) for ˆ = d(i, ˆ k), d(i, ˆ k) ˆ = d(i, k), P1 , P2 ∈ I N ∪ b I N , such that d(P, P) = 0, d(i, k) ˆ and d(i, k) < d(i, k).

Let K i j i< j be a family of nonnegative functions defined on the set {d(i, j), ˆ Suppose that K i j is decreasing: K i j [d(i, j)] ≥ K i j [d(i, j)]. ˆ d(i, j)}. Define the decreasing rearrangement f i∗ by ˆ f i∗ (i) = max{ f i (i), f i (i)},

ˆ = min{ f i (i), f i (i)}. ˆ f i∗ (i)

(45)

Finally, for 9 : [−∞, ∞) → R we set Q 9 [ f1, . . . , f N ] =

X ˆ P1 ∈{1,1}

···

X

Y

N X K i j [d(Pi , P j )]9 f k (Pk ) .

PN ∈{N , Nˆ } i< j

k=1

THEOREM 3.8 (Discrete rearrangement inequality) Assume that none of the K i j vanishes identically and that none of the f k is identically −∞. If 9 is convex, then

Q 9 [ f 1 , . . . , f N ] ≤ Q 9 [ f 1∗ , . . . , f N∗ ].

(46)

Let R ∗ be the set of pairs (i, j) for which K i j is strictly decreasing, denote by R1∗ , . . . , R ∗p its connected components, and let R1∗ , . . . , R ∗p be defined as in (38). If


501

9 is strictly convex, then equality in (46) occurs if and only if for each q = 1, . . . , p ˆ fˆk (k) ˆ = we have f k = f k∗ , ∀k ∈ Rq , or fˆk = f k∗ , ∀k ∈ Rq , where fˆk (k) = f k (k), f k (k). Proof For notational convenience, let us momentarily drop the symbols K i j : (i j) stands for ˆ stands for K i j [d(i, j)]. ˆ K i j [d(i, j)], and (i j) b = {eˆ : e ∈ E} and E c = I N \ E. The set of N -tuples For E ⊆ I N , let E ˆ is indexed by {P1 , . . . , PN } with Pi ∈ {i, i} bc , E b ∪ E c } E⊆I . {E ∪ E N −1 For A, B ⊆ I N , let ˙ = (a, b), a < b, a ∈ A, b ∈ B , A×B and for E ⊆ I N , define ˙ ˙ c ) ⊆ I N ×I ˙ N. K E = (E ×E) ∪ (E c ×E Clearly, ˙ N \ K E = (E ×E ˙ c ) ∪ (E c ×E). ˙ K Ec = I N ×I With this notation, and using the definition of K i j , we get h X X Y Y X ˆ Q9 = (ik) (`ˆr ) 9 f k (k) + f k (k) E⊆I N −1 K E

K Ec

Ec

E

+9

X

ˆ + f k (k)

X

i f k (k) .

Ec

E

ˆ with Aik ≥ 0, and note that In this formula, set (ik) = Aik + (i k) Y Y X Y ˆ = (s m). ˆ Aik [Aik + (i k)] KE

R ⊆K E R

K E \R

502

CARLO MORPURGO

˙ N \ R ) the following: Thus, we can write (with R c = I N ×I h X X X Y Y X ˆ Q9 = Aik (`ˆr ) 9 f k (k) + f k (k) E⊆I N −1 R ⊆K E R

Rc

Ec

E

+9

X

ˆ + f k (k)

Ec

E

=

X

Y

i f k (k)

X

Y X h X X ˆ Aik (`ˆr ) 9 f k (k) + f k (k)

˙ N R R ⊆I N ×I

Rc

E⊆I N −1 K E ⊇R

Ec

E

+9

X

ˆ + f k (k)

X Ec

E

=

X

Y

i f k (k)

X Y X X ˆ . Aik (`ˆr ) 9 f k (k) + f k (k)

˙ N R R ⊆I N ×I

Rc

E⊆I N K E ⊇R

˙ N, We are then reduced to proving that for any R ⊆ I N ×I X X X X X X ˆ ≤ ˆ . 9 f k (k) + f k (k) 9 f k∗ (k) + f k∗ (k) E⊆I N K E ⊇R

k∈E

k ∈E /

E⊆I N K E ⊇R

(47)

Ec

E

k∈E

(48)

k ∈E /

We begin with the following lemma. LEMMA 3.9 Let 9 be as in the hypothesis of Theorem 3.8. Then, for E ⊆ I N , |E| ≥ 2, X X X X ˆ ≤9 ˆ . 9 f k (k) + 9 f k (k) f k∗ (k) + 9 f k∗ (k) k∈E

k∈E

k∈E

(49)

k∈E

If 9 is strictly convex, then equality occurs if and only if f k = f k∗ , ∀k ∈ E, or fˆk = f ∗ , ∀k ∈ E. k

Note. If |E| = 0, 1, the inequality is a trivial identity. Proof of Lemma 3.9 First, observe that a1 ≥ a2 ∧ b1 ≤ b2 =⇒ 9(a1 + b1 ) + 9(a2 + b2 ) ≤ 9(a1 + b2 ) + 9(a2 + b1 ) (50) (and, by symmetry, also if a1 ≤ a2 ∧ b1 ≥ b2 ), which follows easily from the convexity of 9. If 9 is strictly convex, then equality in (50) does not occur if and only if a1 > a2 ∧ b1 < b2 . In this argument, a j , b j ∈ [−∞, +∞), but some a j ’s and some b j ’s are finite.


503

ˆ E 1 = {k ∈ E : f k (k) > f k (k)}, ˆ Now let E 0 = {k ∈ E : f k (k) = f k (k)}, ˆ E 2 = {k ∈ E : f k (k) < f k (k)}, and X X X X f k (k) = f k (k) + f k (k) + f k (k) := c + a1 + b1 , k∈E

k∈E 0

k∈E 1

X

X

X

ˆ = f k (k)

k∈E

ˆ + f k (k)

k∈E 0

k∈E 2

X

ˆ + f k (k)

k∈E 1

ˆ := c + a2 + b2 . f k (k)

k∈E 2

Since none of the f k is identically −∞, c must be finite. Inequality (49) follows by applying (50) to the convex function 9(c + · ). If 9 is strictly convex, then equality occurs if and only if E 1 = ∅ or E 2 = ∅; in the first case, fˆk = f k∗ for all k ∈ E, and in the second case, f k = f k∗ for all k ∈ E. ˙ N . If R = ∅, Continuing the proof of Theorem 3.8, we now show (48). Let R ⊆ I N ×I then the left-hand side of (48) reduces to X X X ˆ , 9 f k (k) + f k (k) E⊆I N

k ∈E /

k∈E

which is invariant under replacement of f k with f k∗ . Sp If R 6= ∅, then let R = 1 R` , where the R` are the connected components of R , and let R, R1 , . . . , R p be defined as in (38). Clearly, R c , R1 , . . . , R p form a partition of I N , where R c = I N \ R. ˙ ˙ c ) ⊇ R are Then all sets E ⊆ I N with the property that K E = (E ×E) ∪ (E c ×E of the form [ E= R` ∪ R 0 , R 0 ⊆ R c , L ⊆ {1, . . . , p}. `∈L

Thus, we can rewrite the left-hand side of (48) as follows: X X X ˆ 9 f k (k) + f k (k) E⊆I N K E ⊇R

k ∈E /

k∈E

=

X

9

X

+

ˆ f k (k)+

k∈R c \R 0

k∈R 0

L⊆{1,..., p} R 0 ⊆R c

X

f k (k) +

XX

f k (k) +

`∈L k∈R`

X X

ˆ , f k (k)

`∈L c k∈R`

where L c = {1, 2, . . . , p} \ L. For given L, X X X XX X X ˆ + ˆ 9 f k (k) + f k (k) f k (k) + f k (k) R 0 ⊆R c

=

R 0 ⊆R c

9

`∈L k∈R`

k∈R c \R 0

k∈R 0

X

X k∈R 0

f k∗ (k) +

X k∈R c \R 0

ˆ + f k∗ (k)

XX `∈L k∈R`

`∈L c k∈R`

f k (k) +

X X `∈L c k∈R`

ˆ . f k (k)

504

CARLO MORPURGO

Now fix R 0 ⊆ R c . If p ≥ 1, and for fixed 1 ≤ q ≤ p, X X X X X X X ˆ + 9 f k (k) + f k (k) f k∗ (k) + `∈L k∈R`

L⊆{1,..., p}

k∈Rq

L⊆{1,..., p} L63q

k∈Rq

n X X o ˆ + 3 L ,q,R 0 , 9 f k∗ (k) + 3 L ,q,R 0 + 9 f k∗ (k)

X

≤

k∈R c \R 0

k∈R 0

n X X o ˆ + 3 L ,q,R 0 9 f k (k) + 3 L ,q,R 0 + 9 f k (k)

X

=

`∈L c k∈R`

ˆ f k∗ (k)

k∈Rq

L⊆{1,..., p} L63q

k∈Rq

(51) where 3 L ,q,R 0 =

X X

f k (k) +

`∈L k∈R` `6=q

X X

ˆ + f k (k)

`∈L c k∈R` `6=q

X

f k∗ (k) +

X

ˆ f k∗ (k).

k∈R c \R 0

k∈R 0

The first formula in (51) holds since all possible subsets L of {1, . . . , p} are obtained from those of {1, . . . , p} \ {q} by adding or not adding the point q. The inequality is due to Lemma 3.9 since the function 9(u + 3 L ,q,R 0 ) is convex if 3 L ,q,R 0 is finite, or identically zero if 3 L ,q,R 0 = −∞. Now apply (51), starting first with q = 1; note that the last member of (51) is equal to the first one, provided we substitute f k∗ in place of f k for all k ∈ R1 . Then apply (51) again for q = 2 to this new expression, and so on, until we reach q = p and all functions are symmetrized. To discuss equality, assume that 9 is strictly convex, and let R ∗ be the set of pairs (i, j) for which K i j is strictly decreasing. Note that if (`, r ) ∈ / R ∗ , then (`ˆr ) > 0 due to our assumptions on K i j . First, note that the sum in (47) is over all R ⊆ R ∗ . Since each term of (47) satisfies the inequality, then equality for Q 9 implies equality for all terms and in particular for the one corresponding to R ∗ . If R1∗ , . . . , R ∗p denote the connected components of R ∗ and R1∗ , . . . , R ∗p are defined as in (38), then we must have equality of the corresponding expressions inside braces in (51) for each 1 ≤ q ≤ p, each L ⊆ {1, . . . , p} \ {q}, and each R 0 ⊆ R c . Take q = p, and consider the last inequality in the chain described above. By choosing R 0 = R c and L = {1, 2, 3 . . . p − 1}, we get 3

L , p,R c

=

p−1 X X `=1 k∈R`

f k∗ (k) +

X

f k∗ (k),

k∈R c

which must be finite since by assumption none of the f k is identically −∞ (so that f k∗ (k) > −∞). Hence, we can apply the equality statement of Lemma 3.9 to the


505

strictly convex function 9(u + 3 L , p,R c ) and deduce that either f k = f k∗ for all k ∈ R p , or fˆk = f k∗ for all k ∈ R p . Proceeding in this fashion for q = p−1, p−2 . . . , we easily get the desired conclusion. Conversely, if one such condition is true for all q ≤ p, then equality in (51) holds for all q ≤ p, and it is easy to see that it also holds for all the terms of (47). (Given R ⊆ R ∗ , the connected components of R are included in the ones of R ∗ .) This concludes the proof of Theorem 3.8. Proof of Lemma 3.7 Apply Theorem 3.8 to the integrand in (44), with I N = {x1 , . . . , x N } and IˆN = {ρ H x1 , . . . , ρ H x N }, for fixed (x1 , . . . , x N ) ∈ (H + ) N . Applying the inequality in the integrand for a.e. x1 , . . . , x N yields the inequality (43) since the 2-point rearrangement (45) corresponds exactly to the hyperplane rearrangement. To analyze equality, let j := {x ∈ Rn : f j (x) > −∞},

Hj := ( j ∪ ρ H j ) ∩ H + ,

H e H := 1H × · · · × N ,

which are sets of positive measure, and notice that the integrand in (44) is zero if x j ∈ H + \ Hj for any given j. Hence, the integral in (44) can be evaluated on the e H . The same is true for the same integral evaluated at f H , . . . , f H since outside set N 1 H j we have f j = f j ◦ ρ H = f jH = f jH ◦ ρ H = −∞. Hence, equality in (43) occurs if and only if the integrand in (44) is equal to the corresponding quantity evaluated at e H . Now suppose that 9 is strictly convex and f 1H , . . . , f NH , for a.e. (x1 , . . . , x N ) ∈ Q 9 is finite. Let R1 be a set of pairs (i, j) so that K i j is strictly decreasing in (0, ∞), and assume that R1 is connected; also, let R1 be as in (38). If equality occurs in (43), e H of the set then Theorem 3.8 implies that the complement in e H : f j (x j ) = f jH (x j ), ∀ j ∈ R1 , (x1 , . . . , x N ) ∈ or f j (ρ H x j ) = f jH (x j ), ∀ j ∈ R1

(52)

has zero Lebesgue measure. If we let A j = {x ∈ Hj : f j (x) < f j (ρ H x)} and e H of the set in (52) B j = {x ∈ Hj : f j (x) > f j (ρ H x)}, then the complement in S H can be written as F`1 × · · · × F` N , where F` j = A j , B j , j , the union being over all sets with exactly one A j and one Bk . But this set has measure zero if and only P if j6=k |A j ||Bk | = 0. This sum is zero if and only if |A j | = 0 for all j ∈ R1 , or |B j | = 0 for all j ∈ R1 , or |A j | = |B j | = 0 for |R1 | − 1 values of j ∈ R1 ; this implies the first half of the equality statement. Conversely, if f j = f jH a.e. in H + for all j ≤ N , or f j ◦ ρ H = f jH a.e. in H + for all j ≤ N , or f j = f j ◦ ρ H a.e. in H + e H of the set in (52) with R1 = I N for N − 1 values of j, then the complement in has zero measure and this implies that equality occurs in (43).

506

CARLO MORPURGO

For later use we note the following inequality. 3.10 Under the same hypothesis as that of Theorem 3.5, for any H ∈ H , if we define COROLLARY

K iHj (xi , x j ) = K i j (|xi − x j |) − K i j (|xi − ρ H x j |) and H Q9 [ f1, . . . , f N ] Z N N N h X X i Y = f j (x j ) + 9 f j (ρ H x j ) d x1 · · · d x N , K iHj (|xi − x j |) 9 (H + ) N i< j

j=1

j=1

then, for a connected set of pairs R1 so that R1 = {1, 2, . . . , N }, H H Q 9 [ f 1H , . . . , f NH ] − Q 9 [ f 1 , . . . , f N ] ≥ Q 9 [ f 1H , . . . , f NH ] − Q 9 [ f1, . . . , f N ] ≥ 0 (53) with the same conditions for equality as in Lemma 3.7.

Proof Rewrite the integrand of Q 9 [ f 1 , . . . , f N ] using (44); then rewrite that expression ˆ = f k (ρ H xk ). Do using (47) with Aik = K ikH (xi , xk ), f k (k) = f k (xk ), and f k (k) the same for the integrand of Q 9 [ f 1H , . . . , f NH ]. Thus, the left-hand side of (53) can be written as a single integral over (H + ) N of a function I (x1 , . . . , x N ), which is greater than the corresponding integrand for the quantity on the right-hand side of (53). This follows since I (x1 , . . . , x N ) is a sum over all R ⊆ {(i, j) : i < j}, each term of which is nonnegative by (48); thus, we can toss all such terms except the one corresponding to R = R1 and get the integrand on the right-hand side of (53). Proof of Theorem 3.5 Inequality. The step from the inequality of Lemma 3.7 to that of Theorem 3.5 follows the same line of reasoning as in [BT], with just a few technical modifications; details are entirely omitted. Equality. Since f jH is equimeasurable to f j , we have Q 9 [ f 1 , . . . , f N ] ≤ Q 9 [ f 1H , . . . , f NH ] ≤ Q 9 [ f 1∗ , . . . , f N∗ ]. Hence, equality in (40) implies equality in (43), and therefore the functions f j satisfy


507

the property that

for each H 63 0

 H  a.e. ∀ j ∈ R1 , or   fj = fj , H f j = f j ◦ ρ H , a.e. ∀ j ∈ R1 , or   f = f ◦ρ for at least |R1 | − 1 values of j. j j H

We show that if (54) happens, then ( f j = f j∗ ( · + a), a.e. ∀ j ∈ R1 , or f j = Const.,

a.e. for at least |R1 | − 1 values of j.

(54)

(55)

Assume first that the f j are nonnegative and continuous. LEMMA 3.11 Let f : Rn → [0, ∞) be continuous and not a.e. a translate of f ∗ . Then there exist a line ` and an open interval I ⊂ ` \ {0}, so that for each H passing through I and orthogonal to `, we have f 6 = f H and f 6 = f H ◦ ρ H on sets of positive measure in H + . If n = 1, then H is just a point contained in an interval I ⊂ R \ {0}.

Proof Let L t = {x : f (x) > t} be the (open) level set corresponding to t > 0. There are two cases: (a) there is a t > 0 such that L t 6 = ∅ and L t is not a ball a.e., and (b) L t is a.e. a ball or L t = ∅ for all t > 0. Suppose first n = 1. In case (a), L t is not a.e. an interval. Decompose L t in a.e. connected components (i.e., connected sets minus a set of zero measure). Each component I is a.e. an interval with finite positive measure since L t is open with finite measure, and for each open interval I 0 ⊃ I (I 0 6 = I ), we have |I 0 ∩ L ct | > 0. Pick two components I1 , I2 , with I1 lying to the left of I2 , and call J the interval between them (necessarily with positive measure); assume |I1 | ≤ |I2 |, and let m be the midpoint of J . If δ > 0 is small enough (e.g., δ < |I1 |/2), then for each H ∈ (m − δ, m) it holds that |ρ H I1 ∩ L ct | > 0 and |ρ H I2 ∩ L ct | > 0. But this implies that, for all H 6= 0 in a small enough interval, f (x) < f (ρ H x) on ρ H I1 ∩ L ct and f (x) > f (ρ H x) on I2 ∩ ρ H L ct , and these two sets are both in H + and have positive measure. The argument is similar when |I1 | ≥ |I2 |. This implies the lemma in case (a). In case (b) we can find t1 < t2 such that L t1 and L t2 are intervals a.e., nonempty, and not with the same center. (Otherwise, f would be a translate of f ∗ !.) Clearly, L t2 ⊂ L t1 . Since f is continuous, the set L t1 \ L t2 = {x : t1 < f (x) ≤ t2 } is the union of two open nonempty intervals I1 , I2 , which are separated by L t2 . The proof goes as in case (a).

508

CARLO MORPURGO

Let now n ≥ 2. In case (a), we have two subcases: (a.1) there is a line ` such that ` ∩ L t is nonempty and is not a.e. an interval on `, and (a.2) ` ∩ L t is either a.e. an interval or empty, for every line `. (Note: since L t is open, if ` ∩ L t is nonempty, then it has positive measure on `.) In subcase (a.1), since ` ∩ L t is bounded and open in the R-topology of `, we can proceed as in the one-dimensional case (a), and find an interval I on ` so that for each h ∈ I the function f /` satisfies the desired one-dimensional property. Since the set L t is open, the conclusion follows also for f by taking hyperplanes through I orthogonal to `. In subcase (a.2), we use the following. CLAIM

Let B ⊂ Rn be a bounded open set that is not a.e. a ball and so that for each line ` either B ∩ ` = ∅ or B ∩ ` is a.e. an interval on `. Then we can find two distinct parallel lines `1 and `2 that intersect B in a.e. intervals I1 and I2 , with the property that the orthogonal projection of I2 on `1 is not concentric to I1 . This claim can be shown by induction on the dimension in a more or less straightforward way; we leave the details of the proof as an exercise. Returning to the case (a.2), let `1 , `2 be the lines of the claim corresponding to B = L t . Project the intervals I j = ` j ∩ L t orthogonally on a given line ` parallel to `1 (e.g., `1 itself), and let I be the nonempty open interval between the midpoints of such a projected interval. If necessary, restrict I so that 0 ∈ / I . Given any hyperplane H through I and perpendicular to `, either `1 ∩ ρ H L t ∩ L ct and `2 ∩ ρ H L ct ∩ L t have positive measure in `1 , `2 , or vice versa (i.e., switching `1 with `2 ). In either case we can then use the fact that L t is open to conclude the proof of the lemma in case (a) when n ≥ 2. To treat case (b), that is, dimension n ≥ 2 and for every t > 0, L t is a.e. a ball or empty, we can find t1 < t2 such that L t1 and L t2 are a.e. balls, nonempty, and not with the same center. Clearly, L t2 ⊂ L t1 . We can apply the one-dimensional argument of case (b) to the restriction of f along the line joining the centers of L t1 , L t2 , and then finish up the proof using the continuity of f . Continuing the proof of the equality case of Theorem 3.5, suppose that (55) is false; suppose without loss of generality that 1 ∈ R1 , that f 1 is not a translate of its decreasing rearrangement, and that f j do not vanish for j ∈ R1 \ {1}. (Recall our assumption that f j ≥ 0 and is continuous.) By Lemma 3.11 we find hyperplanes H such that the first two conditions of (54) are false. To see that the third condition must also be false,


509

let first n = 1, and suppose that a function f is continuous and satisfies f = f ◦ ρ H for every H in an open interval I . It is then easy to see that for each H, H 0 ∈ I the function f must be periodic of period 2(H − H 0 ), and this implies that f must be constant. Thus, since we assumed that the f j are not constant when j ∈ R1 \ {1}, the third condition of (54) cannot be true. If n ≥ 2, f is continuous, and f = f ◦ ρ H for every H orthogonal to a line `, passing through an open interval I ⊂ `, then we deduce that f must be constant along every line parallel to `. Applying this to the functions f j , we deduce that if the third condition of (54) is true, then all of the f j with j ∈ R1 \ {1} must vanish in Rn , against our assumption. This concludes the proof of the equality statement in the case when all the f j are nonnegative and continuous. Now let f j ≥ 0 a.e. and in L 1 . Let φ : Rn → [0, ∞) be continuous with compact R support, radially symmetric, decreasing, and with Rn φ = 1. Let φ (x) = −n φ(x/) so that f j = φ ∗ f j → f j in L 1 and a.e. in Rn , as → 0 (more precisely, along an appropriate sequence m → 0). If the functions f j satisfy (54), then so do the continuous functions f j ; indeed, Z f j (x) − f j (ρ H x) = φ (x − y) − φ (x − ρ H y) f j (y) − f j (ρ H y) dy, H+

and this expression is nonnegative, nonpositive, zero if, respectively, f j = f jH , f j = f jH ◦ ρ H , f j = f j ◦ ρ H a.e. Thus, if the f j satisfy (54), then the f j satisfy (55), and by letting → 0, the f j also satisfy (55). If the f j are bounded and nonnegative, let f jk = f j if f j > 1/k, and f jk = 0 otherwise. By assumption, the set where f j > 1/k has finite measure, so that the f jk are in L 1 , and also f jk ↑ f j . If the f j satisfy (54), then the f jk also satisfy (54), and this implies that (55) must hold for the f j . If the functions f j are bounded but not necessarily nonnegative, we apply the previous result to the functions f˜j = f j − ess inf f j , which obviously satisfy (54) if the f j do. Suppose now that the f j are bounded below, and let f jk = min( f j , k). If the f j satisfy (54), then it is easy to see that this happens also for the f jk , which must then satisfy (55). Since ( f jk )∗ ↑ f j∗ , then the f j satisfy (55) also. Finally, if f j is arbitrary, let f jk = max( f j , −k), so that ( f jk )∗ = ( f j∗ )k ↓ f j∗ ; more generally, (ϕ ◦ f )∗ = ϕ ◦ f ∗ if ϕ is decreasing (not necessarily strictly decreasing). The same reasoning as before concludes the proof of Theorem 3.5. Remark. In the case of a single function, the above proof gives the following: If f is a measurable function satisfying (30) and for each H either f = f H a.e. or f = f H ◦ ρ H , then f is equal a.e. to a translation of f ∗ .

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Proof of Theorem 3.1 Inequality. Suppose first that w ∈ L ∞ (Rn ). Inequality (33) follows in this case from Lemma 2.7 and Theorem 3.5 since E d (t, r ) is strictly decreasing in r (e.g., by the subordination principle). Assume now that w is as in the statement of Theorem 3.1, but with w ≥ 0. Let wk = min(w, k) ↑ w, so that I9 [wk , t] ≤ I9 [wk∗ , t]. Now we proceed as Rt Rt in the proof of Theorem 3.5. First, note that 0 wk ↑ 0 w; and the same holds for wk∗ , w∗ . Next, if I9 [w, t] is finite, then 9(0) = 0 by (35), and one has three cases: 9 ≥ 0, 9 ≤ 0, 9 changes sign. In the first two cases the monotone convergence theorem gives (33), and in the third case we write 9 = 9+ − 9− , and I9 [wk , t] = I9+ [wk , t] − I9− [wk , t]. The function 9+ is increasing, so that the monotone convergence theorem gives I9+ [wk , t] ↑ I9+ [w, t], and, similarly, I9+ [wk∗ , t] ↑ I9+ [w ∗ , t]. Next, the function 9− is continuous and concave in some interval [0, C], and there is a point M ∈ (0, C) so that 9− is increasing in [0, M]. Let A = {x : w(x) > M/t}, and let C (A) = {B ∈ D : Bv ∈ A, some 0 ≤ v < t}. We note that C (A) = {B : D A (B) < t}, where D A (B) = inf{u ≥ 0 : Bu ∈ A}, is known as the first entrance time in A. It is a known result that C (A) is measurable in F ⊂ Ft (see [Chu]). Then we can write Z Z o Z t n Z I9− [wk , t] = + + 9− wk (Bv ) dv d x dµx,x,t . Ac ×C (A)c

A×D

Ac ×C (A)

0

The first term converges to the corresponding integral evaluated at w, by the monotone convergence theorem. The second term is bounded above by a constant times |A| < ∞, so that the dominated convergence theorem applies. To handle the third term, we use the following lemma. LEMMA 3.12 Let A be a Borel set in Rn with finite measure, and let

C (A) = {B ∈ D : Bv ∈ A for some 0 ≤ v < t},

a measurable set in Ft . Then Z

d x d Px,x,t (B) ≤ |A|.

