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O. Choose v E N such that Ifnk (z)- f(z)1 < E/2, Ifnk (w) - f(w)1 < E/2, and /3 fn k < B + E for all k ;::: v. Then
If(z) - f(w)1 < E + /3 fn k P(Z, w) < E(l
+ p(z, w)) + Bp(z, w).
JOEL COHEN AND FLAVIA COLONNA
16
Letting E --) 0, we obtain If(z) - f(w)1 :::; Bp(z, w). By Theorem 3.1, f is a Bloch 0 function, and {3f :::; B. We now give another description of the Bloch semi-norm. THEOREM 3.3. Let fEB. Then {3f =
z~in II (:~ (z)(1-l z 11 2), ... , :~ (Z)(1- IZn I2 )) II·
For the proof we need the following lemma. LEMMA 3.4. Let Wk E C and let ak > 0 for k
IL~=1 Wk Ukl 2 Ij;.Ij2'1 L~=1 akl u kl 2
=
= 1, ... , n.
Then
~ IWkl2 ~ ~.
PROOF. The above equality is obvious if each Wk = O. So assume that at least one ofthe Wk is nonzero. Choose the arguments of U1, ... ,Un so that I L~=1 WkUk I = L~=1Iwkllukl. Next observe that
0 there exists polynomial P such that sup {IP(z)1 : zED} ~ 1 2000 Mathematics Subject Classification. Primary 32U20.
23
OLEG EROSHKIN
24
and sup{IP(z)1 : z E X} :S
EdegP .
en
THEOREM 1.2. Let M c be a totally real submanifold of Gevrey class GS. Let X be a compact subset of M. If dim M = m and ms < n, then for every h < ~s and every N > No = No(h) there exists a non-constant polynomial P E Z[Zl, Z2, ... , zn], deg P :S N with coefficients bounded by exp(N h ), such that sup {IP(z)1 : z E X}
(1.1 )
< exp( _N h )
.
This result is similar to the construction of an auxiliary function in transcendental number theory (cf.[18] Proposition 4.10). The Theorem 1.1 gives some information about polynomially convex hulls of manifolds of Gevrey class. Recall, that the polynomially convex hull X of X consists of all z E em such that IP(z)1 :S sup IP(OI , (EX
for all polynomials P. It is well known, that the polynomially convex hull of a pluripolar compact set is pluripolar (this follows immediately from Theorem 4.3.4 in [11]). We also introduce the notion of Kolmogorov dimension for a compact subset X c (denoted K-dim X) with the following properties.
en
(1) O:S K-dim X :S n. (2) K-dim (3)
X=
K-dim X. m
K-dim
UXj = max{K-dim Xj : j = 1, ... ,m} . j=l
en
(4) If D c is a domain, XeD and ¢ : D then K-dim ¢(X) :S K-dim X. (5) If K-dim X < n, then X is pluripolar.
->
ek
is a holomorphic map,
The main result of this paper is the following estimate of the Kolmogorov dimension of totally real sub manifolds of Gevrey class.
en
THEOREM 1.3. Let M c be a totally real submanifold of Gevrey class GS. Let X be a compact subset of M. If dim M = m then K-dim X :S ms. REMARK 1.4. This estimate is sharp. The similar estimates hold for more general class of CR-manifolds. These issues will be addressed in the forthcoming paper. In the next section we recall the definition and basic properties of functions of Gevrey class. The notion of Kolmogorov dimension of X is defined in terms of E-entropy of traces on X of bounded holomorphic functions. The definition and basic properties of E-entropy are given in Section 3. In Section 4 we discuss the notion of Kolmogorov dimension. The proof of Theorem 1.3 is sketched in Section 5. The author wishes to thank the referee for useful comments and numerous suggestions.
PLURIPOLARITY OF MANIFOLDS
25
2. Gevrey Class We need to introduce some notation first. For a multi-indices 0: = (0:1,0:2, ... ,O:m), f3 = (f31, f32, ... ,f3m) we define 10:1 = 2:j=l O:j, o:! = ITj=l O:j!, and
(~) -
(0:
_0:~)!f3! .
For an integer k we define 0: + k = (0:1 + k, 0:2 + k, ... , O:m + k). For a point x E ]Rm we define x = IT;:l x;j. If f E coo(]Rm) we denote
of=
011 f· 01 Xl ... o'mxm
Let U be an open set in]Rm and s ~ 1. A function f E COO(U) is said to belong to Gevrey class GS(U) if for every compact K c U there exists a constant C K > 0 such that (2.1)
sup 10 f(x)1 :::; C~I+l(o:W , xEK
for every multi-index 0:. The class GS forms an algebra. The Gevrey class GS is closed with respect to composition and the Implicit Function Theorem holds for GS [13], thus one may define manifolds (and submanifolds) of Gevrey class GS in the usual way.
3. The notion of c-entropy Let (E, p) be a totally bounded metric space. A family of sets {Cj } of diameter not greater than 2c is called an c-covering of E if E ~ UCj . Let Nc;(E) be the smallest cardinality of the c-covering. A set Y ~ E is called c-distinguishable if the distance between any two points in Y is greater than c: p(x, y) > c for all x, y E Y, x -=I- y. Let Mc;(E) be the largest cardinality of an c-distinguishable set. For a nonempty totally bounded set E the natural logarithm
J(c;(E) = log Nc;(E) is called the c-entropy. The notion of c-entropy was introduced by A. N. Kolmogorov in the 1950's. Kolmogorov was motivated by Vitushkin's work on Hilbert's 13th problem and Shannon's information theory. Note that Kolmogorov's original definition (see [12]) is slightly different from ours (he used the logarithm to base 2). Here we follow
[14]. We will need some basic properties of the c-entropy. LEMMA
3.1. (see [12], Theorem IV) For each totally bounded space E and each
c>O (3.1) LEMMA 3.2. Let { (Ej, pj) : j = 1,2, ... , k} be a family of totally bounded metric spaces. Let (E, p) be a Cartesian product with a sup-metric, i.e.
E = E1
X
E2
X ... X
Ek ,
P((X1' X2,···, Xk), (Yl, Y2,"" Yk)) = maxjpj(xj, Yj) .
OLEG EROSHKIN
26
Then
:Hg(E) :::;
L :Hg(Ej) j
PROOF.
Let {CjL} l = 1, ... N j be an E-covering of E j • Then the family
{C ll ! x C2l2 is an E-covering of E.
X ... X
Ckl k
:
lj = 1, ... ,Nj }
o
We also need upper bounds for E-entropy of a ball in finite-dimensional Coo space. Let IR~ be IR n with the sup-norm:
II(Xl,X2, ... ,x n )1100 =maxixJl· J
LEMMA
3.3. Let Br be a ball oj radius r in :Hg ( Br)
IR~.
Then
< n log (~ + 1) .
PROOF. The inequality is obvious for n from Lemma (3.2).
=
1. The general case then follows
0
4. Kolmogorov Dimension
en.
Let X be a compact subset of a domain D c Let A ~ be a set of traces on X of functions analytic in D and bounded by 1. So J E A ~ if and only if there exists a function F holomorphic on D such that sup IF(z)1 :::; 1 zED
and J(z) = F(z) for every z E X. By Montel's theorem A~ is a compact subset of C(X). The connections between the asymptotics of E-entropy and the pluripotential theory were predicted by Kolmogorov, who conjectured that in the one dimensional case lim :Hg(A~) = C(X, D) g~O 10g2(1/E) (21l')' where C(X, D) is the condenser capacity. This conjecture was proved simultaneously by K. I. Babenko [2] and V. D. Erokhin [8] for simply-connected domain D and connected compact X (see also [9]). For more general pairs (X, D) the conjecture was proved by Widom [19] (simplified proof can be found in [10]). In the multidimensional case Kolmogorov asked to prove the existence of the limit . :Hg(A~) 1lm---,-'-;-'-,---- 2 and P E PN sup IP(z)1 2 zEX
max ZEX1/N,N
1(1 )N
IP(z)12 2
-
N
Therefore by Corollary 4.9 K-dim X = n. To finish the proof of Theorem 4.6 we need the following well-known result. LEMMA 4.11. Let X be a compact subset ofC n . If there exists a sequence {ad, ak > 0 and a family of polynomials P k E Pk such that sup IPk(Z)1 :::; e- ak
(4.10)
and
,
zEX
. ak hm k
(4.11)
= 00,
then X is pluripolar. PROOF. Let Vk(Z) = a1k 10gPk(z) and v(z) = limsupvk(z). We will show that v 2 -2/3 on a dense set. Let ( E C n and 0 < 8 < 1. Suppose that ~((, R) :J ~(O, 1). We will show that there exists a nested sequence of closed polydisks ~m = ~(Wm' 8m ) with ~l = ~((, 8), and an increasing sequence of positive integers kl = 1 < k2 < ... < k m < ... such that Vkrn 2 -2/3 on ~m for m > 1. Given ~ = ~m = ~(Wm' 8m ) by Lemma 4.3 for any given k there exists W E ~ such that (4.12) Choose k
IPk(w)1
= kmH > k m such
2(R8~ 8)
k
that
ak
R+8
k2210g~.
Then by (4.12) Vk(W) 2 -1/2. Choose WmH = w. Because the function Vk is continuous at w, there exists a closed polydisk ~m+ 1 = ~ (Wm+1, 8m+d C ~m' such that Vk 2 -2/3 on ~mH' Therefore v 2 -2/3 on a dense set. By (4.10), vlx :::; -1 and so X is a negligible set. By [5], negligible sets are pluripolar and result follows. D REMARK 4.12. This lemma and the converse follow from Theorem 2.1 in [1].
PLURIPOLARITY OF MANIFOLDS
31
5. Manifolds of Gevrey Class In view of Theorem 4.8 and Theorem 4.6 (6), Theorems 1.2 and 1.1 are corollaries of Theorem 1.3. In this section we prove Theorem 1.3. Let M c en be an m-dimensional totally real sub manifold of Gevrey class GS. Let X c M be a compact subset. Fix p EM. There exist holomorphic coordinates (z, w) = (x + iy, w) E en, X,y E lR m , wE e n- m near p, vanishing at p, real-valued functions of class GS h1' h 2,... , hm' and complex valued functions of class GS H 1, H 2 , •.. ,Hn - m such that h~ (0) = h;(O) = ... = h~(O) = 0, H~ (0) = H~(O) = ... = H~_m(O) = 0, and locally
(5.1)
M = {(x + iy, w) : Yj = hj(x), Wk = Hk(X)} .
For smooth manifold the existence of such coordinates is well known (see, for example [3], Proposition 1.3.8). Note, that functions h j and Hk are defined by Implicit Function Theorem, and so by [13J are of class GS. We fix such coordinates and choose r sufficiently small. In view of Theorem 4.6 (5), it is sufficient to prove Theorem 1.3 for X c .6.(p, r). Put D = .6.(p, 1). To estimate w(X, D) we will cover X by small balls, approximate functions in A~ by Taylor polynomials, and then replace in these polynomials terms w A and yV by Taylor polynomials of functions H A and hV. To estimate the Taylor coefficients for powers of functions of Gevrey class we need the following lemma. LEMMA 5.1. If f E GS(K) and If I ::::; 1 on K, then there exist a constant C such that for any positive integer k and any multi-index a the following inequality holds on K
lanfkl::::;
(5.2) Recall, that a
Clnl(a+~-l)(a!)s.
+ k = (a1 + k, a2 + k, ... , am + k).
PROOF. We will argue by induction on k. Because If I < 1, there exists a constant C, such that an f::::; C1nl(aW and (5.2) holds for k
= 1.