(56)

Ac ×C (A)

Recall that µx,x,t = E d (t, 0)Px,x,t . The proof of this lemma is outlined in the appendix, but the idea is essentially that loops starting outside A and touching A can be counted as loops starting inside A and touching Ac .


511

Remark. The lemma also holds more generally for a Feller process with a symmetric transition density. Remark. Lemma 3.12 justifies (35). Indeed, assume for simplicity that w ≥ 0 a.e., 9(0) > 0, and 9+ = 9. By continuity, 9(u) ≥ m > 0 for 0 ≤ u ≤ . Letting A = {x : w(x) > /t}, then C (A)c = {B : Bv ∈ Ac , ∀v ∈ [0, t)}, and Z Z I9 [w, t] ≥ m d x d Px,x,t (B) = ∞, Ac

C (A)c

which follows from Lemma 3.12 and the fact that Ac × D has measure |Ac | = ∞. Assuming this lemma, we can again apply the dominated convergence theorem and conclude that I9− [wk , t] → I9− [w, t] as k ↑ ∞, and also that I9− [wk∗ , t] → I9− [w ∗ , t]. All this together yields (33). Rt Observe that the proof works also if 0 w(Bv ) dv = +∞ for B in a set of positive measure. Suppose now that I9 [w, t] is not finite, in which case we assume that 9 has constant sign. If 9(0) 6 = 0, then both sides of (33) are either +∞ or −∞; the same holds even if 9(0) = 0, in this case by the monotone convergence theorem applied to the sequence wk . Next, if ess inf w > −∞, then we can apply the previous point to the functions e (u) = 9(u + t · ess inf w), using the hypothesis w˜ = w − ess inf w ≥ 0 and 9 Rt 0 |w(Bv )| dv < ∞ if w changes sign on sets of positive measure. If ess inf w = −∞ and 9(−∞) = 0, then 9 is nonnegative and increasing. Suppose first that w = −∞ outside a compact set F and that w is bounded above. Arguing as before, I9 [w, t] is finite, +∞, −∞ if and only if 9(−∞) = 0, 9(−∞) > 0, 9(−∞) < 0. Suppose that I9 [w, t] is finite, so that 9(−∞) = 0, and that 9 is nonnegative and increasing. Let wk = max(w, −k) ↓ w, and let 9k (u) = 9(u) − 9(−tk), a convex function of u. Thus, I9k [wk , t] ≤ I9k [wk∗ , t] by the reRt sults just proved. The function B → 9k 0 wk (Bv ) dv vanishes in the set C (F)c , so that, again using Lemma 3.12 and the dominated convergence theorem, we can conclude that I9k [wk , t] → I9 [w, t] and, similarly, that I9k [wk∗ , t] → I9 [w∗ , t]. Finally, suppose that ess inf w = −∞, not necessarily bounded above, and let wk = min(w, k) on B(0, k) and wk = −∞ on B(0, k)c . Then wk ↑ w, and the inequality follows from the monotone convergence theorem. Equality. For every hyperplane H 6 3 0, the function w H is equimeasurable to w; hence, I9 [w, t] ≤ I9 [w H , t] ≤ I9 [w ∗ , t].

512

CARLO MORPURGO

Thus, it is enough to estimate the quantity I9 [w H , t] − I9 [w, t] from below; the following proposition gives the first result we need. PROPOSITION 3.13 For a given hyperplane H 63 0, the function

E dH (t, x, y) = E d (t, |x − y|) − E d (t, |x − ρ H y|),

x, y ∈ H + ,

defines a Feller semigroup on C0 (H + ). The Feller process corresponding to this semigroup induces a probability measure PxH on the space D H of functions B : [0, ∞) → H∂+ that are right continuous and with left limits, valued in the one-point compactH , and ification of H + . Denote the pinned measures for this Feller process by Px,y,t H H H µx,y,t = E d (t, x, y)Px,y,t . Under the same hypothesis as that of Theorem 3.1, define Z Z h Z t Z t i H H I9 [w] = 9 w(Bv ) dv + 9 w(ρ H Bv ) dv d x dµx,x,t (B); H+

DH

0

0

then I9 [w H , t] − I9 [w, t] ≥ I9H [w H , t] − I9H [w, t].

(57)

H ) that satisfy Moreover, for almost all paths B ∈ D H (with respect to d x dµx,x,t Bv ∈ ∪ ρ H for a.e. v ∈ [0, t], we have

9

t

Z 0

Z t w(Bv ) dv + 9 w(ρ H Bv ) dv 0 Z t Z t ≤9 w H (Bv ) dv + 9 w H (ρ H Bv ) dv ; (58) 0

0

if 9 is strictly convex, then equality occurs if and only if either w(Bv ) ≤ w(ρ H Bv ) for a.e. v ∈ [0, t], or w(Bv ) ≥ w(ρ H Bv ) for a.e. v ∈ [0, t]. Remark. In the case d = 2 the process of the above proposition coincides with the Brownian motion killed upon exiting H + . In the case d < 2, however, this process does not coincide with the d-stable process killed outside H . Note. Inequality (58) (and the corresponding equality statement) is the infinitedimensional analogue of Lemma 3.9. Proof Note that E dH (t, x, y) is a sub-Markovian transition density, which means that it satR isfies the properties of a transition density with H + E dH (t, |x − y|) dy < 1; by a


513

standard trick one can extend this density to a Markovian one in the space H∂+ (see [Chu]). Observe that the space C0 (H∂+ ) of continuous functions, vanishing at the point at infinity ∂, is identified with the space of continuous functions f on H + so that limx→∂ H + f (x) = 0. The generator of the semigroup is found easily by noting R R that H + E dH (t, x, y) f (y) dy = Rn E d (t, |x − y|)R H f (y) dy, where R H f is the odd extension of f to Rn . The existence of the corresponding Feller and pinned processes follows in the same way as described in Section 2. To prove (57) we proceed, once again, by approximation. Suppose first that w is bounded. By Lemma 2.7, formula (29), we can write I9 [w H , t] − I9 [w, t] = lim N →∞ (Q N [w H , t] − Q N [w, t]), where N Y

Z Q N [w, t] =

(Rn ) N j=1

E d (t/N , |x j − x j+1 |)9

N t X w(x j ) d x1 · · · d x N . N j=1

By (53) H H Q N [w H , t] − Q N [w, t] ≥ Q N [w H , t] − Q N [w, t],

where H QN [w, t] =

Z

N Y

(H + ) N j=1

E dH (t/N , x j , x j+1 )

N N h t X t X i × 9 w(x j ) + 9 w(ρ H x j ) d x1 · · · d x N . N N j=1

j=1

H [w, t] = I H [w, t]. This follows in the same way as in Now we have lim N →∞ Q N 9 Lemma 2.7, using the Feynman-Kac formula and the Trotter product formula for the H L 2 (H + )-semigroup e−t (1d +λw) . This implies (57) for bounded w. To show (57) for w bounded below, we let wk = min(w, k) ↑ w; as shown in the proof of inequality (33), I9 [wk , t] → I9 [w, t], and since (wk ) H = (w H )k ↑ w H , we also have I9 [wkH , t] → I9 [w H , t]. The same is true for the functional I9H by adapting the arguments to the process in H + . Finally, the proof for general w is obtained by approximating with functions wk bounded below, using the convergence arguments in the proof of the inequality (33). The proof of (58) is similar to the one in the discrete case of Lemma 3.9. First, R observe that the hypothesis of Theorem 3.1, L(t,B) |w(Bv )| dv < ∞ up to a set of zero d x dµx,x,t -measure on Rn × D , implies easily that |w(Bv )| is integrable on {v : Bv ∈ } and that |w(ρ H Bv )| is integrable on {v : ρ H Bv ∈ }, up to a set of H -measure in H + × D ; this follows essentially because E H (t, x, y) ≤ zero d x dµx,x,t H d E d (t, |x − y|). So from now on we work outside this exceptional set of zero measure.

514

CARLO MORPURGO

Let Bv ∈ ∪ ρ H for a.e. v ∈ [0, t], and let I = {v ∈ [0, t] : w(Bv ) > w(ρ H Bv )}, J = {v ∈ [0, t] : w(Bv ) < w(ρ H Bv )}, K = {v ∈ [0, t] : w(Bv ) = w(ρ H Bv )}; by the right continuity of B these sets are all Borel measurable in [0, t]. Let Z t Z Z Z w(Bv ) dv = w(Bv ) dv + w(Bv ) dv + w(Bv ) dv := a1 + b1 + c, 0 I J K Z t Z Z w(ρ H Bv ) dv = w(ρ H Bv ) dv + w(ρ H Bv ) dv J 0 I Z + w(ρ H Bv ) dv := a2 + b2 + c. K

Since w(Bv ) is finite for v ∈ I , by hypothesis we also have a1 finite, and the same is true for b2 . Since w(Bv ) and w(ρ H Bv ) cannot be both −∞, we also have c finite. The proof of the inequality follows from (50) applied to the convex function 9( · + c). If 9 is strictly convex, then equality does not hold if and only if a1 > a2 and b1 < b2 , which means that for equality we need |I | = 0 or |J | = 0. To continue the proof of the equality statement of Theorem 3.1, suppose that 9 is strictly convex and that it is not true that w is a translate of w∗ ; for the moment we do not assume that is open and connected. By Lemma 3.11 we can find a hyperplane H 6 3 0 so that the sets A1 = {x ∈ H + : f (x) < f (ρ H x)} and A2 = {x ∈ H + : f (x) > f (ρ H x)} have positive Lebesgue measure. Write Z Z H I9H [w H , t] − I9H [w, t] = J9H [w H , t, B] − J9H [w, t, B] d x dµx,x,t (B), H+

where J9H [w, t, B] = 9

DH

t

Z 0

(59)

Z t w(Bv ) dv + 9 w(ρ H Bv ) dv . 0

We want to show that in this situation (59) is positive. To do that, if w is finite a.e., that is, if |c | = 0, it is enough to show, using Proposition 3.13, that the set of paths H -measure (i.e., there that stay in A1 and A2 for a positive time has positive d x dµx,x,t are enough bridges where the integrand in (59) is positive). But if w = −∞ on a set of positive measure, then we need to make sure that the paths above do not stay in the complement of ∪ ρ H for a positive time; otherwise, the integrand in (59) vanishes. To this end we need analogues of (36) and (37). For B ∈ D H , let Z t n o H := ( ∪ ρ H ) ∩ H + , S H (B) = inf t > 0 : χ H (Bv ) dv < t . 0

The quantity

Rt 0

χ H (Bv ) dv is the time spent by the process inside H , up to time t.


515

Even in this case we have, with PxH -probability 1, B : Bv ∈ H , a.e. v ∈ [0, t] = S H ≥ t = S H > t . As in Lemma 3.3, we have that a.e. x ∈ / H is S-regular for H + \ H , that is, H (S PxH (S H = 0) = 1; hence, Px,y,t / H and a.e. y ∈ H + . H > t) = 0 for a.e. x ∈ This implies that I9H [w, t] Z Z =

h Z t Z t i H 9 w(Bv ) dv + 9 w(ρ H Bv ) dv d x dµx,x,t (B),

H ×{S H >t}

0

0

and the same holds if w is replaced by w H . This is because for a.e. x ∈ / H almost every loop that starts and ends at x must spend positive time outside H , killing the integrand above. Note that the set H constructed from w H is the same as the one obtained from w. Therefore, (59) can be written as I9H [w H , t] − I9H [w, t] Z Z =

H (B), (60) J9H [w H , t, B] − J9H [w, t, B] d x dµx,x,t

H ×{S H >t}

and the integrand is positive in the set of paths that stay in both A1 and A2 for a positive time and that do not leave H for a positive time (i.e., stay within {S H > t}). H -measure, we consider the process in To show that this set has positive d x dµx,x,t H + H (with transition function E d (t, x, y)) killed when it penetrates H + \ H . This transition function is given by H (S H > t), E dH (t, x, y; H ) := µx,y,t

x, y ∈ H ,

satisfies the Chapman-Kolmogorov identity, is symmetric in x, y, and satisfies Z H E dH (t, x, y; H ) dy Px (Bt ∈ F, S H > t) =

(61)

(62)

F

for any Borel set F (see [DvC, App. C, Sec. 2, App. D, Ths. D.3 and D.4, and Rem. 2

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CARLO MORPURGO

on p. 403]). Here H can be any Borel set inside H + . Hence, for any x ∈ Rn , Z Z tZ t H dµx,x,t χ{u:Bu ∈A1 } (u)χ{v:Bv ∈A2 } (v) dv du 0

{S H >t} t

Z

0 t

Z

H µx,x,t (Bu ∈ A1 , Bv ∈ A2 , S H > t) dv Z u Z Z du dv E dH (t − u, x, z; H )

du

= 0

Z

0

t

≥ 0

×

0

E dH (u

A1

A2 H

− v, z, y; )E dH (v, y, x; H ) dy dz.

Thus, if we know that E dH (t, x, y; H ) > 0, ∀t > 0 and all (or a.e.) x, y ∈ H , then the set {(x, B) ∈ H × {S H > t} : |{u : Bu ∈ A1 }| > 0 and |{v : Bv ∈ A2 | > 0} H -measure, and on this set the integrand of (59) is positive by has positive d x dµx,x,t Proposition 3.13. It is only at this point that we assume that is open and connected, for in this case H is also open and connected in H + and we can use the following. PROPOSITION 3.14 H (S > t) is jointly If D is open and connected in H + , then E dH (t, x, y; D) := µt,x,y D H H continuous, symmetric, and positive in (0, ∞) × × for all x, y ∈ D.

Proof The continuity result is true for general Feller processes having a continuous and symmetric density, and killed outside any open set D (see [DvC, App. D, Th. D.5, and Remark 2 on p. 403]). The positivity result is known for the symmetric d-stable process on Rn killed upon hitting D c , with D open and connected in Rn . This means that if TD = inf{s > 0 : Bs ∈ D c } is the first exit time from D, then the corresponded (t, x, y; D) := µt,x,y (TD > t), which satisfies the ing killed process has density E semigroup property and (62), and is symmetric, jointly continuous, and positive for all x, y ∈ D. For d = 2, see S. Port and C. Stone [PS]; for 0 < d < 2 and smooth open connected D, see [CS, Thm. 2.4]. Note that for general D, given x, y ∈ D, one can always find a smooth domain D 0 ⊆ D containing x, y and apply monotonicity of heat kernels.) Similarly, we can prove that if we kill the process in H + given by E dH upon hitting D c , with D open and connected in H + , then we still get a positive eH . The proof of this fact is completely similar to the one given in transition function E d [PS] and [CS]. For example, in case d < 2, one has the following asymptotic result: lim r n+d+`

r →∞

∂` 0(n + d + `) E d (1, r ) = (−1)` C(n, d). ` 0(n + d) ∂r

(63)

The constant C(n, d) is positive for 0 < d < 2 (and zero for d = 2) and was computed explicitly in [BG] (which derived (63) for ` = 0, extending a 1-dimensional


517

result of G. Pólya). The same proof, with minor modifications, gives the general case. Now, if 0 < r1 < |x − y| < r2 (x, y ∈ D), then by (63) there is a t0 > 0 small enough so that for 0 < t ≤ t0 , E dH (t, x, y) −1/d |x−ρZ H y|t

=

t −n/d − ∂u E d (1, u) du ≥ c t |x − y|−n−d − |x − ρ H y|−n−d

|x−y|t −1/d

and E dH (t, x, y) ≤ 2c t |x − y|−n−d . If in addition x, y belong to a small enough ball, then E dH (t, x, y) ≥ c0 t |x − y|−n−d . These estimates are exactly the same as the ones used in [CS, Th. 2.4], and the rest of the proof proceeds unchanged. A similar argument can be made in the simpler case d = 2. Finally, PxH (S D ≥ TD ) = 1 ∀x ∈ eH (t, x, y; D) > 0 for all t > 0 and all x, y ∈ H + . H + ; hence, E dH (t, x, y; D) ≥ E d This concludes the proof of Proposition 3.14 and at the same time concludes the proof of the equality statement of Theorem 3.1. Remark. If w = 0 inside and w = −∞ outside , and if we denote 1H = ∩ H + and 2H = ρ H ∩ H + , then from (57) and (60) we easily obtain Z I9 [w ∗ , t] − I9 [w, t] ≥ 29(0) µx,x,t (S H > t, S H ≤ t, S H ≤ t) d x, (64) H

1

2

namely, the d x dµx,x,t -measure of the set of bridges that are inside H for almost all times, and stay in H \ 1H for a positive time and in H \ 2H for a positive time. Note that 9(0) > 9(−∞) = 0 by strict convexity. This lower bound should be compared with the one in [BS, 3.39]. By the assumption that w is not a.e. a translate of w, the proof of Theorem 3.1 yields that the right-hand side of (64) is positive. We end this section about rearrangement with a couple of inequalities related to certain Dini-type integral norms, which are used later in this work (Section 5). THEOREM 3.15 Under the hypothesis of Theorem 3.1, if we define Z Z h Z t Z t i 1 h9 [w, t] = 9 tw(Bv ) dv − 9 w(Bv ) dv d Px,x,t d x, 0 Rn D t 0

then h9 [w, t] ≥ h9 [w∗ , t].

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518

CARLO MORPURGO

Under the hypothesis of Theorem 3.5, if we define h9 [ f 1 , . . . , f N ] Z N N N h1 X X i Y = K i j (|xi − x j |) 9 N f j (x j ) − 9 f j (x j ) d x1 · · · d x N , N (Rn ) N i< j

j=1

j=1

then h9 [ f 1 , . . . , f N ] ≥ h9 [ f 1∗ , . . . , f N∗ ].

(66)

If the left-hand sides of (65) or (66) are finite, then equality holds under the same conditions as those in Theorems 3.1 and 3.5. We note that the integrands entering in the definition of the functionals are nonnegative, by Jensen’s inequality, and that if 9 is concave, the reversed inequalities hold. Proof The proofs of the above inequalities are identical to those of Theorems 3.1 and 3.5, PN and are based on the discrete inequality corresponding to (46) with 9 f k (Pk ) 1 P PN replaced by N −1 1N 9(N f k (Pk )) − 9 1 f k (Pk ) . In turn, this inequality (with the equality case) follows as in the proof of Theorem 3.8 since the additional quantity is invariant under discrete rearrangement. R We note that the finiteness of h9 does not imply the finiteness of Rn 9(tw(x)) d x (in which case the inequalities are direct consequences of Theorems 3.1, 3.5). A case that interests us later is the function w = |Jπ |d/n , and K 12 (|x − y|) = |x − y|2d−2n , which, with ψ(u) = u 2 , satisfies hψ [w] = hψ [w, . . . , w] < ∞ when (n − 2) < 2d < n (Lemma 5.8), but it is not always an L 2 -function (e.g., when n = 2 and d ≤ 1/2). We end this section about rearrangements with a few remarks about the S n version of the symmetrization results just presented. For measurable functions F on S n , one can define the symmetric decreasing rearrangement F ∗ in terms of the “great circle” distance on the sphere. The hyperplane rearrangement F H is defined as in Rn but for hyperplanes through the center of the sphere, and where H + , H − are the two corresponding hemispheres. All the machinery developed for Rn works also for S n ; in particular, the spherical versions of Theorems 3.5 and 3.6, Lemma 3.7, and Corollary 3.10 are still true. Theorem 3.1 is true on S n , provided one checks that the processes related to exp(−t Ad ) on the sphere satisfy the properties used in the proof. Applications to traces of Schrödinger semigroups Suppose that is an open domain of Rn , not necessarily with finite measure. As we mentioned in the proof of Proposition 3.14, the first exit time for the d-stable process


519

on Rn is defined as T (B) = inf{t > 0 : Bt ∈ / }, and the first penetration time S in c is defined as in (36). From the definition, we have Px (S ≥ T ) = 1 for all x ∈ Rn . As it is well known, one can define two semigroups killed outside by killing using penetration times or killing using hitting times. Denote the corresponding transition densities as E d (t, x, y; ) := µx,y,t (S > t),

ed (t, x, y; ) := µx,y,t (T > t), E

ed (t, x, y; ) ≤ E d (t, x, y; ) and are jointly continuous and symwhich satisfy E metric. The corresponding semigroups act on L 2 (), and their generators are de^ d/2 ) , also called pseudo-Dirichlet and Dirichlet Laplanoted by (1d/2 ) and (1

cian, respectively (see [DvC], [AB]). These operators are both valid selfadjoint extensions of 1d/2 as acting on Cc∞ (), and they correspond to the quadratic form R Q(u, v) = u1d/2 v d x, but with respective domains H d/2 (Rn ) ∩ L 2 () = u ∈ d/2 L 2 (Rn ) : Q(u, u) < ∞, u = 0 a.e on c for (1d/2 ) , and H0 (), the closure 1/2 ^ d/2 ) . References here are [HZ], which of C ∞ () under Q(u, u) + k f k2 for (1 c

2

did the case d = 2 but which is also extendable to d ≤ 2, and [CS] or [Ok, (4.2)], and [HJ, Prop. 6.2]. d/2 It is easy to see that H0 () ⊆ H d/2 (Rn ) ∩ L 2 (), but in general equality does not hold unless c is Kac regular, that is, unless Px (S = T ) = 1 for each x ∈ c , as in this case we can just replace S with T and get the same semigroups (cf. [HZ], where the necessity of the condition is also proved for d = 2, but presumably the argument works also if d < 2). A sufficient condition for Kac regularity is that each T -regular point for c is a T -regular point for (c )◦ (see [DvC, Prop. 2.24]). Recall that a point x is T -regular for c if Px (T = 0) = 1. It is also known for Brownian motions that if x ∈ Rn satisfies the usual cone condition with respect to c , then x is T -regular, as shown in [PS, Prop. 3.3]. The same proof, with minor changes, yields the same result for d-symmetric stable processes in Rn , in particular, for smooth domains S = T with probability 1. ] d/2 ) + V For V ∈ L ∞ () the Schrödinger operators (1d/2 ) + V and (1 have the same form domain as the respective Dirichlet Laplacians, and they generate strongly continuous and positivity-preserving L 2 ()-semigroups having transition densities Z Z t −t ((1d/2 ) +V ) e (x, y) := exp − V (Bv ) dv dµx,y,t , {S >t}

e

−t ((] 1d/2 ) +V )

(x, y) :=

Z

0

{T >t}

t

Z

exp − 0

V (Bv ) dv dµx,y,t .

520

CARLO MORPURGO

(See [Ok, Prop. 4.2, 4.3] and [DvC, Th. 2.27].) Clearly, e−t ((1

] d/2 ) +V )

(x, y) ≤ e−t ((1

d/2 ) +V )

(x, y)

with equality in the case of smooth (or Kac regular) domains. When || < ∞, both semigroups are compact since the corresponding kernels are easily seen to be Hilbert-Schmidt. Therefore the Schrödinger operators above have a discrete spectrum, and their heat trace is finite. Moreover, if we set ( V (x) if x ∈ , e(x) = V +∞ if x ∈ / , then, from (37) with 9(u) = e−u , we obtain Tr[e−t ((1

] d/2 ) +V )

] ≤ Tr[e−t ((1 ) +V ) ] Z Z Z t e(Bv ) dv d x dµx,x,t . = exp − V d/2

Rn

D

0

Thus, we can apply Theorem 3.1 and obtain the following. COROLLARY 3.16 Let ⊂ Rn be an

open domain with || < ∞, and let V ∈ L ∞ (). Then Tr[e−t ((1

d/2 )

+V )

] ≤ Tr[e−t ((1

d/2 )

+V∗ ) ∗

(67)

]

and Tr[e−t ((1

] d/2 ) +V )

] ≤ Tr[e−t ((1

] d/2 ) +V ) ∗ ∗

] = Tr[e−t ((1

d/2 )

∗

+V∗ )

],

(68)

where ∗ is the ball centered at the origin and with volume ||, and where V∗ : e)∗ (x) ∗ → R is the symmetric increasing rearrangement of V (i.e., V∗ (x) = −(−V ∗ e for x ∈ , where V is defined as above). Equality occurs in (67) if and only if V is a.e. a translate of V∗ , that is, if is a.e. a ball and V = V∗ a.e. in . If equality occurs in (68), then V is a.e. a translate of V∗ . Clearly, one could extend the above result to more general potentials V , modulo justifying the relations between the Feynman-Kac generators and the extensions of the corresponding Schrödinger operators. The equality in (68) is due to the smoothness of the ball. Note that if the domain is smooth, or just Kac regular, (68) coincides with (67). But in the general case one cannot conclude that if V is a.e. a translate of V∗ , then equality in (68) should hold.


521

This is because removing a set of positive capacity has the effect of strictly increasing the first eigenvalue (see [GZ], where a proof of this fact is given for the case d = 2). In their independent work Burchard and Schmuckenschläger [BS] give a more general version of the above corollary in the special case when V ≡ 0 and d = 2. In particular, they consider the case when is a general Borel set, and they characterize the equality case in terms of polar sets; this last fact is essentially an immediate consequence of the lower bound (64). In [BS] the reader may also find interesting symmetrization results for exit times of Brownian motions. The novelty in Corollary 3.16 is the equality statement, for the inequality itself appears, for example, in [Lu] (in the case of the Laplace operator, that is, Brownian motions) and is derived using the Trotter product formula and Riesz’s inequality (42). The equality statement, however, is not a direct consequence of Theorem 3.15. 4. Sharp inequalities for the zeta and 9-functions In this section we state and prove the first main trace inequalities of this work. Recall that the functionals J9 [W ], I9 [w], Js [W ], Is [w] were defined in (22) – (25) and that their relations with the zeta and 9-functions are given in Proposition 2.6. THEOREM 4.1 R1 Let 9 be convex and continuous, and let 0 t −n/d−1 |9(t)| dt < ∞. Suppose that n/d n/d 0 ≤ W ∈ L n/d (S n ), 0 ≤ w ∈ L n/d (Rn ), and kW kn/d = kwkn/d = |S n |. Then, for 0 < d ≤ 2 and 0 < d < n, J9 [W ] ≤ J9 [1], I9 [w] ≤ I9 |Jπ |d/n . (69)

If s > n/d, with s an integer if d > 2, then Js [W ] ≤ Js [1],

Is [w] ≤ Is |Jπ |d/n .

Equality in the above inequalities holds if and only if W = |Jτ |d/n a.e. for some d/n τ ∈ CON(S n ), or w = (|Jπ | ◦ h)|Jh | a.e. for some h ∈ CON(Rn ). Note that from Proposition 2.6 we have J9 [1] = I9 |Jπ |d/n = Z 9 (Ad ), which is finite under the given integral condition on 9, from Lemma 2.4. Also, by an easy calculation, the conformal images of the function |Jπ | are the translations and dilations of the function itself. In the special case s = 2 we obtain the sharp Hardy-Littlewood-Sobolev inequality (5), and in the case when s is any integer we obtain a special case of a multilinear inequality due to Beckner [Bec3]. By Proposition 2.6 we can spell out the above results in terms of traces and zeta functions; on S n we obtain the following.

522

CARLO MORPURGO

COROLLARY 4.2 Under the hypothesis of Theorem 4.1, we have

Z 9 Ad (W ) ≤ Z 9 (Ad ),

Z s, Ad (W ) ≤ Z (s, Ad ),

∀s >

n , d

with the restriction s an integer if d > 2. In particular, λ0,d = λ0 (1) ≤ λ0 (W ). Equality in these inequalities occurs if and only if W = |Jτ |2/n a.e. for some τ ∈ CON(S n ). In particular, among all smooth metrics of given volume in the conformal class of the standard round metric of S n , the maximum of the zeta function is achieved precisely at the round metric, up to isometries. Proof of Theorem 4.1 We assume 0 < d ≤ 2 and give the proof of (69); the proof of the remaining case of Theorem 4.1 is completely similar. We follow closely the procedure used in [CL1] (see also [LL, Chap. 4] for more details). We report the the argument in this setting, in part for the convenience of the reader and also because some modification of it is used later on (proof of Theorem 5.10). Consider the rotation on S n with n ≥ 2, given as 2(ξ ) = (ξ1 , . . . , ξn+1 , −ξn ), where ξ = (ξ1 , . . . , ξn+1 ) ∈ S n ; this 90◦ rotation sends the north pole to the point (0, . . . , 0, 1). When n = 1, let 2 be any fixed rotation by an irrational multiple of π. e := π −1 ◦2◦π, The conformal transformation on Rn corresponding to 2 is given by 2 n/d n e and the action of 2 on functions w ∈ L (R ) is denoted by d/n e w := w2 e) |J2 2 . e = (w ◦ 2 e|

e |Jπ |d/n = |Jπ |d/n , and only the radial functions f such that As observed in [CL1], 2 e w is also radial have the form w = C |Jπ |d/n for some positive constant C. 2 If w ∈ L n/d (Rn ), denote Rw(x) = w∗ (x). Then e )k w → wk := (R 2

kwkn/d |Jπ |d/n |S n |d/n

(70)

strongly in L n/d (see [CL1] and [LL, Theorem 4.6]). n/d Let now 0 ≤ w ∈ L n/d (Rn ), kwkn/d = |S n |, and recall that I9 [w] = Tr[L9 (w1/2 1−d/2 w1/2 )]. We want to show that I9 [w] ≤ I9 [|Jπ |d/n ] = d/n , by the monotone Tr[L9 (A−1 d )]. To do this, it is enough to assume that w ≤ C|Jπ | convergence theorem; this condition guarantees that I9 [w] < ∞. From (34) we have I9 [w] ≤ I9 [Rw] with equality if and only if Rw is a.e. e w]. This a translate of w, and from Proposition 2.3 we also have I9 [w] = I9 [2


523

implies that if we define the sequence wk as in (70), then I9 [wk ] ≤ I9 [wk+1 ]. We pass to a subsequence (still denoted by wk ) so that wk → |Jπ |d/n a.e., and since wk ≤ C|Jπ |d/n (see [CL1, Theorem 3.1]), uniformly in k, we conclude by the dominated convergence theorem that I9 [w] ≤ I9 [wk ] ↑ I9 [|Jπ |d/n ]. To prove the equality statement, suppose that equality holds in (20) for a given e w] = I9 [R 2 e w]. This implies that 2 ew w ∈ L n/d (Rn ), so that I9 [w] = I9 [2 e e is a translation of the function R 2w; that is, R 2w is a conformal transformation of w, and by iteration every wk (constructed from the given w) is a conformal transformation of w: wk = h k w := wh k . Thus, by the conformal invariance of the norms, d/n → w in L n/d . As shown in [LL, Theorem 4.7], the conformal images of h −1 k |Jπ | the function |Jπ |d/n are the dilations and the translations of the function itself, and d/n → e hence h −1 h|Jπ |d/n pointwise for some e h ∈ CON(Rn ), which implies k |Jπ | d/n w =e h|Jπ | a.e., that is, a conformal image of the stereographic Jacobian.

5. Zeta regularization and (logarithmic) Sobolev inequalities The purpose of this section is to “unravel” the zeta function around its largest pole, s = n/d, and to obtain explicit computations and extremal results for its analytic continuation. The sphere setting plays no role in the actual computations, so we treat a general compact manifold and more general operators. Regularized zeta functions: General formulas Throughout this section, M denotes a compact Riemannian manifold with metric g and without boundary. The geodesic distance in M is denoted by d(x, y). Recall that a (polyhomogeneous) pseudodifferential operator of order d on Rn is an operator P : Cc∞ (Rn ) → C ∞ (Rn ) such that there exists a sequence of functions pd− j (x, ξ ) ∈ C ∞ (R2n ), j = 0, 1, . . . , satisfying the following conditions: (a) pd− j (x, tξ ) = t d− j pd− j (x, ξ ) for t, |ξ | ≥ 1; β (b) |Dx Dξα pd− j (x, ξ )| ≤ C(1 + |ξ |)d−|α| with x ∈ K compact, ξ ∈ Rn , and C independent of x, ξ ; (c) for any λ ∈ R, integer k, and f ∈ Cc∞ (Rn ), k

X

Op( pd− j ) f

P f − j=0

where, with standard notation, Z Z Op( p) f (x) = Rn

Rn

L 2λ−d+k+1

≤ Ck f k L 2 , λ

e2πi(x−y)·ξ p(x, ξ ) f (y) dy dξ.

The principal symbol of P is the function pd (x, ξ ).

524

CARLO MORPURGO

A pseudodifferential operator of order d on M is an operator P : C ∞ (M) → such that, for any local chart (U, h) and any φ, ψ ∈ Cc∞ (U ), the operator h ∗ (φ Pψ) is a pseudodifferential operator of order d in Rn . The leading symbol of P is the leading symbol of the localized operator h ∗ (φ Pφ), where φ = 1 around a given point x ∈ M in Rn . In what follows we consider a positive, selfadjoint 9DO of order d > 0 on d/2 2 L (M, d x), with the same principal symbol as w−1 1g , where 1g is the positive Laplace-Beltrami operator in the metric g and where 0 < w ∈ C ∞ (M). If Pw is such an operator, then in normal local coordinates y = expx Y , with x, y ∈ M and Y ∈ Rn , the symbol can be chosen so that

C ∞ (M)

pd (0, ξ ) = w(x)−1 (2π|ξ |)d ,

|ξ | ≥ 1,

(71)

modulo a modification on the set |ξ | ≤ 1. The heat semigroup e−t Pw on L 2 (M, d x) has a smooth kernel K w (t, x, y), and the operator Pw−s has kernel (Green’s function) Z ∞ 1 G w (s, x, y) = t s−1 K w (t, x, y) dt, Re s > 0, 0(s) 0 which is smooth in x, y and analytic in s (actually in the whole complex plane), away from the diagonal. This has been proved by R. Seeley [Se]. If 0 < λ0 (w) ≤ λ1 (w) ≤ · · · ↑ ∞ are the eigenvalues of P w , counted with multiplicity, and {φ wj } the corresponding eigenfunctions, then K w (t, x, y) = P∞ −tλ (w) w P −s w w j φ j (x)φ wj (y), and G w (s, x, y) = ∞ 0 e 0 λ j (w)φ j (x)φ j (y). The zeta function of Pw is defined as Z ∞ X −s −s Z (s, Pw ) = λ j (w) = Tr[Pw ] = G w (s, x, x) d x. j=0

M

It is known (see [Se]) that G w (s, x, x), and hence Z (s, Pw ), is regular for Re s > n/d and can be analytically continued to a meromorphic function having simple poles only at s = (n − j)/d with j = 0, 1, 2, . . . , or s = (n − 2 j)/d in the case of differential operators of even order d (see [Gre]). It turns out that Z (s, Pw ) is always regular at s = 0 (see [Se], [Gre]). In this section we consider operators having as principal symbol a multiple of the power Laplacian. We see that for such operators the second largest pole is in fact at s = (n − 2)/d, and we derive explicit representations of Z (s, Pw ) between s = n/d and s = (n − 2)/d; this is done by finding the the off-diagonal asymptotics of the Green function G w and then by regularizing. These formulas are then used to compare the regularized zeta functions for Pw with the one for P1 (i.e., setting w ≡ 1).


525

THEOREM 5.1 Let w > 0, w ∈ C ∞ (M), and let Pw be a positive, selfadjoint, pseudodifferential d/2 operator of order d > 0, with the same leading symbol as w−1 1g . Then, for x ∈ M,

2wn/d (x) 1 · + 8w (s, x), (72) n 0(n)|S | sd − n where 8w (s, x) is analytic in Re s > (n − 2)/d and jointly continuous in (s, x). If y ∈ B(x, ), x 6= y, and (n − 2)/d < Re s < n/d, then G w (s, x, x) =

ws (x) 2wn/d (x) 1 d(x, y)sd−n + · + 8w (s, x, y), n γ (sd) 0(n)|S | sd − n 2wn/d (x) 1 G w (n/d, x, y) = log + 9w (x, y), 0(n)|S n | d(x, y) G w (s, x, y) =

(73) (74)

where 8w (s, x, y) is defined for (n − 2)/d < Re s < (n + 2)/d and y ∈ B(x, ), and in such domain is analytic in s for given y and continuous in y for given s, and where 9w (x, y) is continuous in y. If (n − 2)/d < Re s < n/d, i h ws (x) d(x, y)sd−n (75) G w (s, x, x) = G w (s, x, y) − y→x γ (sd) and h

2wn/d (x) 1 i 0(n)|S n | sd − n s→n/d 2wn/d (x) = log w(x) + αn wn/d (x) d0(n)|S n | h 2wn/d (x) 1 i + G w (n/d, x, y) − log , 0(n)|S n | d(x, y) y→x

G w (s, x, x) −

(76)

where

h n i 1 − γ + log 4 + ψ . 0(n)|S n | 2 Here γ equals Euler’s constant, and ψ(z) = 0 0 (z)/ 0(z). αn =

Recall that γ (α) is the constant entering in the Riesz kernel given as in (7). Note that h n i 1 1 2 = − − γ + log 4 + ψ + 8(s) (77) n γ (sd) 0(n)|S | sd − n 2 with 8(s) analytic when Re s < (n + 2)/d, 8(n/d) = 0, and so when x 6= y the sum of the first two terms of (73) is analytic when 0 < Re s < (n + 2)/d; this is in agreement with the fact that when x 6= y, the function G w (s, x, y) is analytic in the region Re s > 0.

526

CARLO MORPURGO

Proof The following general estimates about heat kernels are used (for the proofs, see, e.g., G. Grubb [Gru]). Suppose that P is any 9DO as in the hypothesis of the theorem, fix a coordinate patch (U, h), and for x, y ∈ U , let x = h(X ), y = h(Y ), with X, Y ∈ h −1 (U ) ⊆ Rn . The principal symbol of P in these local coordinates is pd (x, ξ ) = w(x)−1 (g i j (x)ξi ξ j )d/2 for |ξ | ≥ 1, and the strictly homogeneous symbol pdh (x, ξ ) is the same function defined for |ξ | ≥ 0. Then the heat kernel of P is smooth in x, y ∈ U and t > 0 and has an expansion K (t, x, y) =

m−1 X

K ` (t, x, y) + Rm (t, x, y),

∀m > n + d,

(78)

`=0

with K 0 (t, x, y) =

Z Rn

e2πi(X −Y )·ξ e−t pd (x,ξ ) dξx

(79)

√ with dξx = dξ/ g(x), the measure in the fiber Tx M; and for t > 0, x, y ∈ U , |K ` (t, x, y)| ≤ Ct (`−n)/d e−tC , 0

0 ≤ ` < n + d, 0

|K n+d (t, x, y)| ≤ Ct (1 + | log t|)e−tC , |K ` (t, x, y)| ≤ Cte

−tC 0

,

` = n + d,

` > n + d, 0

|Rm (t, x, y)| ≤ Ct (1 + | log t|)e−tC ,

(80)

where the constants C, C 0 are independent of x, y, t. In particular, for x, y ∈ U , 0

|K (t, x, y)| ≤ Ce−tC ,

t ≥ 1,

(81)

|K (t, x, y)| ≤ Ct −n/d ,

0 < t ≤ 1,

(82)

|K (t, x, y) − K 0 (t, x, y)| ≤ Ct

(1−n)/d

,

0 < t ≤ 1.

(83)

We remark here that the diagonal estimate on K 1 , and hence (83), can be improved under our assumption on the principal symbol. Indeed, in general one has (see [Gru]) Z K ` (t, x, y) = e2πi(X −Y )·ξ v−` (x, t, ξ ) dξx , Rn

where the symbol v−` can be computed in terms of pd , pd−1 , . . . and their derivatives and satisfies |v−` (x, λ−d t, λξ )| = λ−` v−` (x, t, ξ ) for |ξ | ≥ 1. In particud lar, one shows that |v−` (x, t, ξ )| ≤ Ct|ξ |d−` e−ct|ξ | for |ξ | ≥ 1, and, looking at the construction of the parametrix given, for example, in [Gru], one sees that v−1 (x, t, −ξ ) = −v−1 (x, t, ξ ) when |ξ | ≥ 1, and hence the same is true for the h (x, t, ξ ). From this one easily shows that strictly homogeneous symbol v−1 Z Z h h |K 1 (t, x, x)| = v−1 dξx + |v−1 | + |v−1 | dξx ≤ Ct Rn

|ξ |≤1


and

527

|K (t, x, x) − K 0 (t, x, x)| ≤ Ct (2−n)/d ,

0 < t ≤ 1,

(84)

when 0 < t ≤ 1. It is not hard to show that lim K (t, x, y) = 0

t→0

if x 6 = y.

(85)

To see this, it is enough to show that each term in (78) goes to zero as t → 0, if x 6= y (still in U ). This is clear for K ` with ` > n, by (80), but it can be done also for all ` using the explicit forms of the associated symbols (see [Gru, pp. 374 – 379]). Now we apply this to the operator Pw (in which case all constants involved may depend also on w). Write Z 1 Z ∞ 1 G w (s, x, y) = + t s−1 K w (t, x, y) dt 0(s) 0 1 = G 0w (s, x, y) + G 1w (s, x, y),

Re s > 0, x 6 = y.

(86)

The term G 1w is analytic in s for any given x, y ∈ U , including x = y, and C ∞ in x, y ∈ U , from (81). When x 6 = y, the term G 0w is also defined and analytic by (85). Hence, we recover that if x 6 = y, the function G w (s, x, y) is analytic when Re s > 0. (In fact, it is analytic also in the whole plane.) To analyze further the first piece, choose local normal coordinates around a fixed point x, given as y = expx Y , so that d(x, y) = |Y | ≤ (|Y | = Euclidean norm of the Rn vector Y ) and (71) holds. In the following, |O(t α )| ≤ C(w, x, )t α (i.e., is independent of y) for t ≤ 1. d Since e−t pd (0,ξ ) − e−t (2π |ξ |) /w(x) = O(t) when |ξ | ≤ 1, from (79) and (83) we get K w (t, x, y) = E d t/w(x), Y + O(t (1−n)/d ), 0 ≤ |Y | ≤ . (87) Hence, since E d (a, Y ) = a −n/d E d (1, a −1/d Y ), and also using Z 20(n/d) d E d (1, 0) = e−(2π|ξ |) dξ = d0(n)|S n | Rn and (84), we obtain K w (t, x, x) =

20(n/d)w(x)n/d −n/d t + O(t (2−n)/d ). d0(n)|S n |

Observe also that, by the same argument leading to (85), lim E d t/w(x), Y = 0, Y 6 = 0, t→0

and E d (t/w(x), Y ) ≤ Ct −n/d .

(88)

(89)

528

CARLO MORPURGO

Let now Re s > (n − 2)/d; using (86), (88), and the fact that K w (t, x, x) is continuous in x gives (72). Also, from (87), Z 1 1 t s−1 E d t/w(x), Y dt + 80w (s, x, y), (90) G 0w (s, x, y) = 0(s) 0 where 80w (s, x, y) is analytic in Re s > (n − 2)/d and continuous in y ∈ B(x, ). For 0 < Re s < n/d and Y 6= 0, the integral in (90) is finite, from (89), and so is Z ∞ 1 ws (x) sd−n t s−1 E d t/w(x), Y dt = |Y | , 0(s) 0 γ (sd) which can be easily verified. Hence, Z 1 1 ws (x) sd−n t s−1 E d t/w(x), Y dt = |Y | + I1 (s, x, y) 0(s) 0 γ (sd)

(91)

and Z ∞ 1 I1 (s, x, y) = − t s−1−n/d w(x)n/d E d 1, (w(x)/t)1/d Y dt 0(s) 1 Z ∞ 1 =− t s−1−n/d w(x)n/d E d (1, 0) dt 0(s) 1 Z ∞ 1 + t s−1−n/d w(x)n/d E d (1, 0) − E d 1, (w(x)/t)1/d Y dt 0(s) 1 20(n/d)wn/d (x) 1 = · + I2 (s, x, y). (92) n 0(s)0(n)|S | sd − n It is not hard to see that |E d (1, 0) − E d (1, Z )| ≤ C|Z |2 as |Z | → 0, so one sees that I2 (s, x, y) is holomorphic in the region Re s < (n + 2)/d and also continuous in y ∈ B(x, ). Hence, if (n − 2)/d < Re s < n/d, G 0w (s, x, y) =

ws (x) sd−n 20(n/d)wn/d (x) 1 |Y | + · + 81w (s, x, y) n γ (sd) 0(s)0(n)|S | sd − n

with 81w (s, x, y) analytic in (n −2)/d < Re s < (n +2)/d and continuous in B(x, ). When x 6 = y, using (77), we see that the sum of the first two terms is analytic when (n − 2)/d < Re s < (n + 2)/d, and so is G 0w . This, together with (86), implies (73). Letting s → n/d in (73) gives (74) via (77). Indeed, using 0(z) = z −1 −γ + O(z) as z → 0, and 0(z) = 0(a) 1 + (z − a)ψ(a) + O (z − a)2 as z → a 6 = 0, we deduce (after some algebra) that for A > 0 and Re s < n/d, Asd−n 20(n/d) 1 + · γ (sd) 0(s)0(n)|S n | sd − n h n 2 n i 1 = − 2 log A − γ + log 4 + ψ − ψ + O(sd − n) (93) n 0(n)|S | 2 d d


529

with |O(sd − n)| ≤ C(A)|sd − n|, as s → (n/d)− , and for fixed A (which we choose to be w1/d |Y |). To prove (75), we simply observe that the function h i 2wn/d (x) ws (x) 1 lim G w (s, x, y) − d(x, y)sd−n − · y→x γ (sd) 0(n)|S n | sd − n is analytic in (n − 2)/d < Re s < (n + 2)/d by (73) and coincides with G w (s, x, x) −((2wn/d (x))/(0(n)|S n |)) · 1/(sd − n) when n/d < Re s < (n + 2)/d, which is analytic in this range. Hence (73) implies that 8w (s, x, x) = 8(s, x) whenever (n − 2)/d < Re s < (n + 2)/d, and this gives (75). For (76), observe that if s ∼ n/d, then 1/ 0(s) ∼ 1/ 0(n/d) − (sd − n)ψ(n/d)/(d0(n/d)); and so h 2ψ n/d wn/d (x) 2wn/d (x) 1 i G w (s, x, x) − · − 0(n)|S n | sd − n s→n/d d0(n)|S n | Z Z h 1 i ∞ 1 1 = t s−1 K w (t, x, x) dt − wn/d (x)t s−1−n/d E d (1, 0) dt s→n/d 0(s) 0 0(s) 0 Z 1 1 = t n/d−1 K w (t, x, x) − E d t/w(x), 0 dt 0(n/d) 0 Z ∞ 1 + t n/d−1 K w (t, x, x) dt 0(n/d) 1 Z 1 h 1 = t n/d−1 K w (t, x, y) − E d t/w(x), Y dt 0(n/d) 0 Z ∞ i 1 + t n/d−1 K w (t, x, y) dt y→x 0(n/d) 1 Z 1 h i 1 = G w (n/d, x, y) − . (94) t n/d−1 E d (t/w(x), Y ) dt y→x 0(n/d) 0 The third equality is possible via the expansion (88). From (91), (92), (89), and (93) (with A = w1/d Y ), we gather that for Y 6 = 0, Z 1 1 t n/d−1 E d t/w(x), Y dt 0(n/d) 0 i 2wn/d (x) h 1 = − log |Y | − log w(x) 0(n)|S n | d 2ψ n/d wn/d (x) n/d + w (x)αn − d0(n)|S n | Z wn/d (x) ∞ −1 E d (1, 0) − E d 1, (w(x)/t)1/d Y dt + t 0(n/d) 1

530

CARLO MORPURGO

and, as we mentioned earlier, the integrand in the last integral is less than or equal to Ct −1−2/d |Y |2 , so that the last term vanishes when Y → 0. Substituting this formula into (94) yields (76). As a consequence of Theorem 5.1 we have the following. COROLLARY 5.2 Using the same hypothesis as that of Theorem 5.1, we have n/d

Z (s, Pw ) = Tr[Pw−s ] =

2kwkn/d

0(n)|S n |

·

1 + 8w (s), sd − n

where 8(s) is analytic when Re s > (n − 2)/d. Moreover, n/d h 2kwkn/d 1 i e Z (n/d, Pw ) := Z (s, Pw ) − · 0(n)|S n | sd − n s→n/d Z Z 2 = αn wn/d d x + wn/d log w d x d0(n)|S n | M M Z h 2wn/d (x) 1 i + G w (n/d, x, y) − log dx n 0(n)|S | d(x, y) y→x M

and Z w (s) =

Z h M

G w (s, x, y) −

i ws (x) d(x, y)sd−n dx y→x γ (sd)

(95)

(96)

for (n − 2)/d < Re s < n/d. The functional e Z w (n/d, Pw ) defined as in (95) is called the regularized zeta function at s = n/d. Proof The formulas follow immediately from (72), (76), (75), and the fact that 8w (s, x) in (72) is jointly continuous in (s, x) for Re s > (n − 2)/d, so that the function R M 8w (s, x) d x is analytic in that range. We now compare the regularized zeta function of an operator Pw with that of an d/2 operator P having the same leading symbol as 1g . We assume further that the isometries of (M, g) leave P invariant and act transitively on M; in practice, the heat kernel of P is constant on the diagonal x = y at any given time t. Under this assumption the formulas below have a slightly simpler look than in the more general case.


531

COROLLARY 5.3 Given the same hypothesis as that of Theorem 5.1, assume in addition that P is an opd/2 erator with the same leading symbol as that of 1g and that the isometries of (M, g) act transitively on M and leave P invariant: T ∗ P(T −1 )∗ = P for any isometry T . Then, for (n − 2)/d < Re s < n/d, Z kwkss Z (s, Pw ) = Z (s, P) + G w (s, x, y) − ws (x)G(s, x, y) y→x d x, (97) |M| M

where G(s, x, y) is the kernel of P −s and |M| the volume of (M, g). Also, n/d Z kwkn/d 2 e Z (n/d, Pw ) = e Z (n/d, P) + wn/d log w d x |M| d0(n)|S n | M Z + G w (n/d, x, y) − wn/d (x)G(n/d, x, y) y→x d x.

(98)

M

Proof Observe that the integrands in (97) and (98) are continuous in x; this follows from (72) – (76). (Observe that 8w (s, x, x) is continuous in x.) Hence, Z Z Z (s, Pw ) = G w (s, x, x) d x = G w (s, x, y) − ws (x)G(s, x, y) y→x d x M M Z h i 1 + ws (x) G(s, x, y) − d(x, y)sd−n d x. (99) y→x γ (sd) M The given hypothesis on the group of isometries of (M, g) implies that K (t, x, x) is independent on x for any given t > 0, where K (t, x, y) is the heat kernel of P. Hence, the meromorphic function G(s, x, x) is also independent on x for given s, and we can take G(s, x, x) = Z (s, P)/|M| out of the last integral in (99) to obtain (97). The proof of (98) is similar. Another consequence of Theorem 5.1 is the following predictable Green’s function estimate. COROLLARY 5.4 Using the same hypothesis as that of Corollary 5.3, (97), we have

c1 d(x, y)sd−n ≤ G w (s, x, y) ≤ c2 d(x, y)sd−n ,

∀x, y ∈ M,

and under the same hypothesis as (98), c3 1 + | log d(x, y)| ≤ G w (n/d, x, y) ≤ c4 1 + | log d(x, y)| ,

∀x, y ∈ M.