Suppose (5.2) holds for 1,2, ... , k, then
o REMARK 5.2. The same proof holds for the product of k different functions, provided that they satisfy the Gevrey class condition (2.1) with the same constant CK· Let t > s 2: 1 and N be a large integer, which will tend to infinity later. Fix positive a < t - s. Put 0 = N 1 - t and c = N-aN. We may cover X by less than
32
OLEG EROSHKIN
(l/o)m balls ofradius o. Let Q be one of these balls and K be the set ofrestrictions on Q offunctions in A ~. We claim that any function I in K may be approximated by polynomials in Xl, X2, ... ,Xm of the degree ::; N with coefficients bounded by eN (N!)S-l with error less than 2c:, where constant e depends on X and r only. Let us show how the theorem follows from this claim. The real dimension of the space of polynomials of the degree ::; N is T = 2 (N m ). Consider in the T -dimensional space with the sup-norm 1R~ the ball B of a radius eN (N!)S-I. By Lemma 3.3,
t
1C,,(B) ::; 2 (N ;
m)
log
(eN (~!)8-I + 1) = O(Nm+Ilog N)
.
By the claim c:-covering of B generate 3c:-covering of K, therefore
1C3,,(K) = O(N m +1log N) . Then by (4.2)
(5.3)
1C,,(A~) =
0 (
(~) m Nm+Ilog N)
= O(N mt + 1 10g
N) .
Now we let N tend to infinity. By (5.3), K-dim X = W(X, D) ::; mt. The only restriction imposed on t so far was t > s. Hence K-dim X ::; ms. It remains to prove the claim. We approximate a function I in K in two steps. Consider the Taylor polynomial P of I centered at the center of the ball Q of the degree N . By Cauchy's formula
sg; II - PI < 1 _ r1 _ 0 (O)N 1_ r < c: for sufficiently large N. Suppose P(z, w)
= L
CAI-'VxAyI-'WV. Because I E A~,
ICAI-'vi ::; l. On the next step we approximate yl-' and WV by the Taylor polynomials of the degree N of hI-' and HV. Let (xo, Yo, wo) be the center of the ball Q. Let g be one of the functions hI, h2,... hm' HI, H 2,... ,Hn - m and L ::; N. Then by Taylor formula gL(xo
By Lemma 5.1 for
+ h) =
L
101~N
8 0 I(xo) h~ 0:.
Ilhll oo < 0
IRN(X, h)1 ::;
e N+1 oN
L
(0:
101=N+I
Therefore log IRN(x, h)1
+ RN(X, h)
+: -1)
.
(0:!)8-1 .
= (s - t + o(I))Nlog N and claim follows. References
[1] H. J. Alexander and B. A. Taylor, Comparison of two capacities in en, Math. Z. 186 (1984), no. 3, 407-417. [2] K. I. Babenko, On the entropy of a class of analytic functions, Nauchn. Dokl. Vyssh. Shkol. Ser. Fiz.-Mat. Nauk (1958), no. 2, 9-16. [3] M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. [4] E. Bedford, The operator (ddc)n on complex spaces, Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, Lecture Notes in Math., vol. 919, Springer, Berlin, 1982, pp. 294-323.
PLURIPOLARITY OF MANIFOLDS
33
[5] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1-40. [6] D. Coman, N. Levenberg, and E. A. Poletsky, Quasianalyticity and pluripolarity, J. Amer. Math. Soc. 18 (2005), no. 2, 239-252. [7] K. Diederich and J. E. Fornalss, A smooth curve in e 2 which is not a pluripolar set, Duke Math. J. 49 (1982), no. 4, 931-936. [8] V. D. Erohin, Asymptotic theory of the c-entropy of analytic functions, Dokl. Akad. Nauk SSSR 120 (1958), 949-952. [9] ___ , Best linear approximation of functions analytically continuable from a given continuum to a given region, Uspehi Mat. Nauk 23 (1968), no. 1 (139),91-132. [10] S. D. Fisher and C. A. Micchelli, The n-width of sets of analytic functions, Duke Math. J. 47 (1980), no. 4, 789-80l. [11] L. Hiirmander, An introduction to complex analysis in several variables, third ed., NorthHolland Publishing Co., Amsterdam, 1990. [12] A. N. Kolmogorov and V. M. Tihomirov, c-entropy and c-capacity of sets in functional space, Amer. Math. Soc. Trans!. (2) 17 (1961), 277-364. [13] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 69-72. [14] G. G. Lorentz, M. v. Golitschek, and Y. Makovoz, Constructive approximation, SpringerVerlag, Berlin, 1996, Advanced problems. [15] S. Nivoche, Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the €-entropy, Invent. Math. 158 (2004), no. 2, 413-450. [16] S. I. Pincuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki 15 (1974), 205-212. [17] A. Sadullaev, A boundary uniqueness theorem in en, Mat. Sb. (N.S.) 101(143) (1976), no. 4, 568-583, 639. [18] M. Waldschmidt, Diophantine approximation on linear algebraic groups, Springer-Verlag, Berlin, 2000. [19] H. Widom, Rational approximation and n-dimensional diameter, J. Approximation Theory 5 (1972), 343-36l. [20] V. P. Zahariuta, Spaces of analytic functions and maximal plurisubharmonic functions, Doc. Sci. Thesis, Rostov-on-Don, 1984. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW HAMPSHIRE, DURHAM,
03824 E-mail address: oleg. eroshkinlDunh. edu
NEW HAMPSHIRE
Contemporary Mathematics Volume 454, 2008
On a question of Brezis and Korevaar concerning a class of square-summable sequences Richard Fournier and Luis Salinas ABSTRACT. We give an new proof of a result due to Bn§zis and Nirenberg: klakl 2 is an integer whenever {ad~_oo is a sequence of complex
2::;;"=-00
numbers such that
2::::- 00 akan+k
=
{o 1
if n
i- 0,
if n = 0,
for all integers nand
1. Introduction
We consider sequences
{an}~oo
of complex numbers such that if n =i 0, for all integers n, if n = 0,
(1) and
(2)
00
L
Ikllakl 2 < 00.
k=-oo Under these assumptions it has been proved by Brezis and Nirenberg [3, 4] that the sum of the series 2::~=-00 klak 12 is an integer, a rather unexpected and sparkling result. The motivation of Brezis and Nirenberg while proving this was to extend the notion of degree (i.e., index or winding number) to various classes of maps; their proof was rather indirect and used aspects of duality. In a remarkable paper [8] Korevaar studied what happens to the Bn3zis-Nirenberg result when the absolute convergence of the series in (2) is replaced by various notions of convergence of l:~=-oo klakl 2 . In the same paper, Korevaar asked for a more direct proof of the result and the same question has been recently raised by Brezis during a talk at a meeting (2004) held in honour of Prof. Andrzej Granas on the occasion of his 75th 2000 Mathematics Subject Classification. Primary: 42A16; Secondary: 30BI0, 30A78. Key words and phrases. Fourier coefficients of unimodular functions, Hp spaces, Sobolev spaces. R. Fournier was supported by NSERC and L. Salinas by FONDECYT. Both authors would like to thank Oliver Roth and St. Ruscheweyh for their involvement in this project. ©2008 American Mathematical Society
35
R. FOURNIER AND L. SALINAS
36
birthday. Even more recently, the very same question has been raised by Brezis in
[2]. It is of course not so clear what is meant by a more direct proof. Our goal in this paper is to provide a different proof of the result based on facts more readily evident to "classical" complex analysts. Our work is also related to remarks of L. Boutet de Monvel and O. Gabber to be found in an appendix to the paper [1]. We shall finally also obtain the following THEOREM 1.1. Let {ad8" be a sequence of complex numbers for which
I:akiin+k = k=O
{o
1
I:
i!n: 0, for all positive integers n and Ikllakl 2 < ifn - 0, k=O
00.
Then B(z) := 2::%"=0 akzk is a finite Blaschke product and the number of zeros of B in the unit disc {z Ilzl < I}, including multiplicities, is equal to 2::%:1 klak 12. 2. Another Proof of the Brezis-Nirenberg Result We shall proceed by a number of lemmas. LEMMA
u( 0) :=
2.1.
Under
the
hypothesis
the
(1),
21r-periodic
function
2::%"=-00 akeikIJ is well-defined and unimodular for almost all 0 E [0, 21r).
PROOF. By the Riesz-Fischer theorem, there exists an integrable function u whose Fourier coefficients are the numbers {an}~=_oo and by the famous result of Carleson, this function is almost everywhere equal to its Fourier series. (This may also be established by using an older and weaker result of Fejer [10, p. 65]). Thus, we may assume that u( 0) := 2::%:-00 akeikIJ is the Fourier series of a square summable function. We now define, for < r < 1,
00
ur(O)
=
L
°
akrlkleiklJ,
°S
0 < 21r.
k=-oo
This last series is absolutely and uniformly convergent. We have for each integer n,
Since by Abel's continuity theorem
00 lim "" lakl 2 (1 - r2lk1)
r--+l
L-t
k=-oo
= 0,
37
QUESTION OF BREZIS AND KOREVAAR
00 =
(4)
lim "" akllk+nrlkl+lk+nl
r---+-l
L.-t
k=-oo
(5)
=
~
~ k=-oo
_ = {o
akak+n
if n # 0, If n - 0,
'-
1
the passage from (3) to (4) being justified by the absolute and uniform convergence of the Fourier series u r ((}) while (5) follows again from Abel's continuity theorem. This completes the proof of Lemma 2.1: we have shown that all Fourier coefficients (except for the constant one) of luI 2 are zero and thus luI 2 is constant almost everywhere. This result may not be entirely new since a (weaker) version of it was stated without proof in a 1962 paper by Newman and Shapiro [12]. Moreover, the condition (1) is in fact equivalent to the unimodularity of the associated function u(O): this is also a consequence of Parseval's identity. We may now write
u(O) Let U(O) := [13, p. 328] that
47r 2
= eiU(O) , with
2::%:-00 bke ikO .
f k=-oo
Ikllakl 2 =
U(O) real for almost all 0 E [0,211-)'
o
It is readily seen from the formula of Devinatz
{21r
{21rIU(~~-Ui(tp)12
io io
e
dOdtp
- e 'P
= 4 {21r (21r sin2((U(O) - U(tp))/2) dOd io io le'o - e''P 12 tp 1, we have
Applying the divergence theorem in the form JrCz - g*(z))dz
=
2iJ Jc%z(z-
g*(z))da, we obtain A (G) IEp
For p = 1 we have
AIEl (G)
~
2A (G) ylP(G)'
= Jrlz - g*(z)lds
Now for the upper bound, and any p duality rewrite AlE p (G) as:
~
~ IJr(z - g*(z))dzl
= 2A (G).
1, we will use Corollary 2.2 (i) and by
Since the boundary of the domain is analytic and z is real analytic on r, then by S. Va. Khavinson's results on the regularity of extremal functions (see Theorem 5.13 in [19]) we know that J*(z) is analytic across r. Hence we can express J*(z) as the Cauchy integral of its boundary values, J*(z)
= 2~i
l f~~~)
dw. Substituting this
in the last equality, using Fubini's theorem, and bringing absolute values inside the integral we obtain
APPROXIMATING
"if
IN HARDY AND BERGMAN NORMS
47
Holder's inequality yields
Let Fc(z)
=
~ J [w~zd(T,
z E C. Now, the Ahlfor-Beurling estimate [2] (see [10]
for a simple proof) implies that for a fixed z E C and among domains with the same area, the function IFc(z) I attains its maximum value when the domain is a disk of radius p passing through z, which we denote by Dp. So, IFc(z) I :::; IFDp(Z) I :::;
J
A(Dp) ~
=
j
A(C).
Therefore
rr
'
= n is the zero Junction. (For m :::; n, it is clear that znzm = r 2m z n - m is its own best approximation.)
(ii) Ap(G) = IlznzmlllLp(ds,r) = V"27frP (n+m)+l. (iii) The best approximation to z in lEp (G) is the zero Junction and the p- analytic content oj a disk oj radius r is AJEp(G) = IlzlllLp(ds,r) = V"27fr P +1 .
The proof is trivial, we only sketch it for the reader's convenience. SKETCH OF PROOF. Let G = {z E C : Izl < r}, p > 1 and let J(z) for m > n. The function J(z) annihilates lEp(G) since, for k ~ 0
= Izz:~:t ~:
r (lznzmIP) zkds = r 2rr r (p-l)(n+m)+k+l ei(k+m-n)OdO = O. znzm Jo
Jr
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
48
Set J*(z)
=
,so that
IIfllf(z) Lq(ds,r)
IIJ*II
Lq(ds,r)
= 1. Then with g*(z) = 0 the extremal-
ity condition is satisfied:
For p = 1, let f(z) = _iz m- n - 1 ~= and g*(z) = 0, Then and -izm-n-1(znzm)dz = r 2m d(), Hence J*(z) = J(z)
Ir - izm-n-l+kdz =
IIJ(z) IILl (d.,r)
0, and g*(z) are both
extremaL
= Now , AP(G) P
P Ilznzmll ILp(ds,r) = 1211" Irn+mei(n-rn)OIP rd() = 27rrP(n+m)+1,
o Taking n = 0 and m = 1 we obtain (iii).
0
THEOREM 3.2. Let G be a multiply connected bounded domain with the boundary consisting of n simple closed analytic curves. The zero function is the best approximation to z in lEp (G) if and only if G is a disk. PROOF. Necessity is obvious, For the converse, suppose that 0 is the best approximation to z in lEp(G). Then the extremality condition (2), for p ;:::: 1, can be written as
J*(z)zdz = constlzlPds on each boundary component of the domain G. Without loss of generality we will assume the constant is positive. Dividing by z we can rewrite the equation above as
J*(z) --dz = canstlzl p- 2ds.
(4)
z
Notice that 0 E G, otherwise
Ir r;z)dz
= 0, yet I r lzlp- 2ds =I- 0 since this is a
positive measure, For the same reason J*(O) =I- 0, hence r~z) has a pole at the origin, Because the boundary of the domain is analytic, for each boundary component we can find a Schwarz function S(z) = Z, that is, a unique analytic function which at every point along the boundary component takes on the value z [14], [20]. Now, (ds)2 = dzaz = S'(z)dz 2, so ~= = JS'(z) on r and we obtain that
J*(z)
- e - = canst S(Z)~-lJS'(z). Z2
Squaring both sides yields
d [J*(z)]2 = canst - [S(z)p-l] . zP dz This last equation implies that for each contour S(z)p-l is analytic throughout the
(5)
domain, except at the origin. We will now consider a few cases. CASE 1. p = 1
APPROXIMATING
z
IN HARDY AND BERGMAN NORMS
49
When p = 1, IJ*I :::: 1 in G and IJ*I = 1 on r. Therefore, J*(z) is either a unimodular constant or the cover mapping of G onto the unit disk. Suppose J*(z) is not constant. From Corollary 2.2 we have that J*(z) = e' {) Izlds zdz and 1J*(z)1 = 1 almost everywhere on the boundary of G. By S. Ya. Khavinson's regularity results (Theorem 5.13 in [19]) 1J*(z)1 = 1 everywhere on the boundary and J*(z) extends analytically across each boundary component. Therefore, J*(z) maps G onto the unit disk]])) taking each value in the disk k times, and wrapping each boundary component of r around the unit circle at least once, and always following the same positive direction. If that were not the case, and we suppose that at some point w E r, J* (w) changes direction, at that point (w) = 0, so ~(w) = r!:L. ds = O. dz ds dz Hence, for z near w, J*(z) = J*(w) + O((z - W)2). SO J*(z) maps the "half' neighborhood of w that is in G onto a full neighborhood of J*(w), which means that IJ* 1can be greater than 1 near w, and that is a contradiction. Now, in order to wrap each boundary component of G around the circle, J*(z) has to go around the unit circle k times with n :::: k . If we let i:1 arg J* (z) denote the change in the argument of J* (z) as z goes around the boundary of G, then i:1 arg J*(z) ~ n. Moreover, the tangent vector to r traverses the boundary of G once in the clockwise direction, and n - 1 times in the counterclockwise direction. Hence, remembering that r is analytic and 0 E G, by the argument principle we obtain that
fs-
i:1arg (J*;z) dZ)
= i:1argJ*(z) + i:1arg~ + i:1argdz
~ n - 1 + 2 - n = 1,
while i:1arglzIP-2ds = 0 and from (4) we obtain a contradiction. Hence, for p = 1, J*(z) is a unimodular constant so from the equation preceding (4) we invoke that on r, z~; = eialzl, where a is a real constant. Writing on each boundary component z(s) = r(s)eib(sl, substituting and separating real and imaginary parts yields r' = cos a. Since each component is a closed curve, it cannot be a spiral, cos a must be zero, thus each component is a circle centered at the origin. Moreover, the case of the annulus is ruled out because ~; changes directions between the two boundary circles, hence Izl = canst on r, and G is a disk centered at the origin. CASE 2. p> 1, p rt- N If p is not an integer S(Z)p-l may be multivalued. Yet, since the left hand side of (5) is 0 C1p) near zero, it follows that S(z) is 0 (~) in a neighborhood of the origin. Also notice that if p is not an integer S(z) cannot vanish anywhere in G. If it did it would be possible to obtain an unbounded singularity on the right hand side of (4) by differentiation, while the left hand side would remain bounded. Therefore the Schwarz function for every boundary component of G is analytic in the whole domain and has a simple pole at the origin. Moreover, since remains the same when it is continued analytically throughout G, S(z) has to be the same dz and analytic function for each boundary component. So S(z)p-l = from this we obtain that S(z) = co~st + g(z), where g(z) is analytic in G and is independent of which boundary component we consider. S(z) = z on the boundary. S(z)z = Izl2 is real, positive on the boundary and analytic inside the domain G, hence it is constant. The boundary of the domain is therefore a circle centered at the origin.
[/*;:W
fr [/*J:W
50
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
CASE 3. p> 1, pEN When p is even, i.e. p = 2k, (4) becomes
(6)
J*(z) --k- dz z
= constlzlk-Ids,
which in turn yields
(7)
[J*(Z)]2 Z2k
d
= const dz
[S(z)2k-I] .
(6) implies that S(Z)2k-1 is analytic throughout G and has a pole of order 2k - 1 at the origin in G, so S(z) has to have a simple pole at the origin. Following the same reasoning as in case 2 we can conclude that S(Z)2k-1 is the same for every boundary component. Then z2k-1 = S(z)2k-1 = ~~F~~ + g(z) for g(z) analytic in G. Multiplying through by Z2k-1 we have once again that Izl = const. For p odd, i.e. p = 2k + 1, (5) can be written as
(8)
[J*(Z)]2
z
2k+1
d [
= constd z S(z)
2k]
So S(z)2k = c~~r + g(z), with g(z) analytic in G. Hence, once more, the boundary of the domain is a circle. 0
DEFINITION 3.3. ([6], Ch. 10) Let G be a Jordan domain with rectifiable boundary r, let z = ¢(w) map G onto Iwl < 1. Since ¢' E HI and has no zeros, it has a canonical factorization ¢'(w) = S(w)(w) where S is a singular inner function and is an outer function. G is said to satisfy the Smirnov condition if S ( w) = 1, i. e. if ¢' is purely outer.
It is the case that G is a Smirnov domain if and only if EP (G) coincides with the lLP(r) closure of the polynomials. We will use repeatedly the property that if a function f E EP (G) belongs to JLP (r) with q > p, then f E Eq (G).
REMARK 3.4. For a simply connected domain we can significantly relax the assumption of analyticity of the boundary in Theorem 3.2 and obtain that the domain is a disk invoking the following result from [8].
THEOREM 3.5. (Thm. 3.29 in [8]). Let G be a Jordan domain in ~2 ~ C containing 0 and with the rectifiable boundary r satisfying the Smirnov condition. Suppose the harmonic measure on r with respect to 0 equals cl z I ds for z E r, where ds denotes arclength measure on r, a E Rand c is a positive constant. Then (i) For a = -2, the solutions are precisely all disks G containing O. (ii) For a = -3, -4, -5, ... there are solutions G which are not disks. (iii) For all other values of a, the only solutions are disks centered at O. Q
APPROXIMATING
z
IN HARDY AND BERGMAN NORMS
51
To apply this result in our context we need first to notice that the positive measure f~z) dz = constlzlp-2 ds annihilates all analytic functions vanishing at the origin and hence is, after normalizing by a scalar multiple, a representing measure for analytic functions at the origin. Moreover, since the domain is simply connected, we can separate real and imaginary parts and then conclude that this latter measure is precisely the harmonic measure at O. Because p - 2::::: -1, part (iii) applies and the domain is a disk centered at the origin.
3.6. Let p, q ::::: 1, ~ + ~ = l. Let G be an annulus {z : 0 < r < Izl < R} and r = 1'1 U 1'2 be its boundary. (i) For p > 1 the best analytic approximation to w = znzm in lEp ( G) is unique and * n-m r2m+q(n-m)+~ +R2m+q("-m)+~ equal to g (z) = cz where c(n, m,p) = q(n-mH!l q(n-mH.'l. • r p+R p (ii) For p = 1, and n - m = -1, the set oj Junctions that are closest to w = znzm in lEI (G) consist oj all Junctions oj the Jorm g* (z) = cz n- m where c is any constant such that r2m c R2m. (iii) For p > 1, the distance from znzm to lEp(G) is THEOREM
:s :s
Ap(G)
Ilznz m - g*(z)lllLp(ds,r) _ (rRt- m (R 2m _ r2m) - rq(n-m)+~ + Rq(n-m)+~ :=
Note: For n - m i- -1, we have been unable to find the best approximation in closed form, see the remark at the end of the proof. PROOF.
Consider J(z) =
I
n-7n
en-TTL
zz"zzm__ :zn
m
IP
~~. Then
p(n-m)+q(k+l) 1
2m ,P)127r ei(k+m-n)O dB
r r2m -
C
C
0
= 0, unless k = n - m.
and 27r
- clP + rP(n-m)+1 Ir2m - clP ) ( RP(n-m)+1 IR2m R2m - c r2m - c
if
RP(n-m)+1 IR 2m R2m -
clP C
P _ _ rP(n-m)+l Ir2m - cl r2m - c '
-
=0
52
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
which is only possible if r2m < c < R2m. In that case, (
c _ r2m )P-l
= (!!.)p(n- ml+l
R2m - c
r
and after some algebra we obtain that r2m+q(n-m)+~
c=
rq(n-ml+~
Therefore, f (z) annihilates IEp (G). Now let 1*(z) = IIIII I so that 111*11 Lq(d.)
+ R2m+q(n-m)+~ + Rq(n-m)+'f;
Lq(d.)
= 1 and let g*(z) = cz n- m . Then,
Iznzm - czn-ml P ---'-----------'--_""7 1
Ilzn zm-
czn-mll~
ds,
which is condition (iii) in Corollary 2.2. Therefore 1*(z) and g*(z) are extremal. For p = 1, and n - m = -1, by Corollary 2.2, 1*(z) and g*(z) are extremal if and only if they satisfy that 1*(z)(znz m - g*(z))dz = Iznzm - g*(z)lds on each boundary component of the annulus. Consider 1*(z) = -i and g*(z) = ~. On 11 = {z E C: Izl = r}, with clockwise orientation on the boundary we have
-i(znzm - ~)dz = _(r 2m - c)dB z
and on the other hand
cl
Iznzm - ~ Ids = Ir2m - dB z The same analysis on 12 = {z E C : Izl = R}, where the orientation on the boundary is counterclockwise, yields c
-i(znzm - - )dz = (R2m - c)dB z
and
Iznzm - ~Ids = IR2m -
cl dB.