532

CARLO MORPURGO

Proof The estimates follow from (73), (74), and the fact that G w (s, x, y) = G w (s, T x, T y).

Regularized zeta functions for the intertwining operators, and Dini-type integrals In this section we apply the general results of the previous section to the operators W −1/2 Ad W −1/2 , on S n , with 0 < d < n. First of all, W −1/2 Ad W −1/2 is an elliptic d/2 pseudodifferential operator of order d, with the same leading symbol as W −1 1 S n ; this is evident from (2) and the composition theorem. The first step is to obtain more explicit representations of the functionals involved in Corollary 5.3 and to find their Rn -counterparts. For 0 < d < n, s ≤ n/d, s an integer, and 0 ≤ W ∈ L 1 (S n ), we define the arithmetic-geometric (A-G) Dini integral Hd,s [W ] as (with ξ1 = ξs+1 ) Z s s s i h1 X Y Y Hd,s [W ] = W s (ξi ) − W (ξi ) dξ1 · · · dξs . (100) Nd (ξi − ξi+1 ) s (S n )s i=1

i=1

i=1

In the case 0 < d ≤ 2 and 0 < s ≤ n/d, we extend this definition by letting Hd,s [W ] Z ∞ Z Z hZ 1 Z 1 s i 1 s−1 s = t W (Btv ) dv − W (Btv ) dv dµξ,ξ,t dξ dt. 0(s) 0 Sn D 0 0 (101)

The reason we call this functional Dini is that when d ≤ n/2, Z Z 1 |W (ξ ) − W (η)|2 Hd,2 [W ] = dη dξ, 2γ 2 (d) S n S n |ξ − η|2n−2d which is a well-known Dini integral. In Rn we define the corresponding A-G Dini integrals hs,d [w] as in (100) and (101), with the obvious changes (e.g., in the notation of Theorem 3.15, if d ≤ 2, s < n/d, and 9(u) = u s / 0(s), we let hd,s [w] = h9 [w]). The only case that needs to be treated differently is the case when s = n/d, for in this case the “obvious” Rn -versions of the integrals defining H are always divergent (if w 6≡ 0). We let, for s = n/d an integer, n/d

Z hd,n/d [w] =

(Rn )n/d

×

Y ed (x1 − x2 ) Nd (x1 − x2 ) − N Nd (xi − xi+1 )

n/d hd X

n

i=1

i=2

wn/d (xi ) −

n/d Y i=1

i w(xi ) d x1 · · · d xn/d


with ed (x) = N

1 min |x|d−n , 1 . γ (d)

533

(102)

ed can be chosen differently, as long as it satisfies appropriate The regularizing kernel N conditions, which are clear from the arguments below. If 0 < d ≤ 2 and s = n/d (not necessarily an integer), we let Z ∞ Z Z hZ 1 Z 1 n/d i 1 n/d−1 n/d hd,n/d [w] = t w (Btv ) dv − w(Btv ) dv 0(n/d) 0 Rn D 0 0 × E d (t, 0) − E d (t, 1) d Px,x,t d x dt. In general, the A-G Dini integrals are measures of smoothness; when s = 2 < n/d, we get precisely Sobolev norms. Hence, it becomes natural to investigate the nature of the functionals for general s. Proposition 5.5 shows that, after a power normalization, they are all essentially equivalent. PROPOSITION 5.5 Let n − 2 < sd ≤ n with s an integer if d > 2. Assume 0 ≤ W ∈ L s (S n ) when s ≥ 1, and if s < 1, take W ∈ L 1 (S n ). (1) Hd,s [W ] is well defined and nonnegative (possibly +∞) if s > 1, nonpositive (possibly −∞) if 0 < s < 1, and zero if s = 1. (2) If 0 < d ≤ 2 and s ≤ n/d is an integer, then the two definitions of Hd,s in (100) and (101) agree. (3) If s > 1, there exist constants c1 , c2 > 0 (depending on s) such that for all W ∈ L 2 (S n ), Z Z |W (ξ ) − W (η)|2 c1 Hd,s [W 2/s ] ≤ dξ dη ≤ c2 Hd,s [W 2/s ], (103) |ξ − η|2n−sd Sn Sn

and if s < 1, then the left inequality holds, with −Hd,s in place of Hd,s . In particular, for 1 < s < n/d, kW k2 + Hd,s [W 2/s ] < ∞ if and only if W ∈ H (n−sd)/2,2 (S n ). (4) The functional Js [W ] := kW k1 + α Hd,s [W 1/s ] is a norm, for any constant α, so that α(s − 1) > 0. Moreover, if W ∈ H (n−sd)/2,1 (S n ), then Js [W ] < ∞. The statements in (1) – (4) extend in the obvious way to Rn . In particular, if w ∈ 2 L (Rn ), for 1 < s < n/d, c1 hd,s [w2/s ] ≤

Z

Z Rn

Rn

|w(x) − w(y)|2 d x dy ≤ c2 hd,s [w2/s ], |x − y|2n−sd

534

CARLO MORPURGO

and when s = n/d, we have c1 hd,n/d [w2d/n ] − c2 kwk22 ≤

|w(x) − w(y)|2 d x dy ≤ c3 hd,n/d [w2d/n ]. |x − y|n

Z Z |x−y|≤1

(104) Proof We observe that for F ∈ L 1 (S n ), Z Z 1 Z F(Btv ) dv dµξ,η,t = D

0

Sn

F(ζ )

1

Z

K d (tv, ξ, ζ )K d (t − tv, ζ, η) dv dζ,

0

(105) where K d (t, ξ, η), we recall, denotes the heat kernel for Ad . This formula follows from (17), and a similar version holds in Rn and more general spaces. As a consequence, Z Z Z 1 Z F(Btv ) dv dµξ,ξ,t dξ = K d (t, ξ, ξ )F(ξ ) dξ. (106) D

Sn

Sn

0

R1 In particular, for W ∈ L s (if s ≥ 1; otherwise, take L 1 ), the quantity 0 W s (Btv ) dv is finite for almost every path B and all t > 0, by (106); by the artihmetic-geometric mean or Jensen’s inequality, the integrands of (100) and (101) are well defined (possibly infinite) and with constant sign. This shows (1). To see (2), let 0 < d ≤ 2 and s ≥ 2 an integer, and let, for ξ = (ξ1 , . . . , ξs−1 ), ξ K d,s (t, ξ0 , ξs )

t

Z

u s−1

Z du s−1

= 0

u2

Z du s−2 · · ·

du 1

0

0

s Y

K d (u i − u i−1 , ξi , ξi−1 )

i=1

with u 0 = 0, u s = t. For all t > 0, (106) implies t s−1 0(s)

Z

1

Z Z Sn

D

0

W s (Btv ) dv dµξ,ξ,t dξ s

Z = (S n )s

1X s ξ W (ξi )K d,s (t, ξs , ξs ) dξ1 · · · dξs . s i=1

Similarly, one finds t s−1 0(s)

Z

1

Z Z Sn

D

0

s W (Btv ) dv dµξ,ξ,t dξ Z =

s Y (S n )s i=1

ξ

W (ξi )K d,s (t, ξs , ξs ) dξ1 · · · dξs .


535

R1 Q R1 (To derive this formula, start by writing ( 0 W (Btv ))s = s1 ( 0 W (Btv j )), use symmetry in the v’s, use the properties of the conditional Wiener measure, and make some changes of the v variables.) Now (2) follows since ∞

Z

ξ

K d,s (t, ξs , ξs ) dt =

0

s Y

Nd (ξi − ξi−1 ).

(107)

i=1

The proof of (3) is based on the following two estimates. First, for any f : [0, 1] → [0, +∞] not constant a.e. and with k f k2 < ∞, we have 1≤

k f k22 − k f k22/s k f k22 − k f k21

≤ s − 1,

s ≥ 2,

(108)

whereas the inequalities are reversed if 1 < s < 2. The inequality on the left is obvious, and the one on the right is a simple consequence of the convexity of log k f k1/r . We note that (108) holds on any probability space. Second, if m ≥ 2, then for any a1 , . . . , am ≥ 0 not all equal (see [MPF, Chap. V, Th. 4]), Qm 2/m 1 Pm a 2 − i=1 ai m (109) ≤ mP i=1 i 2 ≤ m. P m 1 m−1 a 2 − 1 m ai i=1 i

m

Third, if 0 < s < so that

1, then k f k22/s −k f k22

≥

m

i=1

β(k f k22/s −k f k21 ) with β

= (1−s)/(2−s),

k f k22/s − k f k22 ≥ (1 − s) k f k22 − k f k21 .

(110)

In the case 0 < d ≤ 2, s > 1, we apply (108) to f (v) = W (Btv ), and observing RR that 2(k f k22 − k f k21 ) = [ f (v) − f (u)]2 du dv, we get that Hs [W 2/s ] is bounded above and below by a constant times 2 0(s)

Z 0

∞

t s−1

Z

1

Z Z Sn

D

v

Z dv

0

0

2 du W (Btu ) − W (Btv ) dµξ,ξ,t dξ dt Z Z 2 e ξ, η) dξ dη = W (ξ ) − W (η) G(s, Sn

Sn

with Z ∞ Z 1 2 s−1 t u K d (t − tu, ξ, η)K d (tu, ξ, η) du dt 0(s) 0 0 Z ∞ Z 1 1 = t s−1 K d (t − tu, ξ, η)K d (tu, ξ, η) du dt 0(s) 0 0 Z ∞Z ∞ 1 (t + r )s−2 K d (t, ξ, η)K d (r, ξ, η) dt dr. = 0(s) 0 0

e ξ, η) : = G(s,

536

CARLO MORPURGO

e is bounded We only need to integrate in the region r < t, which shows that G n n S s above and below by a constant times G (s −1, ξ, η)Nd (ξ −η), G (s, ξ, η) being the Green function of Asd , and (3) follows from Corollary 5.4. (If d(ξ, η) is the geodesic distance on the sphere, then |ξ − η|sd−n − d(ξ, η)sd−n η→ξ = 0 when (n − 2)/d < s < n/d, and also log d(ξ, η) − log |ξ − η| η→ξ = 0.) The cases 0 < s < 1 and d > 2, s an integer, are handled similarly using (109) and (110). We just remark that in the case s an integer, the quantity Z s Y Nd (ξi − ξi−1 ) dξ1 · · · dξr −1 dξr +1 · · · dξm−1 dξm+1 · · · dξs (S n )s−2 i=1

is comparable to |ξr − ξm |sd−n . This follows from (107), the semigroup property, and previous estimates. The last statement of (3) follows from a well-known characterization of the norms in H α,2 . Note, however, that we do not expect kW k2 + (Hd,s [W 2/s ])1/2 to be a norm. Finally, the first part of (4) follows from Minkowski’s inequality (direct for s < 1, reversed for s > 1 or for the geometric mean), and the second part follows from √ √ (3) and the elementary estimate | W (ξ ) − W (η)|2 ≤ |W (ξ ) − W (η)|. The Rn statements follow similarly. ed given in (102) gives, for a fixed integer s > 1, The choice of N Z 2 1 ed (x)Nsd−d (x) d x = − N · + An (s) + O(sd − n) n | sd − n n 0(n)|S R

(111)

as d → n/s, with An (s) = −

h n n(s − 1) i 1 2 log 4 + ψ + ψ − ; 0(n)|S n | 2s 2s n(s − 1)

ed of course lead to different values of this constant. different choices of N We also have Z ∞ 1 2 1 t s−1 E d (t, 1) dt = − · + An + O(sd − n) n 0(s) 0 0(n)|S | sd − n

(112)

as s → n/d, with An = −(γ − log 4 − ψ(n/2))/(0(n)|S n |). For a measurable function w : Rn → [0, ∞), we also define, for s an integer, Z s s Y Y e e hd,s [w] = Nd (x1 − x2 ) Nd (xi − xi+1 ) w(xi ) d x1 · · · d xs Rn s

i=2

i=1

(where, as usual, x1 = xs+1 ) and, for d ≤ 2, Z ∞ Z Z Z 1 s 1 s−1 e hd,s [w] = t w(Btv ) dv E d (t, 1) d Px,x,t d x dt. 0(s) 0 Rn D 0


537

The following proposition gives the main explicit representations for the zeta functionals and their Rn counterparts. THEOREM 5.6 Let W : S n → [0, ∞) and w : Rn → [0, ∞) be related as w = (W ◦ π )|Jπ |d/n . Below, assume further that s is an integer if d > 2. (a) If 0 < W ∈ C ∞ (S n ) and (n − 2) < sd < n, then

kW ks Z s, Ad (W ) = Z (s, Ad ) n s − Hd,s [W ] = −hd,s [w], |S |

(113)

and if s = n/d, then n/d

e Z (n/d, Ad (W )) = e Z (n/d, Ad ) +

2 d|S n |0(n)

=e Z (n/d, + +

kW kn/d |S n | Z W n/d log W dξ − Hd,n/d [W ]

Sn n/d kwkn/d Ad ) |S n |

2 n d|S |0(n)

Z

Rn n/d B(d, n) kwkn/d

wn/d log w d x − hd,n/d [w] +e hd,n/d [w],

(114)

where B(d, n) = An (n/d) as in (111) for d > 2 and n/d an integer, and B(d, n) = An as in (112) for d ≤ 2. (b) If 0 ≤ W ∈ L n/d (S n ) and n − 2 < sd < n, then Hd,s [W ] is finite if and only if hd,s [w] is finite, and in this case the second equality in (113) holds. If R n/d log W | < ∞ and s = n/d, then H d,n/d [W ] is finite if and only if hd,n/d [w] S n |W is finite, and in this case e hd,n/d [w] < ∞ and the second equality in (114) holds (defining W n/d log W = 0 wherever W = 0). (c) Under the same hypothesis as (b), the two functionals in (113) and (114) are invariant under the conformal actions w → (w ◦h)|Jh |d/n and W → (W ◦τ )|Jτ |d/n . Remark 5.7 As shown in [CL2, proof of Th. 1], W n/d log W ∈ L 1 (S n ) if and only if wn/d log w ∈ L 1 (Rn ) and wn/d log |Jπ | ∈ L 1 (Rn ). Remark. Formula (113) says that Z (1, Ad (W )) depends on W only through its mean since Hd,1 [W ] ≡ 0. This is not surprising since the Green function of an operator Pw = w−1/2 Pw−1/2 differs from that of P only by the multiplicative factor [w(x)w(y)]−1/2 .

538

CARLO MORPURGO

Proof We begin by observing that the Green function for (W −1/2 Ad W −1/2 )s is given by n

S GW (s, ξ, η) = W 1/2 (ξ )W 1/2 (η)

1 0(s)

∞

Z

t s−1

1

Z Z D

0

W (Btv ) dv

s−1

0

dµξ,η,t dt

in the case d ≤ 2, and Sn GW (s, ξ, η)

=W

1/2

(ξ )W

1/2

(η)

Z

s Y

(S n )s−1 i=1

Nd (ξi − ξi+1 )

s Y

W (ξk ) dξ2 · · · dξs

k=2

in the case s an integer, where ξ1 = ξ, ξs+1 = η. These formulas are implicit in the proof of Lieb’s formula. n On Rn , the corresponding formulas for the Green function G R w (s, x, y) of (w−1/2 Ad w−1/2 )s are similar to the ones above, and, moreover, from (15) we also have n Sn 1/2 GR |Jπ (y)|1/2 (115) w (s, x, y) = G W s, π(x), π(y) |Jπ (x)| when w = (W ◦ π )|Jπ |d/n . We let n

n

G S (s, ξ, η) = G 1S (s, ξ, η),

n

G R (s, x, y) = Nsd (x − y).

As for (113), we first assume that (n − 1)/d < s < n/d. The first equality in (113) is a consequence of formula (97): Z kW ks n Z s, Ad (W ) = Z (s, Ad ) n s + G W (s, ξ, η) − W s (ξ )G S (s, ξ, η) η→ξ dξ. |S | Sn Assume first that s is an integer; then, with ξs+1 = ξ1 , n G W (s, ξ1 , η) − W s (ξ1 )G S (s, ξ1 , η) η→ξ 1 Z s s s iY hY X 1 s W (ξk ) Nd (ξi − ξi+1 ) dξ2 · · · dξs = W (ξk ) − s (S n )s−1 k=1 k=1 i=1 s Z s Y 1X [W s (ξk ) − W s (ξ1 )] Nd (ξi − ξi+1 ) dξ2 · · · dξs . (116) + s (S n )s−1 k=2

i=1

Now notice that for smooth W > 0 the A-G mean is less than or equal to P P C i,k |W s/2 (ξi ) − W s/2 (ξk )|2 ≤ C j |ξ j − ξ j+1 |2 , by (109) or Taylor’s forP mula, and also that |W s (ξk ) − W s (ξ1 )| ≤ C sj=1 |ξ j − ξ j+1 |; these estimates imply that each integrand in (116) is integrable in ξ1 , . . . , ξs (the first one also when (n − 2)/d < s ≤ (n − 1)/d). The ξ1 -integral of the first term is −Hd,s [W ], whereas the ξ1 -integral of the second term vanishes by symmetry.


h

539

In the case when 0 < d ≤ 2 and s satisfies n − 2 < sd < n, we get, similarly, i n Sn GW (s, ξ, η) − W s (ξ )G S (s, ξ, η) η→ξ Z ∞ Z h Z 1 s−1 Z 1 i s−1 = t W (ξ ) W (Btv ) dv − W s (Btv ) dv dµξ,ξ,t dt D

0

∞

Z

t s−1

+

0

1Z

Z

0

D

0

0

W s (Btv ) − W s (ξ ) dµξ,ξ,t dt dv.

(117)

Again, since W is smooth, we can use Taylor’s formula about ξ to see that all the integrands are bounded at least by C|Btv − ξ | so that each integral is controlled by Z 1Z Z ∞ s−1 |Btv − ξ | dµξ,ξ,t dt dv t D

0

0

∞Z 1Z

Z

t s−1 |ξ − ζ |K (tv, ξ, ζ )K (t − tv, ξ, ζ ) dζ dv dt

= 0

0

Z ≤C Sn

Sn

|ξ − ζ |sd−2n+1 dζ < ∞

for s > (n − 1)/d. The ξ -integral of the last term of (117) vanishes by an application of (105). For the first term we observe that ∞

Z

t

s−1

D

0

0

∞

Z

1

Z Z t s−1

= 0

s W (Btv ) dv dµξ,η,t dt 1

Z Z D

W (ζ )

1

Z

W (Btv ) dv

s−1

0

0

K d (tu, ξ, ζ ) dµζ,η,t,tu dt du,

where dµζ,η,t,tu is the conditional Wiener measure starting at ζ at time tu and ending at η at time t (see (17)), and also, Z

∞

t s−1

Z

0

Z

D

∞

t

= 0

W (ξ )

s−1

Z

Z Z D

0

0 1

1

W (Btv ) dv 1

Z W (ξ ) 0

s−1

dµξ,η,t dt

W (Btv ) dv

s−1

K d (tu, ξ, ζ ) dµζ,η,t,tu dt du.

We subtract the first formula from the second one and notice that the ξ - integral of the resulting quantity vanishes, so that the ξ -integral of the first integral of (117) is −Hd,s [W ]. As for the second equality of (113), using stereographic projection and (115), we rewrite (96) as Z h i ws (x) n GR Z s, Ad (W ) = |x − y|sd−n d x, (118) w (s, x, y) − y→x γ (sd) Rn

540

CARLO MORPURGO

and we proceed as before. The only possible difference is the behavior at infinity, so that we are reduced to showing Z Z s sd−2n w (x + t) − ws (x) d x < ∞ |t| Rn

|t|≥2

when w = (W ◦ π)|Jπ |d/n and W is smooth on S n . This is obvious when |Jπ |sd/n ∈ L 1 (Rn ), that is, for n > 1. The case n = 1 is covered by the following lemma (which is also used later). LEMMA 5.8 Let 0 < W ∈ C ∞ (S n ), w = (W ◦ π)|Jπ |d/n , and s, d    Z C s p s w (x + t) − w (x) d x ≤ C log |t|  Rn  C|t|n−2sd p

> 0, p ≥ 1. Then, for |t| ≥ 2, if 2sdp > n, if 2sd p = n, if n − p < 2sd p < n

for some constant C > 0 (depending possibly on W but independent of t). Remark. Note that the conformal invariance of Z (s, Ad (W )) is clearly displayed in the form (118); this is not the case for the A-G Dini integral form (113) (except for rotations and reflections on the sphere). The proof of the lemma is straightforward, and it is outlined in the appendix. To complete the proof of (a) in the case (n −2)/d < s ≤ (n −1)/d, it is enough to show that the last two members of (113) are analytic functions of s in that range, in the case 0 < d ≤ 2, and in analytic functions of d in that range for fixed integer s. The analyticity of Hd,s [W ] in the above range follows easily from (103) and Morera’s theorem, whereas the analyticity of hd,s [w] follows from the Rn -version of (103), Lemma 5.8, and Morera’s theorem. When d ≤ 2, the function Z (s, Ad ) is analytic in (n − 2)/d < s < n/d by Theorem 5.1, and so it remains to see that in this range it is also analytic in d when d < n and s is a fixed integer. Although this fact seems to be obvious, it is not included in the statement (or proof) of Theorem 5.1. However, it is an easy consequence of the following lemma. LEMMA 5.9 Let 0 < d < n, s > 0, and let λk,d be as in (12). Then, as k → ∞, −sd/2

λ−s k,d − λk,2

=

sd 2 (d − 4)k −sd−2 + O(k −sd−3 ). 24

Proof Use the expansion in [Ol, (5.02), p. 119] for the ratio of Gamma functions.


541

From the lemma we deduce that Z (s, Ad ) − Z (sd/2, A2 ) is analytic in d when s > (n − 2)/d, and we already know that Z (s, A2 ) is analytic in (n − 2)/2 < s < n/2. To show (114), subtract from each member of (113) the quantity n/d

n/d

2kW kn/d 0(n)|S n |(sd − n)

=

2kwkn/d 0(n)|S n |(sd − n)

,

and pass to the limit as s → n/d in the case d ≤ 2 and as d → n/s in the case when s is an integer. The left and middle members converge to the left and middle members of (114). (The convergence of Hd,s follows from the dominated convergence theorem.) To analyze the right member, assume first that s is a fixed integer greater than 1, and for d < n/s, write Z −hd,s [w] =

(Rn )s

×

s Y e Nd (x1 − x2 ) − Nd (x1 − x2 ) Nd (xi − xi+1 ) i=2

s h1 X

s

ws (xi ) −

i=1

Z − (Rn )s

+ kwkss

s Y

i w(xi ) d x1 · · · d xs

i=1

ed (x1 − x2 ) N

s Y

Nd (xi − xi+1 )

i=2

Z Rn

s Y

w(xi ) d x1 · · · d xs

i=1

ed (x)Nsd−d (x) d x. N

As d → n/s, the first two terms converge to −hn/s,s [w] and e hn/s,s [w], respectively; therefore, by (111), n/d

− hd,s [w] −

2kwkn/d

∼ −hn/s,s [w] + e hn/s,s [w] Z 2s + ws log w d x + C(n/s, s, n) kwkss n0(n)|S n | Rn

0(n)|S n |(sd − n)

as d → n/s, which yields (114), with s in place of n/d. The case d ≤ 2 is handled similarly. This time it is easier to fix d and let s ↑ n/d; we leave the details to the reader. For the proof of (b), we proceed as follows. Let w = f 1/s , and let φ > 0 be in the R Schwartz class of Rn , with φ = 1, and let φ (x) = −n φ(x/), f = φ ∗ f > 0, 1/s w = f , W = (w ◦ π −1 )|Jπ −1 |d/n . We have, as → 0, f → f in L p (Rn ), p ≥ 1, and we can assume the convergence is also a.e. (Otherwise, pass to a sequence n .) R 1 1/s R R1 By Minkowski’s inequality ( 0 f (Btv ) dv)s ≤ Rn φ (z)( 0 f 1/s (Btv −z) dv)s dz 1/s if s > 1, and the reverse when s < 1. Hence (for d ≤ 2), hd,s [ f ] ≤ hd,s [ f 1/s ] when s > 1, and the inequality is reversed when s < 1 (essentially a generalized

542

CARLO MORPURGO

Minkowski inequality for hd,s ); the case s an integer is similar. Since kW − W ks = k( f − f )|Jπ |1−sd/n k1 ≤ Ck f − f k1 → 0, we deduce that, for s > 1, −Z (s, Ad )

n o kW kss kW kss + H [W ] ≤ lim inf − Z (s, A ) + H [W ] d,s d d,s →0 |S n | |S n | = lim inf hd,s [we ] ≤ hd,s [w], →0

(119)

and the reversed inequality when s < 1. (Observe that, in fact, hd,s [w ] → hd,s [w].) On the other hand, we can get the reversed inequality in (119) (and hence (113)) for s > 1 by letting 8 > 0 be any positive smooth approximation of the identity on S n , and F = W s , F = 8 ∗ F, with convolution on S n (more precisely on O(n), R R using the formula S n Fdξ = O(n) F(R N )d R with d R being the Haar measure on O(n), N being the north pole). The reasoning is the same as above and is similar for s < 1. The proof of (114) proceeds along similar lines. First, let us observe that if R F → F a.e. on S n , and 0 < F, F ≤ C for < 1, then S n F log F → R Conversely, if f → f a.e. on S n F log F as → 0 and the integrals are finite. R R Rn and 0 < f, f ≤ C|Jπ | for < 1, then Rn f log f → Rn f log f with the integrals finite (by the dominated convergence theorem since f ≤ C 0 , so that | f log f | ≤ C|Jπ |α ∈ L 1 (Rn ) with 1/2 < α < 1). Assume now that 0 < W ≤ C, so that 0 < w ≤ C|Jπ |d/n , and let f = wn/d , f = φ ∗ f , with w , W as above. Then f ≤ C 0 |Jπ | for 0 < < 1. This follows at once from the estimate 1 + |x|2 1 ≤ 4 , 1 + |x − t|2 1 + |t|2

x, t ∈ Rn ,

and the fact that φ is Schwartz. Letting Z (W ), z(w) be the middle and right members of (114), respectively, we then have −Z (W ) ≤ − lim inf Z (W ) = − lim inf z(w ) ≤ −z(w).