_(r 2m _ c)dB = Ir2m -
cl dB
(R2m _ c)dB = IR2m -
cl dB.
z
Which means and These two equations hold simultaneously for any constant c in the interval [r2m, R2m]. Finally we compute Ap(G).
APPROXIMATING
z
IN HARDY AND BERGMAN NORMS
For p > 1, recalling that c(n,m,p)
There "lore Ap (G) -Now, for p
r
=
2m+q(n-=H'l. r r
(rR)n-m(R2m_r2m) q(n-mH!l q(n-mH!l p
= 1 and n -
+R
m
q(n-=H~+
R2=+q(n-=H'l. q(n-mH!l P,
p+R
p
2"'" [(r~R%(1-n-m))P "
53
we have
+ (R~r%(1-n-m))P].
p
=
Ai(G)
-1
=
Ii
=
11211' r2m + R2mdoI
=
27r(r2m
znzmj*(z)dzl
+ R2m).
o
The proof of Theorem 3.6 is now complete.
REMARK 3.7. When p = 1 and n - m =1= -1, because the boundary is analytic and J*(z) is continuous on G, 1J*(z)1 = 1 everywhere on the boundary. Therefore, f* (z) is either constant or a k-sheeted covering of the unit disk. It is not a constant since znzmdz = 0 unless n = -1. So j*(z) maps G onto a k-sheeted cover of the unit disk with k ;:::: n. Hence the best approximation to znzm cannot be a monomial cz n - m . Moreover, it follows from the duality relations that f* (z) has to be a transcendental function.
Ir
By letting n
m
= 0 and m =
1 we have the following corollary.
COROLLARY 3.8. Let ~ + ~ = 1. Let G be an annulus {z : 0 < r < Izl < R}. (i) For p > 1 the best analytic approximation to w = z in IEp (G) is g* (z) = r~ . (ii) For p = 1, all functions g* (z) = ~ for any constant c E [r 2, R2], serve as the best approximation to z in lEi (G) . (iii) For p;:::: 1 the p-analytic content of G is AJEp(G) = (R - r)(27r(R + r))~
Notice that the best approximation to of p!
z in IEp(G)
is g*(z)
=
r~ independent
Next we will prove a partial converse for Theorem 3.6 in the case when p = 1. For that we will need the following lemma.
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
54
LEMMA 3.9. Let G be a multiply connected domain in C with analytic boundary consisting of n components. If g* (z) = ~ is the best approximation to z in lE1 (G) and z does not coincide with ~ on any of the boundary components then: i) j*(z), the extremal function in lE~(G) for which sup IJrzf(z)dzl is atfEIE~(G)
tained, is a unimodular constant. ii) The number n of boundary components of G is 2.
PROOF. Replicating the argument used in Theorem 3.2, case 1, we can show that unless j*(z) is a constant, it is a k-sheeted covering of the unit disk, with k :::: n, thus tlargj*(z) :::: n. Moreover, the tangent vector to r goes along the boundary of G once in the clockwise direction, and n - 1 times in the counterclockwise direction. So tlargj*(z)dz :::: n + 2 - n = 2. Now, since the boundary of the domain is analytic and we are assuming that ~ is analytic in G, ~ has no poles in G. By the argument principle we can say that
_ C)
tlarg (z - z
= tlarg
Izl2 z
C=
2 tlarg(lzl - c)
+ tlarg -z1 =
O.
So tlarg(Z - ~)j*dz = tlargJ*(z)dz + tlarg (z -~) :::: 2. Yet Corollary 2.2 (iii) yields that tlarg (z - ~) j* dz = 0 since it has constant argument on r, so we have a contradiction. Hence, j*(z) has to be constant. With f* (z) constant we have that tlarg (z - ~) j* dz = tlargdz 2- n = 0 therefore the number of boundary components of Gis n = 2. o
THEOREM 3.10. Let G be a multiply connected domain in C with analytic boundary r. If g* (z) = ~ is the best approximation to Z in lE1 (G) and the hypotheses of Lemma 3.9 are satisfied, then G is an annulus. PROOF. (That the best analytic approximation to Z in lE1 of the annulus is g*(z) = ~ follows from Corollary 3.8.) We infer from Lemma 3.9 that j*(z) = eia.. By the duality relations we obtain AIEl =
J
zdz =
Jez -
;)dz
: ; JIz - ; 1
ds
= AIEl· (9)
r r r Therefore equality holds throughout. Now, since Izl2 - c is real and the boundary is analytic, (9) implies that arg (dzz) is constant on every boundary component of G. On the other hand we have from Lemma 3.9 (ii) that G has two boundary components 11 and 12, with opposite orientation. So letting z (s) = r (s) e ib (s), with s being the arclength parameter, since I~~ I = 1, by differentiating we obtain
~: were aj, j
= (ir(s)b'(s)
= 1,2 are
+ r'(s))e ibCs ) =
eiaj+ib(s) , j = 1,2
constants on 11 and 12 respectively. This yields that ir(s)b'(s)
+ r'(s)
= eiaj , j = 1,2.
Differentiating again, we obtain r"(s)
+ i(r(s)b'(s))' = 0,
APPROXIMATING z IN HARDY AND BERGMAN NORMS
55
hence (r(s) and b(s) are real-valued functions) r"(s) = 0 and r(s) is a linear function. Recalling that the boundary of the domain consists of two closed curves we conclude that r(s) is linear and periodic, hence it is constant on each boundary component. So the boundary of the domain consists of two concentric circles and the domain is an annulus. 0
REMARK 3.11. In Lemma 3.9, if z does coincide with ~ on one of the boundary components, say /0, i.e. if that component is a circle, then on that boundary component 11* 1 ::; 1 while on the remaining components 11* 1 = 1. In this case we can only infer that ~ arg1* 2': n - 1 and the argument above fails. We conjecture r\l'o that Theorem 3.10 holds for all p 2': 1 and without the additional hypothesis in Lemma 3.9. Yet, we have not been able to prove it.
4. The Bergman Space case: Characterization of disks and annuli in terms of the best analytic approximation to z in Ap norm. Let da be area measure on G. We use the standard notation w1,q(G) and w~,q(G) for Sobolev spaces and Sobolev spaces with vanishing boundary values. The reader may consult [9, Ch.5], [1] for details. Khavin's lemma (see [20]) describes the annihilator of Ap(G) as follows: For p > I,
Ann(Ap(G))
{f E lLq(da,G): Lfgda
:=
=
{~,
uE
=
0 for all 9 E Ap(G)}
w~,q(G)}.
For p = I,
Ann(Al(G)):= {weak(*) closure of
~,
U
E W1,OO(G), in lLoo(da,G)}.
DEFINITION 4.1. The Bergman p-analytic content of a domain G is
AA(G):= P
inf
gEAp(G)
Ilz-g(z)lllL(d17G)' p ,
By the Hahn-Banach theorem,
AA (G) p
=
max
fEAnn(Ap(G»,llfI19
11
G
zfdal·
A similar result to Corollary 2.2 holds in the context of Bergman spaces; we state it as Corollary 4.2 for completeness. See [17] Theorem 3.1, Remarks (i) and (iv). Also see [18] p. 940.
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
56 COROLLARY
4.2. Let ~
+ i = 1,
and let w(z) E lLp(dO", G). Then the following
hold: (i)
inf
gEAp(G)
Ilw(z) -g(z)111L
(dC7G)
=
p,
sup
fEAnn(A p (G)),llfI19
11
G
w(Z)fdO"I·
(ii) There exist extremal functions g*(z) E Ap(G) and J*(z) E Ann(Ap(G)) for which the infimum and the supremum are attained in (i). (iii) When p > 1, g*(z) E Ap(G) and J*(z) E Ann(Ap(G)) are extremal if and only if, for some real number 8,
eili J*(z)(w(z) - g*(z))
>
A~p 1J*(zW
o in G, Iw(z) - g*(z)IP in G,
where AAp = Ilw(z) - g*(z)ll lLp (d 1 the best approximations g*(z) E Ap(G) and J*(z) E Ann(Ap(G)) are always unique. For p = 1 and w(z) continuous in G, the best approximation g*(z) E Ap(G) is unique. For discontinuous w(z) the best approximation need not be unique. Also, in the case where p = 1 the duality condition in (iii) implies that J*(z) E Ann(Al(G)) is unique, up to a unimodular constant, provided that w(z) does not coincide with an analytic function on a set of positive area measure. REMARK 4.3. For the case of the disk][)) = {z E C : Izl < r} it was shown in [17] Proposition 2.3, that the best rational approximation in Ap(][))) to w = znzm for p:2 1 and m > n is g*(z) = o. When m ~ n, g*(z) = cz n - m , where c = c(n, m,p) is an appropriate constant.
In that case we can compute the Bergman p-analytic content of ][)) as follows: AAp (][)))
=
fo27r foT Ite-it IPtdtdO 211" foT
IW tdt
Following the argument in [17] we find the extremal functions for the case of the annulus. PROPOSITION 4.4. Letp,q:2 1, ~+i = 1. LetG be an annulus {z: r < Izl < R, r for every t E (0,1] and every f,g E L oo [0,1]. Suppose also that lim '1]1111 (t) = 0. Then
°
t->O+
(1) Ilfll ~ Ilflloo for every f E L OO [0,1] (2) Ilfnll --> if and only if Un} is I II-cauchy and fn --> in measure. (3) Xliii c Y (4) (Xliii, Y) is a multiplier pair with pointwise (a.e.) multiplication. (5) .co = Ro = L oo [0,1] and IILtl1 = Ilflloo always holds.
°
°
Proof. (1) This follows from Ilfll = Ilf· 111 ~ Ilflloo 11111 = Ilfll oo ' (2) Suppose Ilfnll --> 0. Clearly, Un} is II II-cauchy. Also if e > {x E [0,1] : Ifn (x)1 ~ e}, then
°
°
and En =
Ilfnll ~ IlfnxE,J ~ e IlxEnll·
°
Since AIIII (t) > for every t E (0,1] it follows that JL (En) --> 0. Hence fn --> in measure. Conversely, suppose Un} is II II-cauchy and fn --> in measure, and Ilfnll -+> 0. By taking a subsequence and normalizing, we can assume that r ~ Ilfnll ~ 1 for every n E N and some r > 1. Choose N so that m, n ~ N implies Ilfn - fmll < 1/3. Let Em = {x E [0,1] : Ifm (x)1 ~ 1/3}. Since fm --> in measure, JL (Em) --> 0. Since lim '1]1111 (t) = 0, we have IlxEm II --> 0. Then we have
° °
t->O+
1 ~ IlfN11 ~ IIUN - fm) (1 - XEm)11 ~ IlfN - fmll ~ 2/3
+ IlxEm1IIIfNII + Ilfm (1- xE,JII
+ IIXEm 1IIIfNIIoo + Ilfm (1 -
+ IlxEm 1IIIfNIIoo -->
XE)ll oo
2/3,
which is a contradiction. (3) It follows from (2) that the inclusion map from Loo [0, IJ into Y extends to a continuous injective map from Xliii into Y. (4) The continuity of the multiplication follows from (2), and the other properties are obvious. (5) Suppose f E .co, r > and E = {x E [0,1] : If (x)1 ~ r}. Then
°
IILtxeII11xE11 ~ IILtxEl1 ~ r IlxEII,
DON HADWIN AND ERIC NORDGREN
66
so if J-L (E) > 0, it follows that IILfl1 2: r. Hence Ilflloo ~ IILfl1 < implies that IILfl1 ~ Ilfll oo ' •
00.
Statement (1)
We say that a norm 1111 on LOO [0,1] (with respect to Lebesgue measure J-L) is a symmetric norm if
(1) 11111 = 1 (2) Illflll = Ilfll for every f E LOO [0,1] (3) Ilf 0 o+
IIX[o,tJ II
2
IILfl1 , then the essential range of 1 contains a number Awith IAI > IILfll· For every E > 0 the set E (E) = {x: 11 (x) - AI < E} has positive measure, so if gE = XE(E)/ IIXE(E) II, then
II(L f - A)gEII :::;
11(1 - A) XE(E)lloo IlgE11 :::; E.
AE a(Lf), which contradicts IAI > IILfll. Thus 1 E IILfll· However, it follows from (1) that IILfl1 :::; 11111 00 , •
Hence
LOO [0,1] and
1111100 :::;
4. Unitarily Invariant Norms on a Finite Factor
Suppose M is a I h factor von Neumann algebra with a faithful normal trace A norm von M is a unitarily invariant norm if v (1) = 1 and v (UTV) = v (T) for every T E M and all unitaries U, V E M. The Russo-Dye theorem tells us that the closed unit ball of M is the norm-closed convex hull of the set of unit aries in M, so we have v (T) :::; IITII for every T E M. 1 Since every T in M has a polar decomposition T = U (T*T) 2 with U unitary, it
T.
follows that v (T)
=v
((T*T) ~) for every T EM. If we expand the set of spectral
projections XE ((T*T) ~ ), with E ranging over intervals of the form [0, s) or [0, s], to a maximal chain of projections {Pt : t E [0, I]} with each T (Pt ) = t, we see that (T*T)~ E {Pt : t E [0, l]}/I. Moreover, {Pt : t E [O,l]}/I is tracially isomorphic to Loo [0, 1]. More precisely, the map X[O,t) --+ Pt extends uniquely to a *-isomorphism 7r : Loo [0,1] --+ {Pt : t E [O,l]}/I such that, for every 1 E LOO [0, 1],
T(7r(1)) =
r
ld/L.
i[O,l]
It follows from results of Huiru Ding and the first author [2] that if 7r1, 7r2 : LOO [0,1] --+ M are unital *-homomorphisms such that TO 7r1 = TO 7r2, then there is a net {UA} of unit aries in M such that, for every 1 E LOO [0,1], V (U~7r1
(1) UA - 7r2 (1)) :::; 11U~7r1 (1) UA
-
7r2 (1)11
--+
0,
so v 0 7r1 = V 0 7r2. Hence the norm v 0 7r is independent of the element T E M or the representation 7r. Moreover, if a : [0,1] --+ [0,1] is a bijective measurable measure-preserving transformation, then, for every 1 E LOO [0,1] ,
T(7r(10a))=
r
i~,~
(1oa)d/L=
r
i~,~
ld/L.
A GENERAL VIEW OF MULTIPLIERS AND COMPOSITION OPERATORS II
69
Hence the norm v 0 7r is a symmetric norm on Loo [0, 1J. It follows from results in [3J and [4J that every symmetric norm on Loo [0, 1J corresponds to a unitarily invariant norm on M. There is also a notion of convergence in measure introduced by Nelson [10J. A net {TA } in M converges in measure to T E M if and only if, for every € > 0 there is a projection P with T (P) < € and there is a .Ao such that
I (TA -
I 0), and I is a collection of Borel subsets of X such that:
(1) 0 < JL(I)
0, there is an E E I with Eel n J, then we get statement (4) for free. These conditions hold in most of the classical examples.
Li
Throughout this paper (X, JL,I) will denote a BMO triple. We define lac (JL) to be the collection of all measurable functions I : X ---7 C such that JI III dJL < 00 for every I E For I E lac (JL) we define the average of lover I by
I.
Li
1(1) we define the mean oscillation of
I
= JL!I)
J
(I I - I (I) I), sup I (11- 1(1)1). lET
on I by I
II/II~MO(T}J-) ,
=
IdJL, and we define
BMO AND VMO
77
We define BMO(I, /l) = {f E Li,loc (/l) : Ilfll~MO(T"L) < 00 } . We also define the space V MO(I, /l) to be the set of all functions f E BMO(I, /l) such that lim
JL(I)+diam(I) ~O
I (If - I(f)I) = O.
It is clear that Ilfll~MO(T'JL) = 0 if and only if the function f is constant a.e. (/l) on every I E I, and conditions (3) and (4) on I imply that Ilfll~MO(T'JL) = 0 if and only if the function f is constant a.e. (/l) on X. DEFINITION
2.2. Suppose
f and I are as above. If X E I, we define
IlfIIBMO(T,JL) =
Ilfll~MO(T'JL) + /l (~)
Il
fd/ll·
Otherwise, we define
IlfIIBMO(T,JL) = Ilfll~MO(T'JL) , and in this case, to make IlfIIBMO(T,JL) a norm, we identify functions in BMO (I, /l) that differ by a constant, i.e., we mod out by the subspace of constant functions. It is apparent that f E BMO(I, /l) if and only if Re(f) , Im(f) E BMO(I, /l). It is also simple but a useful fact that the space ofreal-valued BMO(I, /l) functions forms a lattice. In other words, if the real-valued functions f, 9 E BMO(I, /l), then If I ,Igl E BMO(I,/l), and therefore, so do max (f,g) and min (f,g) .
The reader should note that this notion of BMO includes all classical BMO definitions [7], [3], [10]. LEMMA 2.3. Suppose Un} is a Cauchy sequence in BMO(I, /l) and J Let gn = fn - J(fn). Then I(gn) is a Cauchy sequence for every I E I .
PROOF.
E
I.
The proof when J C I follows from the following:
II (gn) - I (gm)1 = II(fn - fm) - J(fn - fm)1 :S J (Ifn - fm - I(fn - fm)!) /l(I) /l(I) :S /l(J) I (Ifn - fm - I(fn - fm)!) :S /l(J) Ilfn - fmIIBMO(T,JL) . The proof when I C J is the same. For the general case choose h = I"" ,1M = J in I as in condition (4) in the definition of I, and note that M-l
II (gn - gm) I = II(gn - gm) - J(gn - gm) I :S
L
Ih(gn - gm) - h+l (gn - gm) I
k=l
o PROPOSITION 1.
(BMO(I, /l),
II'IIBMO(T'JL))
is a Banach space.
DON HADWIN AND HASSAN YOUSEFI
78
PROOF. We only need to show that BMO(I, fL) is complete. Suppose Un} is a Cauchy sequence in BMO(I, fL). First suppose that X fJ. I. Fix 10 E I and let gn = In - IoUn). For every I E I and I E BMO(I, fL) we have: 11/11l,!
~ 111 -
=
11/1 dfL
~
fL(I) II/IIBMO(I,/Ll
IU)I dfL
+ IIU)I fL(I)
+ IIU)I fL(I)·
Since I(lgn - gm - I(gn - gm)1) = I(l/n - 1m - IUn - 1m)!) we have Ilgn - gmIIBMO(I,/Ll = Il/n - ImIIBMO(I,/Ll' From Lemma 2.3 we know that {I (gn)} is a Cauchy sequence. Thus the above inequality with I = gn - gm implies that {gn} is Cauchy in L1 (I) for every I and so is convergent in Ll-norm to a function 9 E L1 (I). We have:
<
0 such that 1I/II BMo (..1,/Ll ~ M 1I/IIBMo(I,/Ll' VI E BMO(I, fL)· In particular, 11'IIBMo(I,/Ll is equivalent to 11·II BMo (..1,/Ll il and only il BMO(I, fL) = BMO(.:J, fL)· PROOF. Let 'P : BMO(I, fL) ---- BMO(.:J, fL) be the identity map. By using the Closed-Graph Theorem we will show that 'P is a linear bounded map. Suppose that In E BMO(I, fL), In ---- I in BMO(I, fL), and that In ---- gin BMO(.:J, fL)· We will show that I = 9 in BMO(I, fL). It is clear that if fL (X) < 00, then BMO(I, fL) = BMO(I U {X}, fL) and BMO(.:J, fL) = BMO(.:J U {X}, fL). Thus without loss of generality we can assume that X E In .:J whenever fL (X) < 00. The rest ofthe proof divides into two cases. First suppose that fL (X) < 00. Then, similar to the proof of the previous proposition, we have X (lin - II) ~ Il/n - IIIBMo(I,/Ll .
BMO AND VMO
79
Thus fn -----7 f in Ll (X) . In the same way, fn -----7 9 in Ll (X) . Therefore f = 9 almost everywhere. Next suppose that J-L (X) = 00. Choose I' E I and l E 3 such that J-L (I' n l) > 0 and let 10 = I' n l. Without loss of generality we can assume that 10 E In 3. By the proof of the previous proposition, it follows that fn - 10 (fn) -----7 f - 10 (f) in Ll (1) for every I E I. A similar proof shows that fn - 10 (fn) -----7 9 - 10 (g) in Ll (J) for every J E 3. Thus f - 10 (f) = 9 - 10 (g) on InJ, almost everywhere, VI E I and VJ E 3. Since X = U n21 I n = U n21 J" for some In E I and I n E 3, it follows that f - 10 (f) = 9 - 10 (g) almost everywhere on X. Therefore f = gin BMO(I, J-L). 0 Let C u (X) denote the set of uniformly continuous functions on X. If I E I, define the measure J-LI as the restriction of J-L to the O'-algebra of Borel subsets of I. LEMMA
2.4. If (X, J-L, I) is a BMO triple, then:
(1) Cu(X) n BMO(I,J-L) c VMO(I,J-L) and VMO(I,J-L) is a closed linear subspace of BMO(I,J-L). (2) There is a countable collection of continuous linear functionals on BMO(I, J-L) that separates the points of BMO(I, J-L). (3) For every f E Loo(p),
IlfIIBMo(I,/L)
~
311fll00 .
In particular, the inclusion map from Loo(J-L) to BMO(I, J-L) is continuous. (1) The inclusion Cu(X) n BMO(I, J-L) c V MO(I, J-L) is easily proved. For each I E I, we define TI : BMO(I, J-L) ----* U (J-LI) by 1
PROOF.
TJ (f) = J-L(I) (f - I(f)) II.
Then IITJ II ~ 1 and f
E
V MO(I, J-L) if and only if
/L(I)+1!:!(I)->0 IITI (f) II
= O.
It easily follows that V MO(I, J-L) is a closed linear subspace of BMO(I, J-L). (2) For every In in the definition of BMO triple there exist continuous linear functionals {¢n,d k2 1 on U (J-LIJ that separate the points of Ll (J-LIJ. Define 'l/J n k : BMO(I,J-L) ----* C by
'l/Jn,k(f) = ¢n,k
(J-L(~n) (f -
In(f)) lIn) .
Note that
I'l/Jn,k
(f)1 ~ II¢n,kll
1
J-L(In)
11(f -
In(f))
IInlil ~ II¢n,kllllfIIBMO(I,/L)'
It now follows that {'l/Jn,k : n, kEN} separates the points of BMO(I, J-L). (3) This is obvious.
o If X is the unit circle, J-L is the normalized arc length, and I is the set of all arcs in X, then we obtain the classical BMO and V MO spaces defined on the unit circle. The following proposition shows that our general versions can be quite different.
DON HADWIN AND HASSAN YOUSEFI
80
If I = {I eX: 0 < JL (I) < 00, I is a Borel set}, then: (1) BMO(I, JL) = LOO(JL), (2) if L = min{llf - >'1100 : >. E C}, then L ::::; IlfIIBMo(I,/L) ::::; 311fll00 for every f E LOO(JL), (3) V MO(I, JL) = Cu(X) n LOO(JL).
PROPOSITION
2.
1) Suppose f
if.
L oo (JL) . Then, for each positive integer n there are complex numbers aI, a2 with lal - a21 ?: 2n + 2 such that PROOF.