(120)

The reversed inequality follows by taking F = W n/d , F = 8 ∗ F, etc., and proceeding in a similar manner. This time we need e h d,n/d [w ] → e h d,n/d [w], which is a d/n e consequence of h d,n/d [|Jπ | ] < ∞. (In turn, one can easily see this by transferring the integral on S n .) In summary, we have proved that (114) holds when 0 < W ≤ C. Let now W > 0, so that W n/d log W ∈ L 1 (S n ) or, equivalently, wn/d log w, wn/d log |Jπ | ∈ L 1 (Rn ). Set Wk = min(W, k), so that wk = min(w, k|Jπ |d/n ). It is not hard to R n/d R see that, under the given hypothesis, we have Wk log Wk → W n/d log W and R n/d R wk log wk → wn/d log w (use Remark 5.7). Similarly, if Hd,n/d [W ] < ∞, then Hd,n/d [Wk ] → Hd,n/d [W ]; indeed, by Proposition 5.5(4) it is enough to show

SHARP INEQUALITIES, FUNCTIONAL INTEGRALS, TRACES n/d

that for F = W n/d , Fk = Wk estimate, by (103),

543

, we have Hd,n/d [(F − Fk )d/n ] → 0. To do this we

|(F − Fk )1/2 (ξ ) − (F − Fk )1/2 (η)|2 dξ dη |ξ − η|n Sn Sn Z Z |W n/2d (ξ ) − W n/2d (η)|2 dξ dη → 0 ≤ |ξ − η|n {W (ξ )>k or W (η)>k}

Hd,n/d [(F − Fk )d/n ] ≤ C

Z

Z

since the last integrand is integrable over (S n )2 , again by (103). The proof of hd,n/d [wk ] → hd,n/d [w] when hd,n/d [w] < ∞ is similar but slightly more involved since the pieces of integrands corresponding to w ≤ k|Jπ |d/n do not simply vanish (the details are left to the reader) and e hd,n/d [wk ] ↑ e hd,n/d [w] by monotone convergence. Now, if hd,n/d [w] < ∞, then we can argue as in (120), deduce simultaneously that Hd,n/d [W ] and e hd,n/d [w] must be finite, and obtain one half of (114). For the other half, it is enough to prove that Hd,n/d [W ] < ∞ =⇒ h d,n/d [w] < ∞. To see this, it is enough to consider the case n/d = 2 (make the change W n/d = F 2 ), and the result follows from the estimate Z Z Z Z |W (ξ ) − W (η)|2 (|Jπ (y)|/|Jπ (x)|)1/2 − 1 2 dξ dη ≥ 2 w (x) d x dy |ξ − η|n |x − y|n Sn Sn |x−y|≤1

Z Z +

|w(x) − w(y)|2 d x dy. |x − y|n

|x−y|≤1

It is easy to see that the first integral on the right member is finite, and so must be the second one. But if hd,n/d [w] < ∞, then we can argue as before to finish up the proof of (114) when W > 0. Finally, the case W ≥ 0 can be settled using the sequence Wk = max(W, k −1 ). The functional (113) is clearly invariant under translations, rotations, and dilations of Rn , and the inversion invariance follows from the reflection invariance in W . Hence the w-functional is invariant under the full conformal group of Rn , and hence the W -functional is invariant under the conformal group of S n . The invariance of (114) is similar, except perhaps for the dilations of Rn , since the w-functional is not “obviously” dilation invariant. However, this can be seen by approximation with smooth functions. (This raises the question of whether one can find an Rn representation of (114) which is clearly translation invariant, rotation invariant, but also dilation invariant.)

544

CARLO MORPURGO

Regularized zeta functions and (log) Sobolev inequalities We can now state and prove the main zeta function inequalities for the (real) values of s so that n − 2 < sd ≤ n. First, let us note that in the special case d = n/2 we get Z 4 e W 2 log W dξ Z 2, An/2 (W ) − e Z (2, An/2 ) = n|S n |0(n) S n Z Z 1 |W (ξ ) − W (η)|2 − dη dξ ≤ 0 (121) |ξ − η|n 2γ 2 (n/2) S n S n by Beckner’s sharp log Sobolev inequality (6). In the next theorem we generalize this inequality using the conformally invariant functionals of Theorem 5.6. THEOREM 5.10 n/d n/d Let 0 < d < n, 0 ≤ W ∈ L n/d (S n ), 0 ≤ w ∈ L n/d (Rn ), and kW kn/d = kwkn/d = |S n |. Below, assume further that s is an integer if d > 2. (a) If n − 2 < sd < n and s > 1, then

Z (s, Ad )

(b)

(c)

kW kss − Hd,s [W ] ≤ Z (s, Ad ), |S n |

whereas if 0 < s < 1, the inequality is reversed. If s = n/d and W n/d log W ∈ L 1 (S n ), then Z 2 W n/d log W dξ ≤ Hd,n/d [W ]. d|S n |0(n) S n

(123)

If n − 2 < sd < n and s > 1, then −Z (s, Ad ) ≤ hd,s [w],

(d)

(122)

whereas if 0 < s < 1, the inequality is reversed. If s = n/d and wn/d log w, wn/d log |Jπ | ∈ L 1 (Rn ), then Z 2 wn/d log w d x + B(d, n) ≤ hd,n/d [w] − e hd,n/d [w] d|S n |0(n) Rn

(124)

(125)

with B(d, n) as in (114). If Hd,s [W ] is finite, then equality in (122) and (123) holds if and only if W = |Jτ |d/n a.e. for some conformal transformation τ of S n , and when hd,s [w] is finite, then equality in (124) and (125) holds if and only if w = (|Jπ |d/n ◦ h)|Jh |d/n a.e., for some conformal transformation h of Rn . Regarding the sign of Z (s, Ad ), in (122) and (124), we have the following (the proof of which is postponed to the appendix).


545

PROPOSITION 5.11 If (n − 2) < sd < n and s > 1, then

Z (s, Ad ) < 0, whereas if 0 < s < 1, the inequality is reversed. 5.12 R Let 0 < d < n and 0 < W ∈ C ∞ (S n ), with S n W n/d = |S n |. Below, assume that s is an integer if d > 2. (a) If n − 2 < sd < n and s > 1, then Z s, Ad (W ) ≤ Z (s, Ad ); (126) COROLLARY

(b)

when s < 1, the inequality is reversed. If s = n/d, then e Z (n/d, Ad (W )) ≤ e Z (n/d, Ad ).

(127)

Equality in (126) or (127) holds if and only if W = |Jτ |d/n for some conformal transformation τ . The proof of Theorem 5.10 is based on the following special cases of (65) and (66). PROPOSITION 5.13 For n − 2 < sd ≤ n, with s an integer if d > 2, s > 1, and 0 ≤ w ∈ L n/d (Rn ), the following rearrangement inequalities hold:

hd,s [w] ≥ hd,s [w∗ ], e hd,s [w] ≤ e hd,s [w∗ ];

(128) (129)

if s < 1, the inequality in (128) is reversed. If hd,s [w] is finite (so that the other quantities are also finite by Theorem 5.6), then equality in (128) or (129) occurs if and only if w(x) = w∗ (x − a) a.e., for some a ∈ Rn . Proof Estimate (128) (and equality cases) follows from (65) with 9(u) = u s / 0(s), or from (66) with 9(u) = eu , K i j = Nd , and f j = log w. In the case s = nd, note that ed ≥ 0 and radially decreasing, and E d (t, 0) − E d (t, 1) ≥ 0. Both inequalities Nd − N are strict on a set of positive measure (e.g., E d (t, 1) → 0 as t → 0). Inequality (129) (and equality cases) follows from Theorem 3.1 and the Riesz ed is radially decreasing.) multiple rearrangement inequality. ( N

546

CARLO MORPURGO

Proof of Theorem 5.10 Let n − 2 < sd < n with s > 1 (and s an integer if d > 2). Let w be as in the hypothesis, with hd,s [w] < ∞, and let wk be as in (70) in the proof of Theorem 4.1. Also, let Wk = (wk ◦ π −1 )|Jπ −1 |d/n . By conformal invariance and (128) we have −hd,s [wk ] ≤ −hd,s [wk+1 ]. Hence (since w0 = w, W0 = W ), for each k ≥ 0, −hd,s [w] ≤ −hd,s [wk ] ≤ Z (s, Ad )

kWk kss −→ Z (s, Ad ) |S n |

(130)

as k → ∞ since, from (70), Wk → 1 in L n/d (S n ) and hence in L s (S n ). If 0 < s < 1, all the inequalities in (130) are reversed and the limit still holds. This shows (124). As for the equality case, we argue as in the proof of Theorem 4.1, using Proposition 5.13. By Theorem 5.6 we also get (122), with the equality case. Let s = n/d, and let w be as specified in (d). To prove (125) it is enough to assume w ≤ C|Jπ |d/n , by taking the sequence wk = min(w, k|Jπ |d/n ) and approximating all terms as in the proof of (114) from Theorem 5.6. Under this assumption, if wk is the sequence given as in (70), which also satisfies wk ≤ C|Jπ |d/n , then the functional z[w] equal to the right-hand side of (114) satisfies z[wk ] ≤ z[wk+1 ] by conformal invariance and by (128) and (129) (and also by using the well-known fact R that wn/d log w is invariant under rearrangement). Hence, for each k ≥ 0 (note that kwkn/d = kwk kn/d ), Z 2 n/d z[w] ≤ z[wk ] ≤ e Z (n/d, Ad ) + W log Wk −→ e Z (n/d, Ad ) d|S n |0(n) S n k as k → ∞. (The assumption wk ≤ C|Jπ |d/n , that is, Wk ≤ C, is used to guarantee that the limit of the logarithmic integral is zero.) For the equality case we cannot appeal to the single equality cases of Proposition 5.13 since there is a sum of two functionals with different signs. However, in the case d < 2, if we have equality for a given w given as in (d), then we also have −hd,n/d [w] +e hd,n/d [w] = −hd,n/d [w∗ ] +e hd,n/d [w∗ ] (when hd,n/d [w] < ∞), which implies equality for the functional E d (t, 0)

Z

1

Z Z Rn

D

0

w(Btv ) dv

n/d

d Px,x,t d x − E d (t, 0) − E d (t, 1)

Z Rn

wn/d

for any t > 0, and hence implies equality for the first term of this quantity. (Observe that we can split the integral in this case since w ∈ L n/d (Rn ).) The finite-dimensional case (d > 2) is handled similarly, and in both cases we can settle equality using Theorems 3.1 and 3.5.


547

Remark. The inequality (122), when s > 1, can be rewritten as follows: kW k p 1 kW k1 + Hd,s [W 1/s ] ≤ n n 1/ p |S | −Z (s, Ad ) |S |

(131)

with p = n/sd. Clearly, by Hölder’s inequality, |S n |−1 kW k1 ≤ |S n |−1/ p kW k p . Since the right side is actually a norm, (131) represents the imbedding of some Banach subspace of L 1 into L p . Since the Dini norm measures smoothness, it would be interesting to understand the relation of (131) with the Sobolev imbedding H α,1 ,→ L p . Appendix Proof of Lemma 3.12 Suppose first that A is open. We begin by showing that Z Z d x d Px,x,t (B) = d x d Px,x,t (B). Ac ×C (A)

(132)

A×C (Ac )

This formula implies (56) for A open. By right continuity of the sample paths we can write [ C (A) = {B : Br ∈ A}, r ∈Tt

where Tt is countable and dense in [0, t) (see [Chu]). Now let N ≥ 2, r j = jt/N for S −1 j = 0, 1, 2, . . . , N − 1, and let C N (A) = Nj=0 {B : Br j ∈ A}. We have Z Z d x d Px,x,t (B) = d x d Px,x,t (B). Ac ×C N (A)

A×C N (Ac )

c To see this, note that C N (A) = B : Br0 ∈ Ac , . . . , Br N −1 ∈ Ac ; and if we let A0 = A, A1 = Ac , then Z d x d Px,x,t (B) Ac ×C N (A)

X

=

Z

`2 ,...,` N =0,1 not all ` j =1

=

X `2 ,...,` N =0,1 not all ` j =0

Z Ac

Z

d x1

d x2 A` N

A `2

Z

Z d x2 A`2

dxN A` N

N Y

E d t/N , |x j − x j+1 |

j=1

Z

d x1 A

dxN

N Y j=1

1 E d (t, 0)

E d t/N , |x j − x j+1 |

1 E d (t, 0)

with x N +1 = x1 ; this last equality follows by a simple symmetry argument. Next, take T as the set of dyadic points in [0, 1), and let Tt = t T . Also, let Ttm = { jt2−m , j =

548

CARLO MORPURGO

S 0, . . . , 2m − 1}, and let Cm (A) = r ∈Ttm {Br ∈ A}. Then Cm (A) ↑ C (A), and Cm (Ac ) ↑ C (Ac ). Formula (132) then follows by the monotone convergence theorem. Now let A be compact. As shown in [Chu, p. 93], given a sequence of open T T sets An such that An ⊇ A¯ n+1 , A = n An = n A¯ n , then D An ↑ D A a.s., where D A = inf{v ≥ 0 : Bv ∈ A} is the first entrance time. We can take this sequence so that |A1 | < ∞, which implies that |An | ↓ |A|. Using C (A) = {B : D A < t} ⊆ C(An ), for all n, and Fatou’s lemma, Z Z d x d Px,x,t (B) ≤ lim inf d x d Px,x,t (B) n

Ac ×C (A)

Acn ×C (A)

Z ≤ lim inf n

d x d Px,x,t (B)

Acn ×C (An )

≤ lim inf |An | = |A|. n

If A is any Borel set, there exist compact sets K n ⊆ A, K n ⊆ K n+1 , so that D K n ↓ D A , a.s. (see [Chu, p. 93]), and the conclusion follows again from Fatou’s lemma and the previous case: Z Z d x d Px,x,t (B) ≤ lim inf d x d Px,x,t (B) n

Ac ×C (A)

Ac ×C (K n )

Z ≤ lim inf n

d x d Px,x,t (B) ≤ lim inf |K n | n

K nc ×C (K n )

≤ |A|. Proof of Lemma 5.8 We have the following (C being a positive constant that may vary from place to place): Z Z s w (x + t) − ws (x) p d x ≤ C (1 + |x + t|2 )−sd − (1 + |x|2 )−sd p d x Rn Rn Z s W π(x + t) − W s π(x) p (1 + |x|2 )−sd p d x := I + I I. +C Rn

For the first term it is enough to assume that t = λe1 , where e1 = (1, 0, . . . , 0), by rotation. Thus, via changes of variables of type x 7 → −x and x 7→ x + t, we get Z Z Z 2 −sd 2 −sd p I =2 (1 + |x − t| ) − (1 + |x| ) dx = 2 + 2 . 2x·t≥|t|2

|t|2 ≤2x·t≤3|t|2

2x·t≥3|t|2


For the second term, Z Z ≤ 2x·t≥3|t|2

549

h 1 + |x − t|2 sd i p (1 + |x − t|2 )−sd p 1 − 1 + |x|2

2x·t≥3|t|2

Z

(1 + |x − t|2 )−sdp

≤C

h 2x · t − |t|2 i p dx 1 + |x|2

2x·t≥3|t|2

≤ C|t|

n−2sdp

Z

|x − e1 |−2sdp

2x · e − 1 p 1 d x ≤ C|t|n−2sd p |x|2

2x·e1 ≥3

for 2sdp > n − p. The same estimate holds for the first term when 2sdp < n, and when 2sdp ≥ n, Z Z Z 2 −sd p ≤ (1 + |x − t| ) dx = (1 + |x|2 )−sd p d x |t|2 ≤2x·t≤3|t|2

|t|2 ≤2x·t≤3|t|2

Z

−|t|2 ≤2x·t≤|t|2

(1 + x1 + · · · + xn )−2sdp d x

≤C 0≤x1 ≤|t|/2 x2 ,...,xn ≥0

|t|/2

Z

(1 + x1 )n−1−2sd p d x1 ,

=C 0

which gives the desired estimates for I . For I I , note that |W s (π(x + t)) − W s (π(x))| p ≤ C|π(x + t) − π(x)| p = C2 p |t| p (1 + |x + t|2 )− p/2 (1 + |x|2 )− p/2 ; hence, Z I I ≤ C|t| p (1 + |x + t|2 )− p/2 (1 + |x|2 )−sd p− p/2 d x n R Z Z p = C|t| + ≤C |x| n − p. (When |x| < |t|/2, we use (1 + |x + t|2 )− p/2 < C|t|− p ; when |x| ≥ |t|/2, we use (1 + |x|2 )−sd p ≤ C(1 + |x + t|2 )−sdp and (1 + |x|2 )− p/2 ≤ C|t|− p .) Proof of Proposition 5.11 In the given range of s we have (|ξ − η|sd−n − d(ξ, η)sd−n )η→ξ = 0. Hence, from (96), Proposition 2.2, and (11) we obtain, for n > 1, Z h i 1 Z (s, Ad ) = G(s, x, y) − |ξ − η|sd−n dξ η→ξ γ (sd) Sn Z hX ∞ i 2k + n − 1 1 1 (n−1)/2 = − P (ξ · η) dξ. η→ξ (n − 1)|S n | λsk,d λk,sd k Sn k=0

550

CARLO MORPURGO n

(Recall that G s (s, ξ, h) is the Green function of Asd .) We now use the following. A.1 For a > 0 the function [0(a − x)/ 0(a + x)]1/x is increasing in x ∈ [0, a). As a consequence, if 0 < d < n and 1 < s < n/d, then λk,sd < λsk,d for any k ≥ 0 (i.e., Asd < Asd in the operator sense); if 0 < s < 1, the inequality is reversed. LEMMA

Proof We use the formula ∞ Y 0(a − x) 2x 2γ x =e 1+ e−2x/(m+1) 0(a + x) m+a−x m=0

(see [EMOT, Sec. 1.3 (2)]). To finish the proof, we raise to the power 1/x and check that the functions (1 + (2x/m + a − x))1/x are increasing in x ∈ (0, a) for any m ≥ 0. ν+1 Continuing the proof of Proposition 5.11, recall that (Pkν )0 = 2ν Pk−1 for k ≥ 1, ν ν 0 (P0 ) = 0, and Pk (1) = 0(k + 2ν)/[k!0(2ν)]. This means that when η approaches (n−1)/2 (n−1)/2 ξ along a diameter, Pk (η · ξ ) ↑ Pk (1) and the monotone convergence theorem implies ∞ X 1 1 Z (s, Ad ) = − mk λsk,d λk,sd k=0

with m k = ((2k + n − 1)(k + n − 2)!)/((n − 1)!k!), the multiplicity of λk,d . This formula and Lemma A.1 imply Proposition 5.11 in the case n > 1. The case n = 1 is similar (see Remark 1). Acknowledgments. I am deeply grateful to Bill Beckner for his very valuable help, suggestions, and encouragement. I also would like to thank all the people who have helped in some way or who have discussed issues of this work with me. In particular, I wish to thank Al Baernstein, Tom Branson, Almut Burchard, Eric Carlen, E. B. Davies, Gerd Grubb, Mourad Ismail, Rick Laugesen, Michael Loss, Stanley Sawyer, Jan Van Casteren, Misha Vishik, and Zhen-Qing Chen. References [AB]

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Academic Press, New York, 1974. MR 55:8655 540 [O]

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(Chicago, 1966), Proc. Sympos. Pure Math. 10, Amer. Math. Soc., Providence, 1967, 288 – 307. MR 38:6220 524 B. SIMON, Functional Integration and Quantum Physics, Pure Appl. Math. 86, Academic Press, New York, 1979. MR 84m:81066 486, 493 D. STROOCK, The Kac Approach to potential theory, I, J. Math. Mech. 16 (1967), 829 – 852. MR 34:8499 497

Department of Mathematics, University of Missouri at Columbia, 121 Mathematical Sciences Building, Columbia, Missouri 65211, USA; [email protected]; and Dipartimento di Matematica e Applicazioni, Universitá di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milan, Italy; [email protected]


L p -ESTIMATES FOR SINGULAR INTEGRALS AND MAXIMAL OPERATORS ASSOCIATED WITH FLAT CURVES ON THE HEISENBERG GROUP JOONIL KIM

Abstract The maximal function along a curve (t, γ (t), tγ (t)) on the Heisenberg group is discussed. The L p -boundedness of this operator is shown under the doubling condition of γ 0 for convex γ in R+ . This condition also applies to the singular integrals when γ is extended as an even or odd function. The proof is based on angular LittlewoodPaley decompositions in the Heisenberg group. 1. Introduction Let 0 be a C 1 -mapping (x, t) 7 → 0(x, t) defined in a neighborhood of the origin in Rn × Rk taking values in Rn with 0(x, 0) ≡ x. Let K be a Calderón-Zygmund kernel in Rk . By this we mean that K ∈ C 1 (Rk − {0}) is homogeneous of degree −k and R satisfies |t|=1 K (t) dt = 0. Then we form the singular Radon transform H0 defined in C0∞ by Z H0 f (x) = p.v. f 0(x, t) K (t) dt, (1.1) |t| 1 such that w(Dt) ≥ 2w(t) for t > 0. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 114, No. 3, Received 18 October 2000. Revision received 9 October 2001. 2000 Mathematics Subject Classification. Primary 42B20, 42B25. 555

556

JOONIL KIM

Let us look at the translation invariant case. Suppose that 0(x, t) in (1.1) and (1.2) is given by (x1 −t, x2 −γ (t)), where γ is a convex function on R+ 7→ R+ and γ (0) = 0. Define an auxiliary function on R+ by h(t) = tγ 0 (t) − γ (t). It is known that if γ 0 has bounded doubling time, then h has bounded doubling time, but the reverse is not true (see [17]). In [17] and [18], L 2 -theorems were proved. THEOREM 1 Let γ be extended as an odd curve in R1 . Then H0 is bounded in L 2 (R2 ) if and only if h has bounded doubling time. Let γ be extended to be an even curve in R1 . Then H0 is bounded in L 2 (R2 ) if and only if γ 0 has bounded doubling time. Moreover, M0 is bounded in L 2 (R2 ) if h has bounded doubling time.

In [5], an L p -theorem was given by the following. THEOREM 2 Let γ be extended as an even or odd function in R1 . Suppose that γ 0 has bounded doubling time; then H0 is bounded on L p (R2 ) for 1 < p < ∞, and M0 is bounded for 1 < p ≤ ∞.

In this paper we consider H0 and M0 defined on the Heisenberg group as a variable coefficient model. Let H1 be the three-dimensional Heisenberg group identified with R2 × R. The group multiplication is defined by (x1 , x2 , x3 ) · (y1 , y2 , y3 ) = x1 + y1 , x2 + y2 , x3 + y3 + 2(x1 y2 − x2 y1 ) , and the inverse is given by (x1 , x2 , x3 )−1 = (−x1 , −x2 , −x3 ). We define a class of operators H0α and M0α by formulas (1.1) and (1.2) with n = 3 and k = 1, and −1 0α (x1 , x2 , x3 , t) = (x1 , x2 , x3 ) · t, γ (t), αtγ (t) for each α ∈ R1 . Here γ : R+ 7→ R+ is a convex function with γ (0) = 0. In [12] the combination of the group representation theory and the almost orthogonality argument of [20] proved the following L 2 -theorem. 3 Let γ be extended as an odd curve in R1 . Then H00 is bounded in L 2 (H1 ) if and only if h has bounded doubling time. Let γ be extended as an even curve in R1 . Then H0α is bounded in L 2 (H1 ) if and only if γ 0 has bounded doubling time. Let γ be an extended odd or even curve. If γ 0 has bounded doubling time, then H0α and M0α are bounded on L 2 (H1 ). Moreover, M00 is bounded in L 2 (H1 ) if h has bounded doubling time. THEOREM

SINGULAR INTEGRALS ON THE HEISENBERG GROUP

557

In [5] the authors derived Theorem 2 from the good decay estimate of the Fourier transform of the singular measure in the complement of the normal direction of γ ; the normal direction of γ was handled by a Littlewood-Paley argument based on a decomposition of the Fourier transform plane into lacunary sectors as in [16]. In this paper we develop the Littlewood-Paley argument of [5] on the Heisenberg group, and we combine this with the techniques of [12] to prove the following result. MAIN THEOREM

Let γ be extended as an even or odd curve in R1 . Suppose that γ 0 has bounded doubling time. Then H0α is bounded on L p (H1 ) for 1 < p < ∞, and M0α is bounded on L p (H1 ) for 1 < p ≤ ∞. Remark 1 (1) The L p (H1 )-norms of H0α and M0α are independent of α (see Sec. 5.3). (2) The assumption that convex function γ is even or odd can be weakened by the balanced biconvex curve in [9] and [12]. (3) In [4] the L p (H1 )-boundedness of H0α and M0α was proved under some stronger assumptions for odd curves γ . The main technique in [2] (which uses the dilation property for the scaling argument) was applied for the proof of [4], and this method has also been developed in [3] for the oscillatory singular integrals considered in [20]. However, the operators neither in the main theorem nor in [20] are known to be bounded in L p under the condition of [2] such that h 0 (t) > h(t)/t for some > 0 with odd γ . (4) Examples. The above four theorems admit the following classes of curves: (i) affine linear curve on [t j+1 , t j ] (where j ∈ Z, t j → 0 as j → ∞, and t j → ∞ as j → −∞) with the ratios t j /t j+1 bounded piecewise linear convex curve plotting points on (t, γ (t)) ; (ii) flat curves defined by γ (t) = exp(−1/t), or γ (t) = exp(− exp(1/t)), and so on, in R+ . This paper is organized as follows. In Section 2, we briefly discuss the group representation theory, which is used for the L 2 -estimates of the convolution-type operators on the Heisenberg group. In Section 3, we give an analogue of the multiparameter Marcinkiewicz multiplier theorem of [13] on the Heisenberg group, which is the essential part for the proof of the Littlewood-Paley theory for angular sectors on H1 . In Sections 4 – 6, we give the proof of the main theorem. Notation Suppose that D is a doubling constant of γ 0 satisfying the condition in Definition 1.