JL({x EX: If(x) - ajl < I}) > 0 for j = 1,2. Since JL is CT-finite and nonatomic, there are subsets Ej C {x EX: If (x) - ajl < I} for j = 1,2 such that 0 < JL (Ed = JL (E 2 ) < 00. If we let 1= EI U E 2 , then
la, ;a, _ I(!)I ~ I~ IE, (a, IlfIIBMO(I,/L) ?: JL
!)d~: ~ IE, (a, - !)d~1 < 1,
~I) l l f -
I (f)1 dJL ?: n.
2) We assume that f is not a constant function. Suppose also that f is a real-valued function and let M and m be the essential supremum and the essential infimum of the function, i.e., m ::::; If (x)1 ::::; M a.e. JL and JL{x : M - E < f(x) ::::; M} > 0 and JL{x: m::::; f(x) < m+E} > 0 for every E > O. For every positive integer n, there exist En > 0, I n > 0, and Borel subsets h,n and 12 ,n with the following properties: JL(h,n) ~ JL(h,n) < 00, M - En ::::; f(x) ::::; M for every x E h,n, and m ::::; f(y) ::::; m + I n for every y E h,n. We can choose En and I n so that they both converge to zero as n goes to infinity. Let In = h,n U hn. It follows that ~ (m + M - En) ::::; In(f) ::::; ~ (M + m + I n ) for every n ?: 1. Therefore:
In(lf - In(f)l)
=
JL(~n) [l"n If - In (f) I dJL + 12n If - In (f) IdJL]
JL(~n) [l"n (f - In (f))dJL + 12,n (In (f) ~ _ 1 [r fdJL - r fdJL] JL(In) JII,n J hn .
f)dJL]
The proof for this case will be completed by noticing that L = ~ (M - m) and the following inequalities:
~ (M 2
m - In
-
[r
r
~
En) ::::; _1_ fdJL fdJL]::::; (M - m). JL(In) JI"n JI2,n 2
The proof of the general case that f is a complex valued function will be apparent if one uses the previous case and the facts that Ilfll oo = sup IIRe(e ili 0: 0 we let: Na
(f)
sup
=
/l(I)+diam(I): 0
U
U
n
(Ij n E) a.e. Let An = Ij. We claim that E = An a.e. To show j21 j21 n21 this, suppose x E An. This means that for every n and some j we have x E Ij.
and E =
n
Now we have
dist(x, E) :::; essdiam(Ij) a.e.
BMO AND YMO and this shows that almost everywhere
87
n
An C E. Therefore E E F and so is
every Borel set. (2). For every n define fn(x) =
L J(1)xJ(x) :S a
where the summation is taken over all JEAn. By defining Fn = a-a 19 < An > we have that Fl c F2 C .... Therefore fn is a martingale relative to {Fn , n::::: I}. Since E(1n)
=
p(~)
i
fndp :S
p(~)
i
adp :S a
< 00,
by the Martingale Convergence Theorem [2], we conclude that lim f n (x) exists almost everywhere and converges to f (x). Therefore f(x) :S a almost everywhere.
o REMARK 4.2. Part (2) of the preceding lemma could be proved without using the Martingale Convergence Theorem. One proof is as follows: Let fn be as in the lemma and define the linear operator Tn : Ll (p) ---> Ll (p) by Tn (1) = f - fn- It is easy to see that Tn (1) ---> 0 for every uniformly continuous function. Therefore {J E Ll (p) : IITn (1)11 ---> o} is a closed linear subspace of Ll (p) that contains every uniformly continuous function. Since the set of uniformly continuous functions is dense in Ll (p) we conclude that Ilf - fnlll ---> 0 for every fELl (p). Therefore fn ---> f a.e .. DEFINITION 4.3. In the application of the preceding lemma to a BMO triple (X, p.I), we will insist that the partitions An are related to I (modulo sets of measure 0). Suppose B is a Borel subset of X, and M > 1. We say that B is M-divisible if there is a null sequence {An} of partitions of B such that
(1) Ao = {B} (2) for every n ::::: 1 and every A E An and C E An+l' with C C A, we have p (A) :S Mp (C) (3) for every n ::::: 1 and every A E An there is an I E I such that I C A and p (A) :S Mp (1). LEMMA 4.4. Let (X, p.I) be a BMO triple and suppose I E I is M -divisible with respect to a null sequence {An} of partitions of I such that each An C I . Let fEU (1) be a positive function such that 1(1) < a. Then there is a finite or infinite sequence {Ij } of disjoint subsets of I in I such that
(1) f:S a almost everywhere on 1\ Uj I j , (2) a :S I j (1) < Ma, (3) LP(1j) :S ~p(1)I(1). PROOF. Let El = {J E Ai : J (1) ::::: a}. And for each n ::::: 1 let
en +, ~ en U { J E A n+, , J (f) 2 aand J n C~" A)
~0}
and let E =Un>lEn = {Il,h, ... }. Statement I follows from Lemma 4.1. For each J E E there is an n ::::: 1 and an A E A n - 1 \E such that J E En and J C A. Then J (1) ::::: a and a > A (1) :::::
DON HADWIN AND HASSAN YOUSEFI
88
IL(~) [JL (J) J (f)] , which implies statement 2. Statement 3 follows immediately from
statement 2.
0
We now prove a general version of the John-Nirenberg Theorem [7]. Our proof is very close to the one in [3] THEOREM 4.5. Let cp E BMO(I, JL) and let J be an M -divisible Borel subset of X. Then for every A > 0 and every a ~ 1,
JL {t
E
-In(a)A
J : Icp(t) - J(cp) I > A} :::; aJL (J) exp ( 6M21IcpIIBMO(I,IL)
)
.
PROOF. Note that if {An} is a null sequence of partitions of a Borel subset of X and if we replace I with .J = I U Un>l An, then it follows from Proposition 4 that BMO(I, JL) = BMO(.J, JL) and 11 1 there exists a constant Ap such that
~~~
(J.LtI) 11 'P - 1('P) IP dJ.L) liP:=:; Ap II'PIIBMO(I,/Ll .
(2) The converse of the John-Nirenberg theorem is also valid. In other words suppose 'P is an integrable function on every I E I . If there are constants C and c such that VI E I, ::lCI E C such that
J.L {tEl: 1'P(t) - cII > A} :=:; Ce-c>-J.L(I) for every A > 0, then 'P E BMO(I, J.L). (3) If X = ]R2, J.L is Lebesgue measure, and I is the set of all disks, then every equilateral triangle is 4-divisible. We can partition a triangle into four triangles by joining the midpoints of the sides. With a little more work it can be shown that every disk is M -divisible for some M > 1. (4) We could obtain more precision by choosing {3, M > 1 and saying that a Borel set B is (M, {3)-divisible if in Definition 4.3 we replace M with {3 in statement 3. In this case the right hand side of the John-Nirenberg inequality would replace M2 with M {3. In the triangle case in the preceding remark, we would get that every equilateral triangle would be (4, 3~)_ divisible, so the M2 = 16 could be replaced with M {3 = 4 3~ ::::; 6.6159. (5) If X is a circle (interval) with I the set of open arcs (intervals), and if J.L is any finite continuous (Le., J.L {(x)} = 0 for every x) measure whose support is X, then every arc (interval) is 2-divisible; therefore the John-Niremberg theorem holds in BMO(I, J.L). (6) If in our John-Nirenberg theorem we have J E I and each An C I , then 40:M2 can be replaced with 20:M in the inequality.
5. Complements of VMO The main result of this section is that the space V MO(I, J.L) is never complemented in BMO(I, J.L). The proof is based on a lemma that is adapted from
[4]. LEMMA 5.1. Suppose W is a normed space that has an uncountable subset B whose elements are linearly independent, and that there exists M > 0 such that for every Xl, x2, ... , Xn in B and every 0:1,0:2, ... , O:n E te,
Suppose also that Y is a topological vector space with continuous linear functionals ---> te, that separate the points of Y. Then there is no injective continuous linear map f : W ---> Y.
'P1, 'P2, ... : Y
DON HADWIN AND HASSAN YOUSEFI
90
PROOF. Suppose, via contradiction, that a map 1 exists. For every n the map 'Pn 01 is a bounded linear functional on W. Let En,k = {x E B : I'Pn (f (x)) I 2:: Since the function 1 is 1-1 and the elements of B are linearly independent, then B = U En,k. Thus there exist no and ko such that Enu,k o is uncountable. Choose disk,n tinct elements Xl, X2, ... E Eno,k o and, for the sake of simplicity, define 'Pno (f(Xk)) =
t} .
n
rk eiOk , X =
L e-
iOk Xk·
Then
Ilxll :::; M
and for every n we have:
k=l
n
M
II'Pnu 01112:: Ilxllll'Pno 01112:
l'Pno(f(x))1 =
L
rk
k=l
2: ; , 0
o
which is a contradiction. THEOREM 5.2. There is no injective continuous linear map
'P: BMO(I, p,)/V MO(I, p,)
---+
BMO(I, p,).
In particular, V MO(I, p,) is not complemented in BMO(I, p,). PROOF. By Lemma 2.4 the points of BMO(I,p,) are separated by count ably many continuous linear functionals. By Lemma 5.1 it is enough to find uncountably many functions on BMO(I, p,)/V MO(I, p,) that are linearly independent and that satisfy in an inequality as in Lemma 5.l. To do so, suppose x E X. By using the second property of I, choose In in B(x; ~ )\B(x; n~l)' and, by the regularity of p" choose compact subsets An and Bn of In so that 1
P,(An) ;.::; P,(Bn) ;.::; 2P,(In). Since In" converges to" {x}, the sets A = Un:::: 1 An and B closed subsets of the space X\ {x}. Define Px on X by Px(x) =0, andpx(Y) = d(
y,
d (y, A) A) d(
+
y,
= Un:::: 1 Bn
are disjoint
B) \fYEX\{X}.
Then the function Px is bounded by 1 (and so belongs to BMO(I,p,)), PxlA = 0, and PxlB = 1 (and so Px 1:- VMO(I,p,)). Thus Px is a nonzero function in the quotient space BMO(I,p,)/VMO(I,p,). It is also easy to see that the function Px is uniformly continuous on X\B (x; c) for every c > O. The set
B ={Px : x
E
X}
is an uncountable subset of BMO(I, p,)/V MO(I, p,) whose elements are linearly independent. By Lemma 2.4, every uniformly continuous function is in V MO(I, p,) and so Px, as a function in BMO(I, p,)/V MO(I, p,), is zero everywhere except on B(x; f) for every f > O.This fact can be used to show that:
t
li k=l ak PXk I BMO(I,/-L)/V MO(I,/-L) : :; 3 max {jail , la21 , ... , lanl} , for every PX1' PX2' ... , PX n in B and every proof.
ai, a2, ... , an
E C. This completes the 0
The following Corollary follows from Proposition 2. COROLLARY 3. C u (X) n L OO (p,) is not complemented in L oo (p,).
SMO AND VMO
91
References [1] R. Coifman and G. Weiss, Extensions of Hardy Spaces and their uses in Analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. [2] J. Doob, What is a Martingale?, Amer. Math. Monthly 78 (1971),451-463. [3] J. Garnett, Bounded Analytic Functions, Academic Press INC., 1981. [4] L. Ge, D. Hadwin, Ultraproducts of C'-algebras, Operator Theory: Advances and Applications, 127 (2001), 305-326. [5] D. Hadwin, Continuity Modulo Sets of Measure Zero, Mathematica Balkanica, Vol. 3 (1989), 430-433. [6] F. John, Rotation and Strain, Comm. Pure Appl. Math. 14 (1961), 391-413. [7] F. John and L. Nirenberg, On Function of Bounded Mean Oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. [8] K. Peterson, Brownian Motion, Hardy Spaces and Bounded Mean Oscillation, Cambridge Univ. Press, Cambridge, 1977. [9] D. Sarason, Functions of Vanishing Mean Oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. [10] D. Sarason, Function Theory on the Unit Circle, Virginia Poly. Inst. and State Univ., Blacksburg 1978. MATHEMATICS DEPARTMENT, UNIVERSITY OF NEW HAMPSHIRE
E-mail address: don Lq(m) is bounded. The converse is false, as shown by Hunziker's theorem and the simple example p = q, \[I = 1, and (z) = z. Let dA(z) denote normalized area measure on the disc. For Q > -1 and p::::: 1, the weighted Bergman space A~ is the set of functions analytic in the disc satisfying
I J II~p
"
Also note that for
Q
=
=
j
D
I J(z) IP
(1- I z
12 r'
dA(z)
'(eiO) II : : : C I w(e ili ) I (1- I If> * (e ili ) 1)-(Q+2)/P holds a.e. [m]. It follows that W/(1- I If> * 1)-(Q+2)/p E Lq(m). iO
o
COROLLARY 1.5. Fix a ::::: -1, p ::::: 1 and q > O. Let n be a natural number. The following are equivalent. (1) WW,4> : A~ ---> Lq(m) is order bounded. (2) Ww,4>n : A~ ---> Lq(m) is order bounded.
Since 11.4 gives the result. PROOF.
I If> * (e ili ) I:::;
1-
I (If>*)n(e ili ) I:::;
n(l-
I If> * (e ili )
I), Theorem 0
Let a ::::: -1 and p, q > O. T. Domenig [1] proved that C4> : A~ ---> Lq (m) is order bounded if and only if 1/(1- I If> * 1)(Q+2)/p E Lq(m). His result is recovered here as the case W = 1. As a consequence of Theorem 1.1 (Hunziker) and Domenig's theorem, C4> : A~ ---> Lq(m) is order bounded for fixed a > -1 if and only if C4> : HP ---> L(Q+2)q(m) is order bounded. A version of this result is possible for the weighted composition operator WW,4>. Suppose that W E LOO and Ww,4> : A~ --->
R. A.
96
Lq(m) is order bounded for some Lq(m). It follows that {27r (
io
I \II I (1- I *
l)l/p
0:
HIBSCHWEILER
> -1. By Theorem 1.4, \II /(1- I * 1)(+2)/p E
)(+2)q dm < I \II I (+1)q (Xl
{27r
io
I \II Iq dm < (1- I * 1)(e>+2)q/p
00
.
Thus \II /(1- I * I)l/p E L(+2)q(m). By Theorem 1.4, WI]!,q, : HP ---+ L(+2)q(m) is order bounded. Suppose that WI]!,q, : HP ---+ L(l3+2)q(m) is order bounded for some (3 > -1 and \II is bounded away from 0, that is, there is a positive constant C such that C ::;1 \II(e iO ) I a.e. [m]. An argument using Theorem 1.4 implies that WI]!,q, : A~ ---+ Lq(m) is order bounded. The details are omitted.
2. Weighted Dirichlet Spaces For I > -1, the weighted Dirichlet space D"I is the Hilbert space of analytic functions f = L~=o anz n , (I z 1< 1) with
- ~ I an 12 < II f 11 2D-Y-~(n+l)"I-l
00
.
The functions e"l,n = (n + 1)("(-1)/2 zn, n = 0,1,2, ... are an orthonormal basis for D"I' Note that D1 is the Hardy space H2 and D"I = A;_2 for I > 1. The operator T : D"I ---+ H2 is Hilbert-Schmidt if and only if (Xl n=O
In [13], J. H. Shapiro and P. Taylor proved that Cq, : H2 ---+ H2 is Hilbert-Schmidt if and only if 1/(1- I * I) E L1(m). H. Jarchow and R. Riedl [7] proved that for (3 > 0, Cq, : D/3 ---+ H2 is Hilbert-Schmidt if and only if Cq, : HP ---+ LP/3(m) is order bounded for every p ~ 1. These ideas will be expanded here to the setting of the weighted Bergman spaces. In the rest of this section, will denote an analytic self-map of D such that I * (e iO ) 1< 1 a. e. [m].
°
THEOREM 2.1. Let 0: ~ -1, (3 > and I = (0: + 2)(3. The following are equivalent. (1) Cq, : A~ ---+ LP/3 (m) is order bounded for some (Jar all) p > 0. (2) Cq, : D"I ---+ H2 is Hilbert-Schmidt. PROOF. Because of Jarchow and Riedl's result, it is enough to prove the corollary in case 0: > -1. Suppose that Cq, : A~ ---+ Lp/3 (m) is order bounded. Domenig's theorem yields 1/(1- I * I) E L(e>+2)/3(m). Hunziker's theorem now yields that Cq, : HP ---+ L(+2)p/3(m) is order bounded. Therefore Cq, : D"I ---+ H2 is Hilbert0 Schmidt. These steps can be reversed to prove the remaining implication.
If \II E L(Xl (m) and if there is a positive constant c such that the inequality c ::;1 \II I holds a.e. [m], then a result analogous to the previous corollary holds for the operator WI]!,q,. The statement is omitted. Fix I > and let (1 - z)-"I = L~=o Anh)zn, I z 1< 1. By Stirling's formula, Anh) ~ (n + 1)"1-1 as n ---+ 00.
°
THEOREM 2.2. Suppose that k E N,o: ~ -1 and \II E L 2k (m). Fix p ~ 1 and let I = 2k(0: + 2)/p. The following are equivalent.
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
97
(1) W W ,4> : A:::' _ L 2k (m) is order bounded. (2) Wwk,4> : D, - H2 is Hilbert-Schmidt. PROOF.
By Theorem 1.4, W w,4> : A:::' - L 2k (m) is order bounded {=}
{=}
(1-
I cI>*
2k
III 12)(o+2)/p E L
1a27r I III 12k ~ An(r) I cI>* 12n
(m) dm
: D, - H2 is Hilbert-Schmidt.
o 3. Weighted Banach Spaces
In this section a connection is drawn between ;3-order boundedness of W W ,4> on the Bergman spaces and on certain Banach spaces defined through the use of a weight function v. DEFINITION
3.1. A weight is a non-increasing, continuous function v:
with the properties v(r) >
[0,1]- R
a for a ::; r < 1 and v(l) = O.
For zED, the notation v(z) will be used to denote v(1
Z
I).
DEFINITION 3.2.
H:; = {f
E
H(D) :
II J Ilv =
sup zED
I J(z) I v(z) < oo}.
For any weight v, H:; is a Banach space. In what follows we will be interested in weights of the form v (r) = (1 - r) k, k > O. A more general version of the following result is due to A. Montes-Rodriguez [11]. LEMMA
3.3. Fix k >
a and let v(r)
= (1-
r)k. For z
E
D,let Ez(f) = J(z) Jor
J E H:;. Then PROOF.
Fix zED. Then
I Ez(f) I = I J(z) I v(z)
v(z) ::;~. v(z)
For the remaining inequality, consider the function Jo(w)
= (l-zw)-k (w
ED).
0
Theorem 3.5 will establish a connection between order bounded ness and boundedness on the spaces H:;. The following lemma is needed in the proof. LEMMA 3.4. For k > only on k such that
a and a ::; r < 1, there is a positive constant C depending
R. A. HIBSCHWEILER
98
PROOF. Let I denote the sum in the previous expression. For n = 0,1,2, ... let In = {m E Z : 2n - 1 :s:; m < 2n +l - 1}. Since there are exactly 2n terms in In, it follows that
n=O
mEln
If mE In, then (m + 1)/2 < 2n :s:; m (X)
L
I ~ r2k~) n=O
+ 1.
It follows that
(m + 1)2k-lr2m+2) for 0 :s:; r < 1.
mEln
Stirling's formula now implies that (X)
I ~ C r2 r2k
L
An(2k) (r 2)n
n=O
o Recall the assumption that I 4>*(e i l.l) I < 1 a. e. [m]. The proof of Theorem 3.5 will use Khinchine's inequality. A statement can be found in Luecking's paper [8]. THEOREM 3.5. Fix k,q > 0 and let v(r) = (1- r)k. Let 0 =I- III following are equivalent.
E
Lq(m). The
(1) WW, : H:;' ---t Lq(m) is order bounded. (2) Ww, : H:;' ---t Lq(m) is bounded. (3) III /(v 0 4>*) E U(m). It will be shown that (1) =} (2) =} (3) =} (1). First assume that WW, : H:;' ---t Lq(m) is order bounded. Thus there exists a positive function h E Lq(m) such that PROOF.
I llI(eil.l) f(4)*(e i l.l)) I :s:; h(eil.l) a.e. [m] for all
f
E
H;:' with
II f I v:s:; II 1lI(J
0
for
f
E
1. It now follows that
4>*)
IILq(m)
I f
II h IILq(m) I f Ilv
H:;'.
Next assume that WW, : H:;' constant K such that
for all
:s:;
E
---t
WW,(J)
Lq(m) is bounded. Thus there is a positive
IILq(m)
:s:; K
I f Ilv
H;:'. Let (X)
f(z) =
L2
kn z 2n ,
Iz I
£p{3(m) is order bounded.
100
R. A. HIBSCHWEILER
Putting W = 1 in Corollary 3.7 yields the Bergman space analogue of Jarchow and Riedl's result, mentioned above. The proof of the corollary is omitted. COROLLARY 3.7. Fix a: ~ -1 and (3 > O. Suppose that W E L 00 (m) and there is a constant c such that c ~ 1 w(e iB ) 1 a.e. [m]. The following are equivalent. (1) Ww, : H~,p ~ p!3(m) is order bounded for some (Jor all) p ~ 1. (2) WW, : A~ ~ p!3(m) is order bounded for some (for all) p ~ 1. 4. A Characterization of Boundedness and Compactness In [6], Hunziker and Jarchow found relationships between order boundedness, boundedness and compactness of the operator C on the Hardy spaces. Analogous results are given here for Ww, on the Bergman spaces. THEOREM 4.1 (Hunziker and Jarchow). Lp!3 (m) is order bounded for some p ~ 1, for all p ~ 1. (2) If C : HP ~ HVY is bounded for some C : HP ~ p!3(m) is order bounded for
(1) If (3 ~ 1 and C : HP ~ then C : HP ~ Hp!3 is compact p ~ 1 and 0 < (3 < 'Y - 1, then all p ~ 1.
The converse of assertion (1) is false. To see this in the case (3 = 1, note that J. H. Shapiro and P. Taylor proved that there are compact composition operators C : H2 ~ H2 which are not Hilbert-Schmidt [13]. By Theorem 3.1 [13], it follows that 1/(1- 1 * I) (j. Ll(m). By Hunziker's result, stated here as Theorem 1.2, C : HP ~ LP(m) is not order bounded. In Corollaries 4.2, 4.3 and 4.4, recall the assumption that is a self-map of D with 1* 1< 1 a. e. [m]. COROLLARY 4.2. Suppose that a: ~ -1, (3 > 0 and C : A~ ~ p!3(m) is order bounded. If (a: + 2){3 ~ 1, then C : HP ~ HP(o.+2)!3 is compact for all p ~ 1. PROOF. Since C : A~ ~ p!3(m) is order bounded, Domenig's theorem [1] yields 1/(1- 1* I) E L(o.+2)!3(m). Hunziker's Theorem (Theorem 1.2) implies that C : HP ~ p(o.+2)!3(m) is order bounded for all p ~ 1. The result now follows by Theorem 4.1 (Part 1). 0
R. Riedl [12] used the classical Nevanlinna counting function to characterize self-maps which induce bounded or compact composition operators C : HP ~ Hq in the case 0 < p ~ q. In [14], W. Smith used the generalized Nevanlinna counting function to characterize bounded or compact composition operators C : A~ ~ Ah in the case 0 < p ~ q. These results will expose further connections between order boundedness, bounded ness and compactness. For a self-map of D, W =I=- (0) and 'Y > 0, N,(w) =
I) log(l/
1
z I) )'
where the sum extends over all z with (z) = w, counting multiplicities. Thus the classical Nevanlinna counting function is Nl (w). Riedl [12] proved that C : HP ~ Hq is bounded in the case 0 < p ~ q if and only if
Nl(w) = O( (1- 1w I)q/p ),
1w
I~ 1.