558

JOONIL KIM ˜

˜

Let D˜ = log 2/log D; then γ 0 (2n t) ≥ 2 Dn γ 0 (t) for t > 0. We also denote D 0 = 2 D . Given two quantities a and b, we write a . b or b & a if there is a constant C (depending only on D or the dimensions of the given space) such that a ≤ Cb. 2. Group Fourier transform In this section we sketch some basic facts about group Fourier transforms in [11] and [22]. Let Hn = {(z, t) : z = ( p + iq) ∈ Cn , t ∈ R1 } be the (2n + 1)-dimensional Heisenberg group with the group multiplication (z 1 , t1 ) · (z 2 , t2 ) = (z 1 + z 2 , t1 + t2 + 2=(z 1 · z 2 )). Right invariant vector fields on Hn can be defined for 1 ≤ j ≤ n: 1 ∂ ∂ + 2q j , 2πi ∂ p j ∂t 1 ∂ ∂ Yj = − 2pj , 2πi ∂q j ∂t 1 ∂ T = . 2πi ∂t

Xj =

(2.1)

Then {X j , Y j , T } forms a basis of the Lie algebra hn corresponding to the Lie group Hn (hn is called the Heisenberg Lie algebra). We also define the operators P j and Q j acting on the Hilbert space, say, L 2 (Rn ), where 1 ∂ f (x) P j f (x) = , Q j f (x) = x j f (x). (2.2) 2πi ∂ x j For each λ ∈ R1 , we define a one-parameter Schrödinger representation by a mapping R λ from the Heisenberg group Hn to the group of unitary operators on L 2 (Rn ) such that, for φ ∈ L 2 (Rn ) and ( p, q, t) ∈ Hn , λ R ( p, q, t)φ (x) = [e2πi( p·P+λq·Q+λt I /4) φ](x) = e2πiλ[q·x+ p·q/2+t/4] φ(x + p), (2.3) where P = (P j ), Q = (Q j ), and I is the identity operator. For the proof of the second equality in (2.3), see [11, Chap. 1]. Let B(L 2 (Rn )) be the space of bounded operators on L 2 (Rn ). The group Fourier transform of f ∈ L 1 (Hn )∩L 2 (Hn ) is defined as an operator-valued function from R1 to B(L 2 (Rn )) such that λ ∈ R1 7 → b f (λ) ∈ B(L 2 (Rn )), given by Z λ b f (λ)φ (x) = R (− p, −q, −t)φ (x) f ( p, q, t) dp dq dt. (2.4) Hn

By (2.1) – (2.4), Xd f (λ) · P j , j f (λ) = b

Yd f (λ) · λQ j , j f (λ) = b

λI Tcf (λ) = b f (λ) · . (2.5) 4


559

From (2.4), b f (λ) is an integral operator on L 2 (Rn ) given by Z λ(x + y) λ b F2,3 f x − y, f (λ)φ (x) = , φ(y) dy, 2 4 Rn

(2.6)

where F2,3 is the Euclidean Fourier transform with respect to the second and third component of f . By (2.4), we can also show that k[ ∗ f (λ) = b k(λ) · b f (λ),

(2.7)

where the multiplication on the right is the composition of operators. From (2.4) and (2.7), we can prove a Fourier inversion formula such as Z tra b f (λ) · R λ ( p, q, t) |λ|n dλ, (2.8) f ( p, q, t) = R1

where tra(T ) denotes the trace of the operator T . For T f (x) = R tra(T ) = L(x, x) d x, and the Plancherel theorem is of the form Z 1 2 k f k L 2 (Hn ) = kb f (λ)k2H S |λ|n dλ, 4 R1

R

L(x, y) f (y) dy,

(2.9)

where k · k H S is a Hilbert-Schmidt norm. From (2.7) and (2.9), the following result can be proved. PROPOSITION 1 Let G be a convolution operator defined by G f = k ∗ f on L 2 (Hn ). Then G is bounded from L 2 (Hn ) to itself if and only if the family of operator norms kb k(λ)k L 2 (Rn )7→ L 2 (Rn ) is essentially bounded. And kGk L 2 (Hn )7→ L 2 (Hn ) is given by (1/2) ess supλ∈R1 kb k(λ)k L 2 (Rn )7→ L 2 (Rn ) .

For the details of the proof, see [22, Chap. 12] and [12]. Proposition 1 applies to vector-valued operators. Let T be the operator defined by Tf = {k j ∗ f j } j∈Z for f = { f j } j∈Z ∈ L 2 (l 2 (H1 )). Then by Proposition 1 we have kTk L 2 (l 2 (Hn ))7→ L 2 (l 2 (Hn )) =

1 ess sup kb T(λ)k L 2 (l 2 (Rn ))7→ L 2 (l 2 (Rn )) , 2 λ∈Rn

(2.10)

where b T(λ)g = {[kbj (λ)] · g j } for g = {g j } ∈ L 2 (l 2 (Rn )).

3. Littlewood-Paley theory in the Heisenberg group 3.1. Dyadic decomposition in Hn Let us choose an even function ψ ∈ C0∞ (−1, 1) such that ψ ≡ 1 on [−1/2, 1/2]. For ν ν each ν = 1, . . . , 2n + 1, let {akν }∞ k=−∞ be the sequence such that ak+c /ak ≥ 2 for

560

JOONIL KIM

ν ν ≥ 2 for all k ∈ Z with some c > 0. We all k ∈ Z with some c > 0, or a−(k+c) /a−k ν ν ν define χk (ξ ) = ψ(ak ξ ) − ψ(ak+1 ξ ) with k ∈ Z. Let ∂xν = (1/(2πi))(∂/∂ xν ) and akν = 2k . Then, on the Euclidean space, a dyadic decomposition of the νth axis can be defined as ∞ ∞ Z X X f = χkν (∂xν ) f = [e2πit∂xν f ](x)[Fχkν ](t) dt, k=−∞

k=−∞

where F is the Fourier transform in R1 . By the Fourier inversion formula and integration by parts, we can show that this formula is the dyadic decomposition of the function f on the Fourier transform side. By analogy we can define dyadic decomposition of each νth axis in Hn as ∞ ∞ Z X X f = χkν (X ν ) f = [e2πit·X ν f ](x)[Fχkν ](t) dt. k=−∞

k=−∞

Here we denote Y j in (2.1) by X j+n and T by X 2n+1 . Then by (2.5), (2.6), and (2.7), χk (X ν ) f = f ∗ L νk such that L νk (y) = [F−1 χkν ](yν )δ( y˜ ν ),

(3.1)

where ν = 1, . . . , 2n + 1 and where F−1 is the inverse Fourier transform in R1 . By y˜ ν we denote a vector in R2n by removing νth component of y, and δ is the Dirac measure at zero in R2n . LEMMA 1 For 1 < p < ∞ and for 1 ≤ ν1 , . . . , νm ≤ 2n + 1,

X 1/2

|L νk11 ∗ · · · ∗ L νkmm ∗ f |2

L p (Hn )

k1 ,...,km

. k f k L p (Hn ) .

Proof We show the proof for the cases ν1 = 1, ν2 = 2, and n = 1. By (3.1), Z L 1k ∗ f (x) = f x2 R2x2 (x1 − y1 , x3 − 2x1 x2 ) [F−1 χk ](y1 ) dy1 .

(3.2)

Here we denote f (x1 , x2 , x3 ) by f x2 (x1 , x3 ) and Ra (v1 , v2 ) = (v1 , av1 + v2 ). Take a sequence of Rademacher functions {rk (s) : k ∈ Z, s ∈ [0, 1]}. Then

X 1/2 p

1 2 2 |L ∗ L ∗ f |

p 1 k l L (H )

k,l

Z 1 Z 1 X X

p 1 2

≈ r (s )L ∗ r (s )L ∗ f k 1 k l 2 l

p 0

0

k

l

L (H1 )

ds1 ds2 .


561

P k k rk (s1 )L 1k ∗ gk L p (H1 ) . kgk L p (H1 ) follows from (3.2) and the Marcinkiewicz multiplier theorem on the one-dimensional Euclidean space. In the same way, P k l rl (s2 )L l2 ∗ f k L p (H1 ) . k f k L p (H1 ) . Let ρ[aκ ,bκ ] be the characteristic function supported on the arbitrary interval [aκ , bκ ] in ν R1 with κ ∈ Zd (d ≥ 1). And let us define, for each ν = 1, . . . , 2n + 1, S[a (y) = κ ,bκ ] −1 ν F (ρ[aκ ,bκ ] )(yν )δ( y˜ ). LEMMA 2 For each ν = 1, . . . , 2n + 1 and for 1 < p < ∞,

X 1/2

ν 2 |S[a ∗ f |

κ κ ,bκ ] κ

L p (Hn )

X 1/2

. | f κ |2

κ

L p (Hn )

.

Proof The above inequality (for ν = 1) follows from (3.2) together with the corresponding vector-valued inequality on the Euclidean space (see [21, Chap. 4, Th. 4]). The same proof applies to the case ν 6= 1. 3.2. Marcinkiewicz theorem for the Kohn-Nirenberg correspondence Let m be a differentiable function away from each of the axes in Rd . In particular, m satisfies the condition that ∂ α m(ξ ) C (3.3) α1 ≤ |ξ1 |α1 · · · |ξd |αd ∂ξ1 · · · ∂ξdαd P for α = dj=1 α j and for ξ = (ξ1 , . . . , ξd ). By the multiparameter Marcinkiewicz multiplier theorem on Rn , the operator defined by m(i∂/∂ x1 , . . . , i∂/∂ xd ) is bounded in L p (Rd ) with 1 < p < ∞ under the condition of (3.3) with 0 ≤ α ≤ d. In this section we show how one can extend this result on the Heisenberg group Hn in an appropriate way. Let us denote (X j )nj=1 by X and (Y j )nj=1 by Y . By using a functional calculus (here we adapt the Kohn-Nirenberg correspondence), we assign an operator m K N on n 1 n C∞ 0 (H ) to each Fm ∈ L (H ): Z m K N f ( p, q, t) = [e2πi x·X e2πi y·Y e2πisT f ]( p, q, t)[Fm](x, y, s) d x dy ds, (3.4) where p, q, x, y ∈ Rn , t, s ∈ R1 , and F is the Euclidean Fourier transform in R2n+1 . We can extend this definition for the distributions m that satisfy (3.3), and we prove the following result.

562

JOONIL KIM

THEOREM 4 Suppose that m satisfies the inequality in (3.3) for d = 2n + 1 and 0 ≤ α ≤ 2n + 1. Then m K N can be extended so that it is bounded on L p (Hn ) for 1 < p < ∞.

Remark 2 (1) The class of m satisfying (3.3) is invariant under a large group of dilations, m(ξ ) 7→ m(δ ◦ ξ ), where (δ ◦ ξ ) = (δ1 ξ1 , . . . , δ2d+1 ξ2d+1 ). (2) Instead of the spectral calculus, we used the Kohn-Nirenberg correspondence in (3.4) since X j and Y j are not commutative ([X j , Y j ] = 2πi T ). Another useful functional calculus for noncommuting operators is the Weyl correspondence (see [1], [10], [11], [24]). Let N be a nilpotent group, and let n be its Lie R algebra with dimension d. Then we define an operator m W = e2πi x·Z [Fm] j=d ·(x) d x, where Z = (Z j ) j=1 and {Z j : 1 ≤ j ≤ d} forms a basis of the Lie algebra n as a vector space. It would also be interesting to investigate the L p -boundedness of m W under the condition of (3.3). P (3) Let L be the sub-Laplacian on Hn of the form L = nj=1 X 2j + Y j2 . Since L and T are self-adjoint and commuting operators, the joint spectral multiplier m(L , T ) is well defined. The L p -boundedness of m(L , T ) is proved under the condition of (3.3) with d = 2 and a = ((2n + 2) + )/2 ( > 0) for 1 < p < ∞, and this theorem can also be extended on the product space of the Nilpotent groups (see [14], [15]). Next, we find the convolution kernel of m K N . LEMMA 3 We have m K N f = f ∗ A with A(x, y, s) = [F−1 m](x, y, s + 2x · y), where F−1 is the inverse Fourier transform in R2n+1 .

Proof By (2.5) we have 2πi y·Y e2πisT f ](λ) = b [e2πi x·X e\ f (λ) · e2πiλy·Q · e2πi x·P · e2πiλs I /4 .

(3.5)

From (2.5) and the Campbell-Hausdorff formula in [22, Chap. 12], we can show the formulas e[2πiλy·Q,2πi x·P]/2 g = eπiλx·y g, e2πiλy·Q e2πi x·Pg = e[2πiλy·Q,2πi x·P]/2 e2πi(λy·Q+x·P) g

(3.6)

for the function g defined in Rn . Insert m K N f in (2.8). Then by (3.5) and (3.6), together with the change of variable s 0 = s + 2x · y in (3.4), we obtain m K N f = f ∗ A.


563

3.3. Proof of Theorem 4 We prove that kA ∗ f k L p (H1 ) . k f k L p (H1 ) with A(x1 , x2 , x3 ) = [F−1 m](x1 , x2 , x3 − 2x1 x2 ) and m satisfying (3.3) with d = 3. First, we consider the case when m is a function of the first two variables such that A(x1 , x2 , x3 ) is given by [F−1 m(x1 , x2 )]δ(x3 − 2x1 x2 ). Choose akν = 2−k for each ν = 1, 2, 3 in the definition of (3.1), and write χkν as χk . We have XX A∗ f = L 1k ∗ L l2 ∗ A ∗ L 1k+m ∗ f. m

k,l

From interpolation and duality, it suffices to show that, for 2 ≤ p < ∞,

X

L 1k ∗ L l2 ∗ A ∗ L 1k+m ∗ f p 1 . k f k L p (H1 ) ,

(3.7)

L (H )

k,l

X

L 1k ∗ L l2 ∗ A ∗ L 1k+m ∗ f

k,l

L 2 (H1 )

. 2−c|m| k f k L 2 (H1 )

(3.8)

for some positive constant c. Proof of (3.7) 2 + L 2 + L 2 ; then L ˜ 2 ∗ L 2 = L 2 almost everywhere in H1 . From this, Put L˜ l2 = L l−1 l l l l l+1 together with the duality and Lemma 1, it suffices to show that

X 1/2

|L l2 ∗ A ∗ L 1k ∗ f |2 (3.9)

p 1 . k f k L p (H1 ) . L (H )

k,l

By (2.6) a group Fourier transform of L l2 ∗ A ∗ L 1k at λ ∈ R1 is the operator defined on L 2 (R1 ): Z 2\ 1 b(ξ ) dξ L l ∗ A ∗ L k (λ)φ (x) = χl (λx)m(ξ, λx)χk (ξ )e2πiξ ·x φ (3.10) for φ ∈ L 2 (R1 ). We may assume that the support of the integral kernel of the above operator is contained in the first quadrant: {2l−1 ≤ λx ≤ 2l+1 , 2k−1 ≤ ξ ≤ 2k+1 }. We have Z ξ Z λx Z ξ 00 m(ξ, λx) = m 1,2 (t1 , t2 ) dt1 dt2 + m 01 (t1 , 2l−1 ) dt1 2k−1 2l−1 2k−1 Z λx + m 02 (2k−1 , t2 ) dt2 + m(2k−1 , 2l−1 ); (3.11) 2l−1

here m 0j indicates the partial derivative of m in the jth variable. By inserting each ∗ A ∗ L 1 (λ)φ](x) = term on the right-hand side of (3.11) in (3.10), we express [ L 2 \ l

k

564

JOONIL KIM

λ,n n=1 [Sl,k φ](x):

P4

λ,1 Sl,k φ(x)

ZZ

= Jk,l

λ,2 Sl,k φ(x) =

Z

λ,3 Sl,k φ(x) =

Z

S[t2

l+1 ] 2 ,2

2 1 ∗ L\ l ∗ S[t

1 ,2

k+1 ]

∗ L 1k (λ)φ (x)m 001,2 (t1 , t2 ) dt1 dt2 ,

1 L l2 ∗ S\ ∗ L 1k (λ)φ (x)m 01 (t1 , 2l−1 ) dt1 , [t ,2k+1 ] 1

Jk

Jl

\ ∗ L l2 ∗ L 1k (λ)φ (x)m 02 (2k−1 , t2 ) dt2 ,

S[t2

l+1 ] 2 ,2

2 λ,4 1 Sl,k u(x) = m(2k−1 , 2l−1 ) L\ l ∗ L k (λ)φ (x), where Jk = [2k−1 , 2k+1 ] and Jk,l = Jk × Jl . By the Fourier inversion formula in (2.8), Z L l2 ∗ A ∗ L 1k ( p, q, t) = tra [ L l2 \ ∗ A ∗ L 1k (λ)] · R λ ( p, q, t) |λ| dλ. R1

We replace [ L l2 \ ∗ A ∗ L 1k (λ)] with P4

2 Lk,l 3 Lk,l

to obtain L l2 ∗ A ∗ L 1k ∗ f (x) =

f (x) for x = (x1 , x2 , x3 ) ∈ ZZ f (x) = S[t2 ,2l+1 ] ∗ L l2 ∗ S[t1 ,2k+1 ] ∗ L 1k ∗ f (x)m 001,2 (t1 , t2 ) dt1 dt2 , 2 1 J Z k,l f (x) = L l2 ∗ S[t1 ,2k+1 ] ∗ L 1k ∗ f (x)m 01 (t1 , 2l−1 ) dt1 , 1 Jk Z f (x) = S[t2 ,2l+1 ] ∗ L l2 ∗ L 1k ∗ f (x)m 02 (2k−1 , t2 ) dt2 ,

n n=1 Lk,l 1 Lk,l

λ,n n=1 Sl,k H1 . Here

P4

Jl

2

4 Lk,l f (x) = m(2k−1 , 2l−1 )L l2 ∗ L 1k ∗ f (x).

(3.12)

1 f (x)| with (3.3), we have By using the Schwarz inequality for |Lk,l X X ZZ L 1 f (x) 2 . S 2 l+1 ∗ L 2 ∗ S 1 k+1 ∗ L 1 ∗ f (x) 2 dt1 dt2 . k,l l k [t2 ,2 ] [t1 ,2 ] |t1 | |t2 | Jk,l l,k

l,k

Apply the change of variables 2l t20 = t2 and 2k t10 = t1 to the above integral. Then P 1 f |2 1/2 k by the Minkowski inequality, we know that k k,l |Lk,l L p (H1 ) is majorized by Z 2 Z 2 X 2 1/2

S l l+1 ∗ L 2 ∗ S 1 k k+1 ∗ L 1 ∗ f 2

p 1 dt1 dt2 l k [2 t ,2 ] [2 t ,2 ] 1/2 1/2

2

L (H )

1

l,k

2 l l+1 ] and for p/2 ≥ 1. We use Lemma 2 for S[a ,bκ ] , where [aκ , bκ ] = [2 t2 , 2 κP 2 2 1 κ = (k, l) ∈ Z × Z. Then we find that k l,k |S[2l t ,2l+1 ] ∗ L l ∗ S[2k t ,2k+1 ] ∗ L 1k ∗ 2

1


565

1/2 f |2 k L p (H1 ) is majorized by

X 1/2

p L 2 ∗ S 1 k k+1 ∗ L 1 ∗ f 2

[2 t1 ,2

l

k

]

L p (H1 )

k,l

Z ≈

1 Z 1 X

0

0

X 1 rl (s2 )L l2 ∗ rk (s1 )S[2 kt

l

p

1 ∗ L ∗ f

p k+1 ] k 1 ,2

Z 1 X p

1 1 . rk (s1 )S[2

k t ,2k+1 ] ∗ L k ∗ f p 1 0

L (H1 )

k

X

1 ≈ |S[2 kt

1

L (H1 )

k

,2k+1 ]

∗ L 1k ∗ f |2

k

X 1/2 p

. |L 1k ∗ f |2

p

L (H1 )

k

ds1 ds2

ds1

1/2 p

p

L (H1 ) p

. k f k L p (H1 ) .

The first inequality follows in the same way as the proof of Lemma 1, the second follows from Lemma 2, and the third follows from Lemma 1. So we proved that, for 2 ≤ p < ∞,

X 1/2

L 1 f 2 . kfk p 1

k,l

k,l

L p (H1 )

L (H )

when we restrict the support of the integral kernel in (4.4) to the first quadrant. And the above argument also applies to the other quadrants. The same results can be proved 2 , L 3 , L 4 in the same way. for Lk,l k,l k,l Proof of (3.8) The group Fourier transform of L 1k ∗ L l2 ∗ A ∗ L 1k+m is the operator-valued mapping λ,m λ ∈ R1 7→ Tk,l ∈ B(L 2 (R1 )), given by Z λ,m b Tk,l φ(x) = L m k,l (x, ξ )φ (ξ ) dξ, where Lm k,l (x, ξ )

ZZ =

χk (η)χl (λy)m(ξ, λy)χk+m (ξ )e2πi[(ξ −η)·y+η·x] dy dη.

From (2.10) with the Cotlar-Stein lemma, it suffices to show that

λ,m ∗ λ,m

[Tk1 ,l1 ] Tk2 ,l2

op

λ,m kTk,l kop ≤ 2−c|m| ,

+ Tkλ,m [Tkλ,m ]∗ ≤ 2−c(|k1 −k2 |+|l1 −l2 |) . 1 ,l1 1 ,l2 op

(3.13) (3.14)

In this paper we may assume, without loss of generality, that λ > 0 for the L 2 estimates of all the operators given by the group Fourier transform of (2.4).

566

JOONIL KIM

Proof of (3.13) λ,m The uniform boundedness of kTk,l kop follows from (3.9). So we can assume that m > 100. Let us fix λ, l, k and take two smooth cutoff functions 51 , 52 satisfying supp 51 ⊂ {|x| > 2|m|/10 2(l−3) /λ} and supp 52 ⊂ {|x| < 2|m|/10 2(l+3) /λ} with λ,m 51 + 52 = 1. By 51 and 52 , we split Tk,l = T 1 + T 2 so that the integral kernel of ν the operator T is L ν (x, ξ ) = 5ν (x)L m k,l (x, ξ ) for ν = 1, 2. By applying integration by parts on dy for the kernel L 1 (x, ξ ), we obtain for fixed x, Z Z Z Z 1 1 L (x, ξ ) dξ . dy dη dξ 2 |y|2 k+m k l |ξ − η| |ξ |≈2 |η|≈2 |y|≈2 /λ λ . m l. (3.15) 2 2 Integration by parts on dy for L 1 (x, ξ ) yields for fixed ξ , Z Z Z Z 1 1 L (x, ξ ) d x . dy dη d x |x|2 2 /λ |η|≈2 |y|≈2 /λ 1 . |m|/10 k . (3.17) 2 2 Integration by parts on dy for L 2 (x, ξ ) yields for fixed x, Z Z Z Z 2 1 L (x, ξ ) dξ . dy dη dξ . 2k . |ξ |≈2k+m |η|≈2k |y|≈2l /λ |ξ − η||y|

(3.18)

Thus kT 2 kop ≤ 2−|m|/20 by (3.17) and (3.18). Therefore (3.13) is proved. Proof of (3.14) c b The uniform boundedness of kc L l2 (λ)· A(λ)· L 1k (λ)kop in λ follows from (3.9). And the same result also holds for its adjoint operator. Hence in showing k[Tkλ,m ]∗ Tkλ,m kop . 1 ,l1 1 ,l2 −c(|k −k |+|l −l |) 1 2 1 2 2 for some c, we have only to prove that

c2 c2 (λ)

[ L (λ)]∗ · [d

≤ 2−c(|k1 −k2 |+|l1 −l2 |) . L 1k1 (λ)]∗ · d L 1k2 (λ) · L (3.19) l1 l2 op


567

If |k1 − k2 | > 2, k[d L 1k1 (λ)]∗ · d L 1k2 (λ)kop = 0. Thus it suffices to consider k1 ≈ k2 . c2 (λ)]∗ · [d c2 (λ)φ(x) is written as By (2.6), [ L L 1 (λ)]∗ · d L 1 (λ) · L l1

k1

Z

k2

l2

χl1 (λx)[F−1 χk1 ] ∗ [F−1 χk2 ](x − y)χl2 (λy)φ(y) dy,

where ∗ is the convolution on R1 . Since |[F−1 χk11 ] ∗ [F−1 χk12 ]| . 1/|x|, we have for any fixed y (or x) and for l1 l2 , Z χl (λx)k ,k (x − y)χl (λy) d x (or dy) . 1 (or 2−|l1 −l2 | ). (3.20) 2 1 1 2 Hence (3.19) follows from (3.20). Next, we show that, for some constant c,

λ,m λ,m ∗ −c(|k1 −k2 |+|l1 −l2 |)

T

. k1 ,l1 [Tk1 ,l2 ] op ≤ 2

(3.21)

The operator norm above is bounded by

c2 \ ∗ ∗ 1 1

L (λ) · A(λ) b · L\ b ∗ c2 l2 k2 +m (λ) · [ L k1 +m (λ)] · [ A(λ)] · [ L l1 (λ)] op .

(3.22)

It suffices to show that (3.22) is bounded by 2−c|l1 −l2 | for k1 = k2 = k. The R operator in (3.22) is represented by Vl1 ,l2 ,k (x, y) f (y) dy, where Vl1 ,l2 ,k (x, y) is Z 2 e2πiξ ·(x−y) χl1 (λx)m(ξ, λx) χk (ξ ) χl2 (λy)m(ξ, λy) dξ. Let us fix l1 l2 . Integration by parts yields for each fixed y (or x), Z Z Z χl1 (λx)χl2 (λy) Vl ,l ,k (x, y) d x (or dy) . dξ d x (or dy) 1 2 k |x − y||ξ | |ξ |≈2 . 1 (or 2−|l1 −l2 | ). Thus (3.21) is proved by the above inequality. Hence (3.14) holds, and this completes the proof of (3.8). Therefore the desired estimation is given for the proof of Theorem 4 when m does not depend on the last component. The proof of Theorem 4 (for n = 1) follows by using another decomposition for the third component of A: XX A∗ f = L 1k ∗ L l2 ∗ L 3j ∗ A ∗ L 1k+m ∗ f. m k,l, j

Here we need eight operators instead of four in (3.12): we use an octant such as {2l−1 ≤ λx ≤ 2l+1 , 2k−1 ≤ ξ ≤ 2k+1 , 2n+1 ≤ λ ≤ 2n+3 }. The proof for Hn is the same as above. Here we just replace L 1k , L l2 , L 3j by L 1k1 ∗ · · · ∗ L nkn , L ln+1 ∗ · · · ∗ L l2n , and L 2n+1 in (3.7) and (3.8). And 22n+1 operaj n 1 tors corresponding to (3.12) are used.

568

JOONIL KIM

3.4. Angular decomposition on the Heisenberg group ν ν For each ν = 1, 2, 3, let {bνj }∞ j=−∞ be the sequence such that b j+c /b j ≥ 2 for all ν ν j ∈ Z with some c > 0, or b−( j+c) /b− j ≥ 2 for all j ∈ Z with some c > 0. For each ν, let us define νj in R2 away from R1 × {0} for j ∈ Z: νj (ξ, η) = ψ

ξ ξ −ψ ν . ν bjη b j+1 η

(3.23)

Let ∂xk = (1/(2πi))(∂/∂xk ) for k = 1, 2. On the Euclidean space R2 , a projection operator to the angular sector {|ξ/η| ≈ bνj } on the Fourier transform side is defined by νj (∂x1 , ∂x2 ). On each coordinate plane of H1 , an analogue of this projection operator can be defined as νj (X, T ), νj (Y, T ), or νj (X, Y ) by using the Kohn-Nirenberg correspondence in (3.4). The convolution kernel is given by Lemma 3: 1j (Y, T ) f = f ∗ A1j ,

A1j (y1 , y2 , y3 ) = [F−1 1j ](y2 , y3 )δ(y1 ),

2j (X, Y ) f = f ∗ A2j ,

A2j (y1 , y2 , y3 ) = [F−1 2j ](y1 , y2 )δ(y3 + 2y1 y2 ),

3j (X, T ) f = f ∗ A3j ,

A3j (y1 , y2 , y3 ) = [F−1 3j ](y1 , y3 )δ(y2 );

(3.24)

here F−1 denotes the inverse Fourier transform with respect to the first and second variables. Since f ∗ g(x1 , x2 , x3 ) = g˜ ∗ f˜(x1 , x2 , −x3 ) (where f˜(x1 , x2 , x3 ) = f (x1 , x2 , −x3 )), we write A2j (y1 , y2 , y3 ) = [F−1 2j ](y1 , y2 )δ(y3 − 2y1 y2 ) and prove the following lemma. LEMMA 4 For 1 < p < ∞ and for ν = 1, 2, 3, ∞

X 1/2

|Aνj ∗ f |2

j=−∞

L p (H1 )

. k f k L p (H1 ) .

Proof First, we show the proof for ν = 1. Let f x1 (x2 , x3 ) = f (x1 , x2 , x3 ) and Ra (v1 , v2 ) = (v1 , av1 + v2 ). Then we have A1j ∗ f (x1 , x2 , x3 ) = f x1 (R2x1 ) ∗R2 [F−1 1j ](x2 , x3 − 2x1 x2 ),

(3.25)

where ∗R2 is convolution on R2 . We use the change of variable x30 = x3 − 2x1 x2 in (3.25), and we apply the Littlewood-Paley inequality of the lacunary sectors in the Euclidean space R2 . For the case ν = 3, the similar proof applies. However, for ν = 2, the Euclidean Littlewood-Paley theory does not apply in the same straightforward way as for ν = 1, 3. By using a sequence of Rademacher functions {r j (t), j ∈ Z, t ∈


569

[0, 1]}, ∞

X 1/2 p

|Aνj ∗ f |2

p

L (H1 )

j=−∞

where At (y1 , y2 , y3 ) =

P∞

j=−∞ r j (t)[F

1

Z ≈ 0

p

kAt ∗ f k L p (H1 ) dt,

−1 2 ](y , y )δ(y 1 2 3 j

∞ s1 s2 X r j (t)2j (ξ1 , ξ2 ) . ∂ξ1 ∂ξ2 j=−∞

− 2y1 y2 ). By (3.23),

1 |ξ1 |s1 |ξ2 |s2

,

where s1 ≥ 0, s2 ≥ 0. From Theorem 4, we have kAt ∗ f k L p (H1 ) . k f k L p (H1 ) . This proves Lemma 4 for ν = 2. 3.5. Some vector-valued or maximal inequalities LEMMA 5 P Suppose that E j (y1 , y2 , y3 ) = k≥ j A1k (y1 , y2 , y3 ). Then for 1 < p < ∞, ∞

X 1/2

2 |E ∗ f |

j j j=−∞

L p (H1 )

∞

X 1/2

. | f j |2

j=−∞

L p (H1 )

.

Proof R Let us define c Sg(ξ, η) = ψ(ξ/η)b g (ξ, η), H (a,b) f (x1 , x2 ) = p.v. f (x1 − at, x2 − bt) dt/t. Then the vector-valued version of the Calderón-Zygmund theory in [21] implies that, for 1 < p < ∞, ∞

X 1/2

(a,b) 2 |H g |

j j=−∞

L p (R2 )

∞

X 1/2

≤ C |g j |2

j=−∞

L p (R2 )

(3.26)

with C independent of a. But by the Fourier inversion formula we can obtain Z (1,y) (0,1) Sg(x1 , x2 ) = H (H g) (x1 , x2 )∂ψ(y) dy, (3.27) where g ∈ C0∞ (R). Thus by applying the Minkowski inequality on (3.27) combined with (3.26), we have ∞

X 1/2

|Sg j |2

j=−∞

L p (R2 )

∞

X 1/2

. |g j |2

j=−∞

L p (R2 )

.

(3.28)

√ √ And we define an operator by D j g(x1 , x2 ) = g(x1 / 2− j , 2− j x2 ). Then for

570

JOONIL KIM

1 < p ≤ 2, there exists u ∈ L q (R2 ) with q = ( p/2)0 so that kuk L q (R2 ) = 1 and ∞

X 1/2

|D j g j |2

j=−∞

L p (R2 )

=

∞ Z X

1/2 |g j |2 D− j u (x) d x

j=−∞

≤

∞ X

kg j k2L p (R2 )

1/2

j=−∞

∞

X 1/2

≤ |g j |2

j=−∞

L p (R2 )

.

(3.29) The first inequality follows from the Hölder inequality for p/2 and q, and the second follows from the reverse Minkowski inequality for 0 < p/2 < 1. Let us recall the P notation f j,x1 (x2 , x3 ) = f j (x1 , x2 , x3 ); then from the fact that k≥ j 1k (ξ, η) = ψ(ξ/(2− j η)), E j f j (x1 , x2 , x3 ) = D− j S D j f j,x1 R2x1 (x2 , x3 − 2x1 x2 ) . Thus we have from (3.28) and (3.29), ∞

X 1/2

|E j f j |2

j=−∞

L p (H1 )

∞

X 1/2

≤ C | f j |2

j=−∞

L p (H1 )

.

This proves Lemma 4 for 1 < p ≤ 2. And the case for 2 < q < ∞ also holds because L q (l 2 ) is the dual space of L p (l 2 ), where 1 < p < 2. LEMMA 6 For ν = 1, 2, 3,

X

L νm ∗ f

sup j

m≥ j

X

A1m ∗ f

sup j

m≥ j

L p (H1 )

L p (H1 )

. k f k L p (H1 ) , . k f k L p (H1 ) .

Proof P For ν = 1, m≥ j χm1 (ξ1 ) = ψ(2 j ξ1 ), and ψ ∈ C0∞ . From this and (3.2), we observe P that sup j | m≥ j L 1m | ∗ f (x) is bounded by the strong maximal function in the x1 variable. The same proof applies to ν = 2, 3. And the second inequality can be proved in view of (3.25) and the corresponding result on the Euclidean space R2 . 7 Suppose that σ j is a positive measure in H1 . We assume that k supk σk ∗ f k L p0 (H1 ) . k f k L p0 (H1 ) and that kσk ∗ f k L 1 (H1 ) . k f k L 1 (H1 ) for some p0 ≤ 2. Then we have for LEMMA


571

1/ p ≤ (1/2)(1 + 1/ p0 ), ∞

X 1/2

|σ j ∗ f j |2

j=−∞

L p (H1 )

∞

X 1/2

. | f j |2

j=−∞

L p (H1 )

.

Proof Let f = { f j } be a vector-valued function, and let R be the operator defined by Rf = {σ j ∗ f j }. The hypotheses above imply that kRfk L p0 (l ∞ (H1 )) . kfk L p0 (l ∞ (H1 )) and kRfk L 1 (l 1 (H1 )) . kfk L 1 (l 1 (H1 )) . The interpolation of the two vector-valued spaces L p0 (l ∞ (H1 )) and L 1 (l 1 (H1 )) completes the proof of Lemma 7. 4. Main estimate for M0α For the proof of the main theorem, we need to know the supports of χ νj in (3.1) and νj in (3.23). Recall that χ νj (ξ ) = ψ(a νj ξ ) − ψ(a νj+1 ξ ). We take a 1j = 2 j , a 2j = P ν γ 0 (2− j )2− j , and a 3j = γ 0 (2− j )2− j 2− j in this section. Then ∞ j=−∞ χ j (ξ ) ≡ ±1 for ξ 6= 0 and supp χ 1j = ξ : 2− j−2 ≤ |ξ | ≤ 2− j , supp χ 2j = ξ : (2γ 0 (2− j )2− j )−1 ≤ |ξ | ≤ (γ 0 (2−( j+1) )2−( j+1) )−1 , supp χ 3j = ξ : (2γ 0 (2− j )2− j 2− j )−1 ≤ |ξ | ≤ (γ 0 (2−( j+1) )2−( j+1) 2−( j+1) )−1 . (4.1) And we choose b1j = 2− j , b2j = γ 0 (2− j ), b3j = 2− j γ 0 (2− j ) in (3.23). In addition, √ √ ψ in (3.23) is replaced by σ for ν = 2, 3, where σ ∈ C0∞ [− D 0 , D 0 ] such that √ √ P j=∞ σ ≡ 1 on [−1/ D 0 , 1/ D 0 ]. (D 0 is defined in Sec. 1.) Then j=−∞ νj (ξ, η) ≡ 1 except on the two axes, and each νj is homogeneous of degree zero for ν = 1, 2, 3. We have n o |ξ | supp 1j = (ξ, η) : 2− j−2 ≤ ≤ 2− j , |η| n 0 −( j+1) γ (2 ) |ξ | √ 0 0 − j o ≤ D γ (2 ) , supp 2j = (ξ, η) : ≤ √ |η| D0 n γ 0 (2−( j+1) )2−( j+1) |ξ | √ 0 0 − j − j o ≤ D γ (2 )2 . (4.2) supp 3j = (ξ, η) : ≤ √ |η| D0 We first consider the case α = 0 in the main theorem. In Section 5.3 the case α 6= 0 is handled by a simple modification of the proof for α = 0. Choose a nonnegative R function ϕ ∈ C0∞ [1/2, 1] such that ϕ dt = 1. Set ϕ j (t) = (1/2− j )ϕ(t/2− j ), and define a measure µ j (y1 , y2 , y3 ) = ϕ j (y1 )δγ (y1 ) (y2 )δ0 (y3 ).

572

JOONIL KIM

We know that |M00 f (x)| . sup j |µ j ∗ f (x)|, so we show the L p -boundedness of M which is defined by M f = sup j |µ j ∗ f |. The group Fourier transform of µ j is the class of one-dimensional oscillatory integral operators given by, for each λ, Z µ cj (λ)g(x) = eiλγ (x−y)(x+y) ϕ j (x − y)g(y) dy. (4.3) Here we do not consider π and some constants such as 1/2, 1/4 in (3.1) without loss of generality. The derivative of the phase function is given by −λ 2yγ 0 (x − y) + h(x − y) . Observe that there is a positive constant β such that 2−β |γ 0 (t)t| ≤ |h(t)| ≤ |γ 0 (t)t|. (β can be chosen depending only on the doubling constant D of γ 0 ; see [17, Lem. 2(ii)].) In [12] the kernel of µ cj (λ) in (4.3) is decomposed into three different regions: {y : |y| ≈ 2− j }, {y : |y| 2− j }, and {y : |y| 2− j }. The first region (where the singularity occurs) is handled by the measure estimate for the union of such sets with j. On the second or third region, the oscillatory term 2yγ 0 (x − y), or h(x − y), gives the good decay for each region. We also apply this decomposition for the L p -estimation. Fix j ∈ Z, and define three measures, E νj for ν = 1, 2, 3, on H1 : E νj (y1 , y2 , y3 ) =

X m∈B νj

A1m (y1 , y2 , y3 ),

¯ B 2 = {m ∈ Z : m < j − 5 D}, ¯ and where B 1j = {m ∈ Z : j − 5 D¯ ≤ m ≤ j + 5 D}, j ¯ Here D¯ = β + D˜ −1 . By (2.6), B 3 = {m ∈ Z : m > j + 5 D}. j

cν (λ)g(x) = E j

λ νm λx, g(x), 4 ν

X

m∈B j

where ν = 1, 2, 3. Since νm is homogeneous of degree zero, we can write 2νj (x) = P P 1 \ν cj (λ)g(x) = ν=3 m∈B ν m (λx, λ/4). Now we split µ ν=1 µ j ∗ E j (λ)g(x), where j

ν µ\ j ∗ E j (λ)g(x) =

Z

eiλγ (x−y)(x+y) ϕ j (x − y)2νj (y)g(y) dy.

(4.4)

By (4.2) we see that supp 21j = {|y| ≈ 2− j }, supp 22j = {|y| 2− j }, and supp 23j = {|y| 2− j }. We now define three maximal operators by using E νj (ν = 1, 2, 3), M ν f = sup |µ j ∗ E νj ∗ f |. j


573

Since M f ≤ M 1 f + M 2 f + M 3 f , we need to prove the L p -boundedness of M ν for each ν = 1, 2, 3. The L p -boundedness of M 1 is immediately obtained from Theorem 3 and Lemma 7 combined with Lemma 1. For 4/3 < p ≤ ∞, ∞

X 1/2

kM 1 f k L p (H1 ) . |µ j ∗ E 1j ∗ f |2

j=−∞ ∞

X

.

|E 1j ∗ f |2

1/2

j=−∞

L p (H1 )

L p (H1 )

. k f k L p (H1 ) .

The bootstrap argument extends the range to 1 < p ≤ ∞. By using L νj ’s in (3.1), let us decompose the measure µ j ∗ E νj for each ν = 2, 3 so that D¯ l=20 X M0ν f = sup µ j ∗ E νj ∗ L νj+l ∗ f , j

ν M∞

l=−∞

l=∞ X f = sup µ j ∗ E νj ∗ L νj+l ∗ f . j

¯ l=20 D+1

ν f , we next show the L p -boundedness of M ν and M ν Since M ν f ≤ M0ν f + M∞ ∞ 0 for each ν = 2, 3.

4.1. Estimate of M0ν We define a measure b j (y1 , y2 , y3 ) = ϕ j (y1 )δ0 (y2 )δ0 (y3 ) in H1 . Let us define P∞ ν ν 2 1/2 (ν = 2, 3); then Glν f (x) = j=−∞ |(µ j − b j ) ∗ E j ∗ L j+l ∗ f | kM0ν

f k L p (H1 ) ≤

l=20 XD¯

ν kGlν f k L p (H1 ) + sup |b j ∗ E νj ∗ L j ∗ f | L p (H1 ) , j

l=−∞ ν

where L j =

Pl=20 D¯

l=−∞

L νj+l . From the fact that E νj ∗ L νj+l ∗ f = L νj+l ∗ E νj ∗ f for ν

ν = 2, 3, k sup j |b j ∗ E νj ∗ L j ∗ f |k L p (H1 ) is bounded by

sup b j ∗ sup |L ν ∗ E ν ∗ f | p 1 . sup |L ν | ∗ sup |E ν ∗ f | p 1 j j j j L (H ) L (H ) j

j

j

j

. sup |E νj ∗ f | L p (H1 ) . k f k L p (H1 )

(4.5)

j

for ν = 2, 3. The first inequality follows from the L p -boundedness of the maximal operator defined by sup j |b j ∗ f |, the second and third inequalities from Lemma 6.

574

JOONIL KIM

For ν = 2, 3, ∞

X 1/2

(µ j − b j ) ∗ E ν ∗ L ν ∗ f 2 kGlν f k L p (H1 ) =

j j+l j=−∞ ∞

X

.

j=−∞ ∞

X

.

|E νj ∗ L νj+l ∗ f |2 |L νj+l ∗ f |2

1/2

1/2

j=−∞

L p (H1 )

L p (H1 )

L p (H1 )

. k f k L p (H1 ) .

(4.6)

The first inequality follows from Theorem 3 with Lemma 7, the second from Lemma 5, and the third from Lemma 1. Now we show that there is a constant c such that kGlν f k L 2 (H1 ) . 2cl k f k L 2 (H1 ) .

(4.7)

By (2.6) and (2.7), a group Fourier transform of (µ j − b j ) ∗ E νj ∗ L νj+l is given by R ν,λ the family of operators G ν,λ S j,l (x, y)g(y) dy for λ ∈ R1 − {0}, where j,l g(x) = iλγ (x−y)(x+y) S 2,λ − 1)ϕ j (x − y)22j (y)χ 2j+l (λy), j,l (x, y) = (e

(4.8)

iλγ (x−y)(x+y) S 3,λ − 1)ϕ j (x − y)23j (y)χ 3j+l (λ). j,l (x, y) = (e

(4.9)

Equation (2.10) implies that (4.7) holds if and only if there exists a positive constant c such that

j=∞ 1/2

X

2 |G ν,λ

2 1 . 2cl kgk L 2 (R1 ) j,l g| j=−∞

L (R )

uniformly in λ. Therefore, to show (4.7), by the Cotlar-Stein lemma we have only to ¯ prove that, for ν = 2, 3 and l ≤ 20 D, cl kG ν,λ j,l kop . 2 ,

(4.10)

ν,λ ν,λ ν,λ∗ −c| j1 − j2 | kG ν,λ∗ . j1 ,l G j2 ,l kop + kG j1 ,l G j2 ,l kop . 2

(4.11)

Proof of (4.10) On the support of S 3,λ j,l (x, y), a mean value property on the exponential function shows that ˜ |eiλγ (x−y)(x+y) − 1| . γ 0 (2− j )2− j 2− j . 2( D+2)l . (4.12) And on the support of S 2,λ j,l (x, y), |eiλγ (x−y)(x+y) − 1| . λγ 0 (2− j )2− j |y| . 2( D+1)l . ˜

(4.13)


575

With (4.12) and (4.13), we apply the Minkowski inequality to the L 2 (R1 )-norm of G ν,λ j,l g. Then cν l kG ν,λ j,l gk L 2 (R1 ) ≤ 2 kgk L 2 (R1 ) ,

where c2 = D˜ + 1 and c3 = D˜ + 2. Therefore (4.10) is proved. Proof of (4.11) We may assume that | j1 − j2 | > 10 without loss of generality. The kernel of ∗ ν,λ [G ν,λ j1 ,l ] G j2 ,l is given by Z ν,λ S ν,λ j1 ,l (x, y)S j2 ,l (x, z) d x. (i) For the case ν = 2, from (4.8), the support of the above integral is contained in (x, y, z) : |x − y| ≈ 2− j1 , |x − z| ≈ 2− j2 , |y| 2− j1 , |z| 2− j2 . By triangular inequality, |y| ≈ |z|; this implies that χ 2j1 +l (λy)χ 2j2 +l (λz) = 0. This, in 2,λ 2,λ ∗ ∗ 2,λ turn, implies that k[G 2,λ j1 ,l ] G j2 ,l kop = 0. By the same reasoning, kG j1 ,l [G j2 ,l ] kop = 0. ∗ 3,λ (ii) For the case ν = 3, χ 3j1 +l (λ)χ 3j2 +l (λ) = 0. From (4.9), k[G 3,λ j1 ,l ] G j2 ,l kop =

3,λ ∗ kG 2,λ j1 ,l [G j3 ,l ] kop = 0. So (4.11) is proved.

Hence we obtain (4.7). By interpolation of (4.6) and (4.7), we also obtain kGlν gk L p (H1 ) . 2cl kgk L p (H1 )

(ν = 2, 3)

for some positive constant c. Hence this and (4.5) yield kM0ν

f k L p (H1 ) .

20 D¯ X

kGlν f k L p (H1 ) + k f k L p (H1 ) . k f k L p (H1 ) .

l=−∞ ν 4.2. Estimate of M∞ We now assume that γ ∈ C 2 for technical reasons. However, this assumption can be removed by using the approximation of identity of γ (see [12, Rem. 5]). Now we set, for ν = 2, 3, ν j (ξ1 , ξ2 )

=

νj (ξ1 , ξ2 ) =

m=X j+10 D¯

νm (ξ1 , ξ2 ),

m= j−10 D¯ ν 1 − j (ξ1 , ξ2 ),

(4.14) (4.15)

576

JOONIL KIM

and we define measures in H1 as follows: 2

2

3

3

A j (y1 , y2 , y3 ) = [F−1 j ](y1 , y2 )δ(y3 − 2y1 y2 ), A j (y1 , y2 , y3 ) = [F−1 j ](y1 , y3 )δ(y2 ), ν

Aνj (y1 , y2 , y3 ) = δ(y1 , y2 , y3 ) − A j (y1 , y2 , y3 ). Let L νj =

Pl=∞

¯ l=20 D+1

(4.16)

ν f (x) ≤ M ν,1 f (x) + M ν,2 f (x), where L νj+l ; then M∞ ∞ ∞ ν

ν,1 M∞ f = sup |µ j ∗ E νj ∗ L νj ∗ A j ∗ f |, j

ν,2 M∞

f = sup |µ j ∗ E νj ∗ L νj ∗ Aνj ∗ f |.

(4.17)

j

ν,1 For ν = 2, 3, kM∞ k L p (H1 ) . k f k L p (H1 ) immediately follows from Theorem 3 and Lemmas 1, 4, 5, and 7. Moreover, ν,2 kM∞

f k L p (H1 ) ≤

l=∞ X ¯ l=20 D+1

∞

X 1/2

|µ j ∗ E νj ∗ L νj+l ∗ Aνj ∗ f |2

j=−∞

L p (H1 )

.

ν,2 By interpolation, in proving the L p -boundedness of M∞ , it suffices to show that there is a positive constant c such that ∞

X 1/2

|µ j ∗ E νj ∗ L νj+l ∗ Aνj ∗ f |2

j=−∞ ∞

X

j=−∞

|µ j ∗ E νj ∗ L νj+l ∗ Aνj ∗ f |2

L p (H1 )

1/2

L 2 (H1 )

. k f k L p (H1 ) ,

(4.18)

. 2−cl k f k L 2 (H1 ) .

(4.19)

Equation (4.18) holds as a consequence of Theorem 3 and Lemmas 1, 4, 5, and 7. From (2.10), together with the Cotlar-Stein lemma, (4.19) is equivalent to ν,λ kT j,l kop ≤ 2−cl ,

ν,λ ∗ ν,λ

[T ] T + T ν,λ [T ν,λ ]∗ ≤ C2−c| j1 − j2 | , j1 ,l j2 ,l op j1 ,l j2 ,l op

(4.20) (4.21)

ν,λ ν (λ) · c cν (λ) · Ld uniformly in λ. Here T j,l =µ cj (λ) · E Aνj (λ) · F−1 , and the kernel j j+l

ν,λ of T j,l for each ν = 2, 3 is given by Z 2,λ K j,l (x, ξ ) = ei(λγ (x−y)(x+y)+ξ y) ϕ j (x − y)22j (y)χ 2j+l (λy)2j (ξ, λy) dy, (4.22) Z K 3,λ (x, ξ ) = ei(λγ (x−y)(x+y)+ξ y) ϕ j (x − y)23j (y)χ 3j+l (λ)3j (ξ, λ) dy. (4.23) j,l

Note that νj plays the role of excluding the region {|ξ | ≈ |∂(λγ (x − y)(x + y))/∂ y|} from the integral in (4.22) and (4.23).


577

Proof of (4.21) For ν = 3, (4.21) follows by the same reasoning as in the proof of (4.11). For ν = 2,

2,λ ∗ 2,λ

[T ] T . 2−c| j1 − j2 | j1 ,l j2 ,l op follows in the same way as in (4.11) combined with the uniform boundedness of kc A2j (λ)kop . Next, we show that kT j2,λ [T j2,λ ]∗ kop . 2−c| j1 − j2 | . As a consequence of 1 ,l 2 ,l Theorem 3, kc µ j (λ)kop is uniformly bounded in λ. Therefore it suffices to show that d [ 2 2 2 k E (λ) · L (λ) · d A2 (λ) · [d A2 (λ)]∗ · [ L[ (λ)]∗ · [d E 2 (λ)]∗ k . 2−c| j1 − j2 | . The j1

j1

j1 +l

j2

j2 +l

j2

op

integral kernel of this operator is written as L 2j1 , j2 (x, y) Z = eiξ(x−y) 22j1 (x)χ 2j1 +l (λx)22j2 (y)χ 2j2 +l (λy)2j1 (ξ, λx)2j2 (ξ, λy) dξ. Let 2− j2 2− j1 . We use (4.15) and then apply integration by parts for the above integral. Then we obtain j1 , j2 (x, y) .

2 L

22j1 (x)χ 2j1 +l (λx)22j2 (y)χ 2j2 +l (λy) (x − y)2 min{λγ 0 (2− j2 −20 )|y|, λγ 0 (2− j1 −20 )|x|}

.

Hence we have for each fixed y, Z 2 −c| j1 − j2 | L , j1 , j2 (x, y) d x . 2 and the analogous estimate for the y-integral also holds. Therefore (4.21) is proved for ν = 2, 3. Proof of (4.20) S 3 When ν = 3, from (4.2) and (4.15), 3j (ξ, λ) is supported on the set P03 ∪ ∞ m=5 Pm with P03 = |ξ | ≤ 2−5 λγ 0 (2− j−1 )2−( j+1) , Pm3 = 2m λγ 0 (2− j )2− j ≤ |ξ | ≤ 2m+1 λγ 0 (2− j )2− j . We split 3,λ T j,l g(x) =

Z

K 3,λ j,l (x, ξ )ρ P 3 (ξ ) f (ξ ) dξ +

= T03 g(x) +

0

∞ X m=5

Tm3 g(x),

∞ Z X m=5

K 3,λ j,l (x, ξ )ρ Pm3 (ξ ) f (ξ ) dξ

578

JOONIL KIM

where ρ A is a characteristic function on the set A. On the support of the integral in (4.23), |∂(λγ (x − y)(x + y) + ξ y)/∂ y| & (|ξ | + λγ 0 (2− j−1 )2− j−1 ). So, integration by parts for (4.23) yields P∞ 3 χ j+l (λ)ρ[−2− j+1 ,2− j+1 ] (x) ρ P03 (ξ ) + m=5 ρ P13 m (ξ ) 3,λ K (x, ξ ) . . j,l 2− j (|ξ | + λγ 0 (2− j−1 )2− j−1 ) So, for any fixed ξ , Z

3,λ K (x, ξ )ρ 3 (ξ ) d x . j,l P 0

Z

3,λ K (x, ξ )ρ j,l

Pm3 (ξ )

dx .

χ 3j+l (λ) λγ 0 (2− j−1 )2− j χ 3j+l (λ) 2m λγ 0 (2− j )2− j

. 2−cl 2− j , . 2−m 2−cl 2− j .

(4.24)

For any fixed x, Z

3,λ K (x, ξ )ρ 3 (ξ ) dξ . 2 j , j,l P

Z

3,λ K (x, ξ )ρ

0

j,l

Pm3 (ξ )

dξ . 2 j .

(4.25)

From (4.24) and (4.25), kT03 kop . 2−cl/2 and kTm3 kop . 2−cl/2 2−m/2 . Hence 3,λ kT j,l kop

≤

kT03 kop

+

m=∞ X

kTm3 kop . 2−cl/2 .

m=5

Now there remains only the proof of (4.20) for the case ν = 2. 2,λ 5. Proof of the L 2 -estimate of T j,l 2,λ In this section we show that kT j,l kop ≤ 2−cl with fixed j, l, λ. The oscillatory term in (4.22) gives a decay such as |∂(λγ (x −y)(x +y)+ξ y)/∂ y| & (|ξ |+λγ 0 (2− j−1 )|x|). However, the support size of the integrand in (4.22) is not small enough to get a desired bound of the L 1 -norm of K 2,λ j,l corresponding to (4.24). So, we need more de− j−1 , 2− j ] = supp ϕ with recompositions of the kernel K 2,λ j j,l . We first decompose [2 0 − j − j−1 − spect to the size of γ . Let 2 0 = 2 , and choose 2 jm for each m ∈ Z+ such that 2− jm = sup{t ∈ [2− j−1 , 2− j ] : γ 0 (t) ≤ 2m γ 0 (2− j0 )} γ 0 (2− jm ) = 2m γ 0 (2− j0 ) if γ 0 is continuous . There are finite numbers of such 2− jm ’s on the interval [2− j−1 , 2− j ]. Let Im = [2− jm−1 , 2− jm ] for 1 ≤ m ≤ n (where 2− jn = 2− j is the last point). We choose n smooth cutoff functions ζm (t) for 1 ≤ m ≤ n, satisfying (1) supp ζm ∈ I˜m = Im−1 ∪ Im ∪ Im+1 , where I0 = [2− j−2 , 2− j0 ] and In+1 = [2− j , 2− j+1 ];


(2) (3)

579

dζm (t)/dt . 1/|Im |, where |Im | is the Lebesgue measure of the interval Im ; Pn − j−1 , 2− j ]. m=1 ζm (t) ≡ 1 on [2 Set ϕm = ζm ϕ j . In order to simplify the notation in (4.22), we let 8(y) = 22j (y)χ 2j+l (λy).

(5.1)

Now we define an operator for each m: ZZ Tm g(x) = ei(λγ (x−y)(x+y)+ξ y) ϕm (x − y)8(y)2j (ξ, λy)g(ξ ) dy dξ. Since

Pn

2,λ T j,l

m=1 ϕm

=

n X m=1

2,λ = ϕ j , we can decompose T j,l so that

X

Tm =

X

Tm +

λγ 0 (2− jm )2− j 2− j 1/3 such that λγ 0 (2− jm )2− j 2− j = 2−am l .

(5.3)

For each m, we decompose supp 8 in (5.2) into small pieces of interval with the same length. Assume that y > 0 in supp 8 without loss of generality. First, take y0 = 1/(2λγ 0 (2− j−l )2− j−l ) (which is the smallest positive number on supp 8), and then choose yk ∈ supp 8 so that yk = y0 + k2− j 2m l with k ∈ N (m is chosen as a number depending only on am later). Let us define a family of cutoff functions P ωk ∈ C0∞ with k ωk ≡ 1 on supp 8 satisfying supp ωk ⊂ [yk − 2− j 2m l , yk + 2− j 2m l ],

∂ω (y) 1 k . − j m l . ∂y 2 2

(5.4)

Let 8k (y) = ωk (y)8(y).

(5.5)

580

JOONIL KIM

P We now decompose Tm = k Tm,k , where ZZ Tm,k g(x) = ei(λγ (x−y)(x+y)+ξ y) ϕ jm (x − y)8k (y)2j (ξ, λy)g(ξ ) dy dξ. P P Let us define two operators Rm = k Rm,k and Em = k Em,k by ZZ Rm,k g(x) = ei(λγ (x−y)(x+y)+ξ y) ϕ jm (x − y)8k (y)2j (ξ, λyk )g(ξ ) dy dξ, ZZ Em,k g(x) = ei(λγ (x−y)2yk +ξ y) ϕ jm (x − y)8k (y)2j (ξ, λyk )g(ξ ) dy dξ. So, Tm splits as Tm = [Tm − Rm ] + [Rm − Em ] + Em . Since |x − y| ≈ 2− j and |y − yk | < 2− j 2m l in the above integral, X kTm − Rm k2op . kTm,k − Rm,k k2op , k

kRm − Em k2op

.

X

.

X

kRm,k − Em,k k2op ,

k

kEm k2op

kEm,k k2op .

(5.6)

k

Proof of the estimate of kTm − Rm kop We write the kernel of [Tm,k − Rm,k ] as Sm,k (x, ξ ): Z Sm,k (x, ξ ) = ei(λγ (x−y)(x+y)+ξ y) ϕ jm (x − y)8k (y)1k (ξ, y) dy,

(5.7)

where 1k (ξ, y) = [2j (ξ, λy) −2j (ξ, λyk )]. For the L 1 -norm estimation of the above integral, we need to know supp 1k (·, y). Note that when y ∈ supp ωk in (5.5), y k (5.8) (1 − 2−100 ) < < (1 + 2−100 ) y for l > 100. Inequality (5.8) holds because |y| ≥ (λγ 0 (2− j−l )2− j−l )−1 > ˜ 2(1+ D)l 2− j 2am l > 2100 2− j 2m l , where m is chosen less than am . From (4.2), (4.15), and (5.8), we observe that supp 1k (·, y) is contained for each y ∈ supp 8k in the set P = P1 ∪ P2 , where Ps is defined by n o √ λγ 0 (2− j+Ds )yk Ps = ξ : (1 − 2−90 ) ≤ |ξ | ≤ (1 + 2−90 ) D 0 λγ 0 (2− j+Ds )yk √ D0 ¯ D2 = −10 D¯ − 1. for s = 1, 2 and D1 = 10 D,


581

By using the mean value property for 2j and the definitions of the integrand in (5.7), we get the estimates − j m l 1k (ξ, y) . 2 2 , yk ∂ 1 1k (ξ, y) . , ∂y yk ∂ 1 8k (y)ϕ jm (x − y) . − j . ∂y 2 |I jm |

(5.9) (5.10) (5.11)

Integration by parts yields Z Sm,k (x, ξ ) . ei(λγ (x−y)(x+y)+ξ y) ∂ ϕ jm (x − y)8k (y)1k (ξ, y) dy . ∂ y ξ + ∂ y (λγ (x − y)(x + y)) By using the monotonicity of γ 0 combined with (5.9) – (5.11), we obtain λ S (x, ξ ) . m,k

2m l ρ P (ξ ) . (λγ 0 (2− jm )yk + |ξ |)yk

(5.12)

Since |x − yk | ≤ 2− j 2m l , we estimate from (5.12) that Z − j 2m l λ S (x, ξ ) d x . 2 2 , m,k λγ 0 (2− jm )yk2 Z m l λ S (x, ξ ) dξ . 2 , m,k yk Ps for each s = 1, 2. By the above inequalities, we get kTm,k − Rm,k k2op . Note that

X 1 k

yk3

=

X k

23m l 2− j λγ 0 (2− jm )yk3

.

1 1 . − j (y0 + k2− j 2m l )3 2 2m l y02

(5.13)

(5.14)

and that y0 > 2(1+ D)l 2am l 2− j in the proof of (5.8). Thus, from (5.13) and (5.14), we obtain ˜

kTm − Rm k2op .

X

23m l 2− j

k

λγ 0 (2− jm )yk3

. 2(2m −am −2− D)l . 2−(am /3)l . ˜

The last inequality above holds when we choose m = am /3.

(5.15)

582

JOONIL KIM

Proof of the estimate of kRm − Em kop R Let us define Mk g(y) = 2j (ξ, λyk )g(ξ )eiξ y dξ. Then [Rm,k − Em,k ]g(x) is Z (eiλγ (x−y)(x+y) − eiλγ (x−y)2yk )ϕ jm (x − y)8k (y)Mk g(y) dy. (5.16) By the mean value property for the exponential function, we obtain iλγ (x−y)(x+y) (e − eiλγ (x−y)2yk ) ≤ λγ (x − y)(x − yk + y − yk ) . λγ 0 (2− jm )2−2 j 2m l = 2(−am +m )l . Now we use the Minkowski inequality to estimate the L 2 -norm of (5.16), and we get

[Rm,k − Em,k ]g 2 1 . 2(−am +m )l k8k Mk gk 2 1 . L (R ) L (R ) Hence from (5.6),

[Rm − Em ]g 2 2

L (R1 )

.

X

22(−am +m )l k8k Mk gk2L 2 (R1 )

k

.2

2(−am +m )l

X

8k sup |Mk g| 2 2 k

k

L (R1 )

. 22(−am +m )l k sup Mk gk2L 2 (R1 ) k

. 22(−am +m )l kgk2L 2 (R1 ) = 2−(4/3)am l kgk2L 2 (R1 ) .

(5.17)

The third inequality above follows from the fact that the support of 8u and 8v is disjoint for |u − v| ≥ 2. Finally, the fourth inequality follows from the fact that supk |Mk g| is bounded by the one-dimensional Hardy-Littlewood maximal function. The equality holds because m is taken as (1/3)am . Proof of the estimate of kEm kop Let us introduce two smooth cutoff functions 31 and 32 so that n λγ 0 (2− jm )yk o supp 31 ⊂ ξ : |ξ | < ∪ ξ : |ξ | > 23 λγ 0 (2− jm )yk , 3 2 n λγ 0 (2− jm )y o k 5 0 − jm supp 32 ⊂ ξ : < |ξ | < 2 λγ (2 )y , k 25 and 31 + 32 ≡ 1. For s = 1, 2, define Z s E g(y) = eiξ y 3s (ξ ) F(8k ) ∗ F(Mk g) (ξ ) dξ,

(5.18) (5.19)

where ∗ refers to the convolution in R1 and where F is the Fourier transform in R1 . 1 + E 2 , where Now Em,k splits as Em,k m,k Z s Em,k g(x) = eiλγ (x−y)2yk ϕ jm (x − y)E s g(y) dy.


583

1 and E 2 have the same form as the convolution operators in the We observe that Em,k m,k Euclidean space which appeared in [5, p. 264]. Let Z m(ξ1 , ξ2 ) = ei(ξ1 t+ξ2 γ (t)) ϕ jm (t) dt.

Then the Plancherel theorem shows that Z 2 1 kEm,k gk2L 2 (R1 ) = m(ξ, 2λyk )31 (ξ )[F(8k ) ∗ F(Mk g)](ξ ) dξ . = .

1 (λγ 0 (2− jm )2− j yk )2

[F(8k ) ∗ F(Mk g)] 2 2

L (R1 )

1 k8k Mk gk2L 2 (R1 ) m 0 − (2 λγ (2 j−1 )2− j yk )2 ˜ 2−2m 2−2(1+ D)l k8k Mk gk2L 2 (R1 ) .

So, X

˜

1 kEm,k gk2L 2 (R1 ) . 2−2m 2−2(1+ D)l

k

X

8k sup |Mk g| 2 2 k

k

L (R1 )

(5.20)

˜

. 2−2m 2−2(1+ D)l kgk2L 2 (R1 ) . R 2 g(x) = U We rewrite Em,k m,k (x, η)g(η) dη so that Um,k (x, η) is given by ZZ ei(λγ (x−y)2yk +ξ y) ϕ jm (x − y)32 (ξ )[F8k ](ξ − η)2j (η, λyk ) dy dξ.

(5.21)

We use two observations. On the support of the integral in (5.21), |ξ − η| ≥ 2−10 λγ 0 (2− jm )yk , [F8k ](ξ − η) .

2− j 2m l . − η|)2

(2− j 2m l |ξ

(5.22) (5.23)

By (4.2) and (4.15), the support of 2j (η, λyk ) is contained in the set

η : |η| > 29 λγ 0 (2− j )yk ∪ η : |η| < 2−9 λγ 0 (2− j−1 )yk .

By this and (5.19), we observe that (5.22) holds. We can obtain (5.23) from integration by parts in the integral defining the Fourier transform of 8k . From (5.21) – (5.23), we have Z 1 Um,k (x, η) dη . , (2− j 2m l )2 λγ 0 (2− jm )yk Z 1 Um,k (x, η) d x . . − j l 2 2 m (λγ 0 (2− jm )yk )2

584

JOONIL KIM

So, we have 2 kEm,k k2op .

From this and the inequality X

2 kEm,k k2op .

k

P

k

1 (2− j 2m l )3 (λγ 0 (2− jm )yk )3

.

1/yk3 . 1/(2− j 2m l y02 ), we get 1

≤ 2(am −4m −2−2 D)l ≤ 2−(1/3)am l , ˜

24m l 2− j y02 (λγ 0 (2− jm )2− j )3

where m is taken as (1/3)am . So, we obtain from the above inequality and (5.15), (5.17), and (5.20), kTm kop = kTm − Rm kop + kRm − Em kop ¯ 2−(1+ D)l −(1/6)am l + kEm kop . 2 + . 2m Therefore we have X

kA kop .

λγ 0 (2− jm )2− j 2− j τm }, and 521 + 522 = 1. We decompose L m (ξ, η) = L m,1 (ξ, η) + L m,2 (ξ, η), where L m,1 (ξ, η) = 521 (ξ −η)L m (ξ, η) and where L m,2 (ξ, η) = 522 (ξ −η)L m (ξ, η). To apply integration by parts, let us define a differential operator Du for each u = x, y, z by 1 ∂ Du = . 0 λ i9u (x, y, z, ξ, η) ∂u Then Du ei9

λ (x,y,z,ξ,η)

λ (x,y,z,ξ,η)

. And the transpose of Du is given by 1 g(x, y, z, ξ, η) . DuT g(x, y, z, ξ, η) = ∂u i9uλ0 (x, y, z, ξ, η) = ei9

Now integration by parts yields ZZZ λ 2 L m,1 (ξ, η) = 51 (ξ − η) ei9 (x,y,z,ξ,η) DzT D yT Vm (x, y, z, ξ, η) dy dz d x, ZZZ λ L m,2 (ξ, η) = 522 (ξ − η) ei9 (x,y,z,ξ,η) DxT DzT D yT Vm (x, y, z, ξ, η) dy dz d x.

586

JOONIL KIM

DzT D yT Vm (x, y, z, ξ, η) is given by

−1 ∂ ∂ λ0 0 9z (x, y, z, ξ, η) (9 yλ (x, y, z, ξ, η))−1 Vm (x, y, z, ξ, η) . ∂z ∂y DxT D yT DzT Vm (x, y, z, ξ, η) is given by

−1 ∂ λ0 9x (x, y, z, ξ, η) D yT DzT Vm (x, y, z, ξ, η) . ∂x We gain the multiple of 1/|Im | or 1/2− j when we take the derivative of Vm (x, y, 0 0 z, ξ, η) in the direction of y, z, or x. By the derivatives of (9 yλ )−1 , (9zλ )−1 , we essentially gain the multiple of 1/2− j . We have Z −1 −1 0 ∂ 9 yλ (x, y, z, ξ, η) dy . |ξ | + λγ 0 (2− jm )|x| , Q ∂y Z m −1 −1 0 ∂ 9zλ (x, y, z, ξ, η) dz . |η| + λγ 0 (2− jm )|x| , Q m ∂z ∂ 0 (9 yλ (x, y, z, ξ, η))−1 = 0, ∂z −1 ∂ ∂ 0 9 yλ (x, y, z, ξ, η) = 0. (5.29) ∂z ∂ y The first two inequalities in (5.29) follow from the monotonicity of γ 0 . We also have Z −1 −1 1 0 ∂ ∂ 9 yλ (x, y, z, ξ, η) dy . − j |ξ | + λγ 0 (2− jm )|x| , 2 Q ∂x ∂y Z m −1 −1 1 0 ∂ ∂ 9zλ (x, y, z, ξ, η) dz . − j |η| + λγ 0 (2− jm )|x| , ∂ x ∂z 2 Qm ∂ 0 (9xλ (x, y, z, ξ, η))−1 = 0. (5.30) ∂x As a consequence of (5.29) and (5.30), we obtain L m,1 (ξ, η) .

Z

L m,2 (ξ, η) .

Z

Q

Q

521 (ξ − η) d x, 2−2 j (|ξ | + λγ 0 (2− jm )|x|)(|η| + λγ 0 (2− jm )|x|) 522 (ξ − η) d x. 2−3 j |ξ − η|(|ξ | + λγ 0 (2− jm )|x|)(|η| + λγ 0 (2− jm )|x|) (5.31)


587

By the first inequality in (5.31), for fixed η, Z τm L m,1 (ξ, η) dξ . −2 j 0 2 (λγ (2− jm ))2 y0 Q˜ ≤ .

2−(2/3)l

1 2− j λγ 0 (2− jm )y0 ˜ 2−( D+1)l + . 2m

2−2 j λγ 0 (2− jm ) 2−(2/3)l 2−am l

+

(5.32)

We also obtain the same bound for fixed ξ : Z ˜ −(2/3)l 2−( D+1)l L m,1 (ξ, η) dη . 2 + . (5.33) 2−am l 2m R From the second inequality in (5.31), for fixed η, |L m,2 (ξ, η)| dξ is majorized by 522 (ξ − η)ρ Q (x)

ZZ

d x dξ

|ξ −η| 0. In showing (6.3), we used the cancellation h j (t) dt = 0. Inequality (6.1) follows by applying Theorem 4. And (6.2) follows from the CotlarStein lemma and (2.10), combined with the following three observations:

\

L 1 (λ) · c N j (λ) − j−n

op

. 2−|n| ,

\

\ −| j1 − j2 | 1

L 1

, − j1 −n (λ) L − j2 −n (λ) op . 2

ν ν [ [ ∗ ∗ d ∗ −| j1 − j2 |

d cν (λ) · L cν N j1 (λ) · E . − j1 (λ) · [ L − j2 (λ)] · [ E 2 (λ)] · [ N j2 (λ)] op . 2 j Now we prove the L p -boundedness of H∞ν for ν = 2, 3. We split H∞ν as ν ν ν H∞ν f = Ha,∞ f + Hb,∞ f + Hc,∞ f,


591

ν , H ν , H ν is defined by where each Ha,∞ c,∞ b,∞ ν Ha,∞ f =

ν Hb,∞ f =

ν Hc,∞ f =

∞ X

ν

ν

A j ∗ H j ∗ E νj ∗ L νj ∗ A j ∗ f,

j=−∞ ∞ X

ν

A j ∗ H j ∗ E νj ∗ L νj ∗ Aνj ∗ f,

j=−∞ ∞ X

∞ X

Aνj ∗ H j ∗ E νj ∗ L νj+l ∗ f .

¯ j=−∞ l=20 D+1

By duality, with Lemmas 1, 4, and 5, ν kHa,∞

f k L p (H1 )

∞

X 1/2 ν

. |H j ∗ E νj ∗L νj ∗ A j ∗ f |2

j=−∞

From (4.18) and (4.19), ∞

X 1/2

ν kHb,∞ f k L p (H1 ) . |H j ∗ E νj ∗L νj ∗ Aνj ∗ f |2

j=−∞

L p (H1 )

L p (H1 )

. k f k L p (H1 ) . (7.5)

. k f k L p (H1 ) . (7.6)

ν , we prove For the estimate of Hc,∞ ∞

X

Aνj ∗ H j ∗ E νj ∗ L νj+l ∗ f

j=−∞

L p (H1 )

∞

X

Aνj ∗ H j ∗ E νj ∗ L νj+l ∗ f

j=−∞

L 2 (H1 )

. k f k L p (H1 ) ,

(7.7)

. 2−cl k f k L 2 (H1 )

(7.8)

Pm=5 3 ν ν for some positive constant c. We defined Lg m=−5 L j+l+m . Then H j ∗ E j ∗ j+l = ν ν ν ν ν ν L νj+l = Lg j+l ∗ H j ∗ E j ∗ L j+l . By this we can write A j ∗ H j ∗ E j ∗ L j+l ∗ f as ν ν ν ν ν ν ∗ f . Then from the duality and Lg ∗ H j ∗ E ∗ L ∗ f − A ∗ Hj ∗ E ∗ L j+l

j

j+l

Lemmas 1, 4, and 5, we obtain

X

ν ν ν Lg

j+l ∗ H j ∗ E j ∗ L j+l ∗

j

f

j

X ν

A j ∗ H j ∗ E νj ∗ L νj+l ∗ f

j

j

j+l

L p (H1 )

X 1/2

. |H j ∗ E νj ∗ L νj+l ∗ f |2

L p (H1 )

. k f k L p (H1 ) ,

X 1/2

. |H j ∗ E νj ∗ L νj+l ∗ f |2

j

j

L p (H1 )

L p (H1 )

. k f k L p (H1 ) . By the above two inequalities, (6.7) is proved. Inequality (6.8) follows from the similar estimation of (4.19). This completes the proof of L p -boundedness of H .

592

JOONIL KIM

Acknowledgments. The author wishes to thank his advisor Andreas Seeger for introducing him to this problem and for his careful suggestions throughout this work. References [1]

R. F. V. ANDERSON, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240 – 267.

[2]

A. CARBERY, M. CHRIST, J. VANCE, S. WAINGER, and D. K. WATSON, Operators associated to flat plane curves: L p estimates via dilation methods, Duke Math.

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´ A. CARBERY and S. PEREZ , Maximal functions and Hilbert transforms along variable

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A. CARBERY, S. WAINGER, and J. WRIGHT, Hilbert transforms and maximal functions

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J. 59 (1989), 675 – 700. MR 91m:42017 557 flat curves, Math. Res. Lett. 6 (1999), 237 – 249. MR 2000f:42008 557 associated to flat curves on the Heisenberg group, J. Amer. Math. Soc. 8 (1995), 141 – 179. MR 95g:43010 557 [5]

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functions and Hilbert transforms along flat convex curves in R2 , Bull. Amer. Math. Soc. (N.S.) 14 (1986), 263 – 267. MR 87f:42044 556, 557, 579, 583 M. CHRIST, Hilbert transforms along curves, I: Nilpotent groups, Ann. of Math. (2) 122 (1985), 575 – 596. MR 87f:42039a M. CHRIST, A. NAGEL, E. M. STEIN, and S. WAINGER, Singular and maximal Radon transforms: Analysis and geometry, Ann. of Math. (2) 150 (1999), 489 – 577. MR 2000j:42023 555 R. R. COIFMAN and C. WEISS, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1976. MR 58:2019 ´ A. CORDOBA and J. L. RUBIO DE FRANCIA, Estimates for Wainger’s singular integrals along curves, Rev. Mat. Iberoamericana 2 (1986), 105 – 117. MR 88d:42027 557 A. S. DYNIN, Pseudodifferential operators on the Heisenberg group (in Russian), Dokl. Akad. Nauk SSSR 225 (1975), 1245 – 1248; English translation in Soviet Math. Dokl. 16 (1975), 1608 – 1612. MR 54:11410 562 G. B. FOLLAND, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, 1989. MR 92k:22017 558, 562 J. KIM, Hilbert transforms along curves in the Heisenberg group, Proc. London. Math. Soc. (3) 80 (2000), 611 – 642. MR 2001f:42022 556, 557, 559, 572, 575, 579 J. MARCINKIEWICZ, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78 – 91. 557 ¨ D. MULLER, F. RICCI, and E. M. STEIN, Marcinkiewicz multipliers and multiparameter structure on Heisenberg (-type) groups, I, Invent. Math. 119 (1995), 199 – 233. MR 96b:43005 562 , Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, II, Math. Z. 221 (1996), 267 – 291. MR 97c:43007 562 A. NAGEL, E. M. STEIN, and S. WAINGER, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 1060 – 1062. MR 57:6349 557


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Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA; [email protected]

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