Smith [14] showed that for 0 < p ~ q, C : A~ ~ Ah is bounded if and only if
N!3+2(W) = O( (1- 1w l)(o.+2)q/p ),
1w
I~ 1.
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
101
The analogous statements hold for compactness if the 'big-oh' condition is replaced by'little-oh'. If 0 < a < , and wED, w i- (O), then N,(w)::; (Nrr(w))'/rr. If is of finite valence, then there is a constant C such that Nrr(w)::; C(N,(w))rr h . Thus (N,(w))rr ~ (Nrr(w))' for such functions. See [14] for a discussion of these inequali ties. COROLLARY 4.3. Let p
~
1, Q:
~
-1 and (3
>
O. Suppose that C : A~
U!3 (m) is order bounded and (Q: + 2){3 ~ l. Then C : A~
for any,
~
-7
-7
A~(a+2)!3 is compact
-l.
PROOF. Since C : A~ -7 LP!3(m) is order bounded, Domenig's result yields 1/(1- 1 * I) E L(a+2)!3(m). By Theorem 1.2, C : HP -7 U(Q+2)!3(m) is order bounded. Since (Q:+2){3 ~ 1, Theorem 4.1 (Part 1) yields that C : HP -7 HP(Q+2)!3 is compact. This completes the proof in the case, = -1. Let, > -1. Riedl's characterization yields
Nt(w) = o( (1- 1 w I)(Q+2)!3 ) as 1w
1-7 1.
Since, + 2 > 1, the remarks before the corollary yield
N,+2(W) = o( (1- 1 w I)(Q+2)!3(T+2) ) as 1w
1-7 1.
By Smith's characterization, C : A~
-7 A~(Q+2)!3
0
is compact.
COROLLARY 4.4. Let Q: ~ -1 and {3 > O. Suppose that is of finite valence and C : A~ -7 A~' is bounded for some p ~ 1 and, > (3 + 1. Then C : A~ - 7 U!3/(Q+2) (m) is order bounded for all p ~ l. PROOF. Because of Theorem 4.1 (Part 2), we may assume Q: C : A~ -7 A~' is bounded,
N + 2 (w) = O( (1- 1w Q
> -l. Since
1)(Q+2)-y ) as 1w 1-7 1. = 0 ( (1- 1wi)' ) as 1w 1-7 1 and thus
The valence hypothesis now yields N 1 ( w) by Riedl's result C : HP -7 HP' is bounded. Since {3 < , - 1, Theorem 4.1 (Part 2) implies that C : HP -7 L!3P(m) is order bounded, and thus 1/(1- * 1)!3 E Ll(m). 1
By Domenig's Theorem [1] this is equivalent to order bounded ness of C : L!3p/(Q+2) (m).
A~
-7
0
In the remainder of this work we assume that III is analytic in D and is an analytic self-map of D. The closing results characterize the weighted composition operators Ww, : A~ -7 Ah which are bounded or compact. Related results for C were given by W. Smith [14] in the case 0 < p ::; q, and by Smith and L. Yang [15] in the case 0 < q < p. Let Q: > -1 and let dAQ(z) denote the measure (1- 1z 12)Q dA(z). Smith and Yang showed that if q < p and Q: > -1, then C : A~
-7
Ah is bounded
¢:}
C : A~
-7 Ah
is compact
102
R.
A. HIBSCHWEILER
Let a E D. In the rest of this section, D(a) denotes the pseudohyberbolic disc centered at a with radius 1/8, that is,
a-z D(a) = {z: I -_- I < 1/8}. 1- az The following lemma is well known. LEMMA 4.5. (2) 1- I w 12
(1) 11 - aw I: : : 1- I a 12 : : : 1- I a 12 for wE D(a).
for wE D(a).
THEOREM 4.6. Let 1 S p S q and let (x, (3 > -1. Assume that W E Ah and let be an analytic self-map of the disc. The following are equivalent. (1) Ww,
D is analytic, then ip E Ca for all a ~ 1. Theorem 1 was proved by Bourdon and Cima in [2] when a = 1 and by Hibschweiler and MacGregor in [8] when a > 1. The arguments rely on the classical result of Herglotz and Riesz about functions having a positive real part and on a 1 generalization concerning the family of functions subordinate to F(z) = ( ) 1- z a Theorem 2. If ip E Ca and (3 > a then ip E C/3. Theorem 2 was proved by Hibschweiler in [7]. A critical step in the argument uses Theorem 1 with a > 1. Theorem 3. If ip is an analytic function that maps D one-to-one onto D, then ip E Ca for all a > o. Theorem 3 was proved by Hibschweiler and MacGregor in [8]. This property of conformal automorphisms of D serves as a lemma for various arguments.
3. Necessary conditions As mentioned earlier, Ca C Fa. Since the Taylor coefficients of a function in Fa satisfy (3)
this yields examples of analytic functions ip : D
---->
D which do not belong to Ca.
00
If 0 < a < 1 we may let ip(z) =
I)nznP
where p is a positive integer depending
n=l 00
on a and {b n } is a suitable sequence with
2:: 1b
n
I::;
1. Likewise (3) implies that
n=l 00
ip ~
Ca for all a(O < a < 1) when ip(z) =
t:
2:: r&z2n (I z 1< 1) and
t:
is sufficiently
n=l
small and t: =1= o. The Taylor coefficients of members of Ca also satisfy the following condition. 00
Theorem 4. If 0
< a < 1, ip E Ca and
ip(z)
= 2::anzn(1
z
1< 1), then
n=O 00
(4) n=O
Theorem 4 is a consequence of the more general result in [6; see Theorem 1, p. 163] that (4) holds if 0 < a < 1 and ip E Hoo n Fa.
4. The case IlipllHoo < 1 For a > 0 let Ba denote the set of functions (5)
11 L:
1
!'(re iIJ )
1
f
that are analytic in D and satisfy
(1 - r)a- 1 d() dr
nzn(1 z 1< 1) and n=O
00
2:(n + 1)1-0: I an
(8)
1< 00.
n=O Then Y? E B(3 for (3 > Q. Proof of Theorem 7: If I > 0 then there is a positive constant A such that
Prk < A - (l-r)'Y
( ) 9 for 0 then
I
< r < 1 and k = 1,2, ....
This implies that if 0
I::::
Q
00
00
y?'(z)
< < 1, I z 1= rand 0 < r < 1
2: n
I
an
I
::::1
r n- 1
al
I
+20:2:(n - 1)O:r n- 1n 1-0:
n=l
I
an
1::::1
a1
I
n=2
20: A 00 ~ 1-0: (1 - r)o: L- n n=2 I an I· Hence the assumption (8) implies that there is a positive constant B such
+
that
I y?'(z) I::::
(1 !r)o:' If (3 >
tJ7r I y?'(re
Jo
ilJ )
I (1 -
Q
this yields
r)(3-1dBdr :::: B
-7r
t (1 - r)(3-0:-1dr
Q. 0 The following theorem gives a sufficient condition for membership in Co: for all Q > 0 and only depends on the Taylor coefficients. 00
Theorem 8. Suppose that the function Y? : V z
1< 1)
---t
V is given by y?(z) = 2:anzn(l n=O
and 00
2:(n + 1) I an
(10)
n=O Then Y? E Co: for all
Q
> O.
1< 00.
llO
T.H. MACGREGOR
Theorem 8 was proved in [10; see p. 200]. The argument relies on Theorem 1 with a > 1 and on a result about the multipliers of :Fa involving Taylor coefficients. It is not known whether (8) implies rp E Ca where 0 < a < 1. Our knowledge of which univalent functions belong to Ca is quite limited. The main fact is stated below. It was proved in [10; see Theorem 9.10, p. 214] using Theorem 5. Theorem 9. Let ao = ~ - 3~0. If the function rp is analytic and univalent in V and sup I rp(z) 1< 1 then rp E Ca for all a > ao· Izl 0, rp E Ca and 'ljJ = brp where I b I:::; 1. Then'ljJ E Ca· Proof of Theorem 10: Suppose that a > 0 and I b I:::; 1. For I z 1< 1 let F(z) = 1 ( ) ' and let I ( 1= 1. Since F is analytic in V there is a probability measure
1-z a J.t E M such that
F(b(z)
=
J
F(o-z)dJ.t(a)
T
for
I z 1< 1 [10;
see p.
21].
The equation
(1 _ 1(bz)a =
J(1-
1
az)a dJ.t(a) (I z
1< 1)
T
and the fact that J.t is a probability measure imply that Suppose that rp E Ca and let M
= Ilrplle.,.
1
II (1 - (bz )a IIF" = 1.
Then F(b(rp(z)) E :Fa and 1
IIF(b(rp(z))IIF~~.~. :::; MIIF(b(z)IIF~ = M. Let'ljJ = brp. We have ( 1 - ('ljJ ) a E:Fa
I (1 _ ~'ljJ)a IIF" :::; M. The last inequality holds for all ((I ( 1= 1). that f 0 'ljJ E :Fa for all f E :Fa [7; see p. 59]. Therefore'ljJ E Ca. 0
and
This implies
Let Ma denote the set of functions f such that fg E :Fa for every 9 E :Fa. If f E Ma then the mapping 9 f--t fg is a continuous linear operator on :Fa. We let IlflIM" denote the norm of this operator. Since the constant function 1 belongs to :Fa for every a > 0, we obtain Ma C :Fa. The family of multipliers Ma has been extensively studied [10; see Chapters 6 and 7]. Members of Ma have a number of properties including being bounded. Theorem 11. Suppose that a > 0, f E M a , f -=f. 0 and b is any complex number 1 such that I b 1< IlflIM", . Then bf E Ca. Proof of Theorem 11: Let M = I filM" . The assumption f -=f. 0 implies that M > o. We have IlfgllF" :::; MIIgIIFn for all 9 E :Fa. The constant function 1 belongs to :Fa and 11111F" = 1. Hence f E :Fa and IlflIF" :::; M. Also P E :Fa and IIPIIF" :::; MllfllF" :::; M2, and, in general, r E:Fa and IlrllFn :::; Mk for k = 1,2, ....
FRACTIONAL CAUCHY TRANSFORMS AND COMPOSITION
Suppose that cients defined by
I b 1
w
112
T.H. MACGREGOR
exists and equals (J(w). If A is non-vacuous then either A is finite or A is countably infinite, and
L
(12)
wEA
1 'Y(w)
~
1+ 1 ip(O) 1 1- 1ip(O) I'
Except for the last sentence in Theorem 12, this result and related facts due to Julia can be found in [1; see p. 7], [3; see p. 23] and [5; see p. 43]. We present an argument which also yields the last assertion. Proof of Theorem 12: Suppose that the function ip : V -7 V is analytic and let 1(j 1= 1. For 1z 1< 1 let
1 p(z) = 1 - (jip - (z )
(13)
Set b = Rep(O) and c = Imp(O). The function p is analytic in V and Rep(z) > 1/2 for 1 z 1< 1. The function
b- 1
(14)
q = 2b - 1
is analytic in V, Req(z) > 1/2 for 1z yields
1< 1 and q(O)
where /-L is a probability measure on T. Let e e > 0 and (15) and (14) yield
(16)
p(z)
=e
1 =
1. The Herglotz-Riesz formula
(I z 1< 1) iTr ~d/-L(() 1- (z
q(z) =
(15)
p - ic
+ 2b -
= 2b -
1 and
f = 1 - b + ic. Then
__ d/-L(() + f (I z 1< 1). iTr_1 1- (z
Suppose that wET. Then
(17)
=e
(1 - wz)p(z)
iTr 11 --
~z d/-L(() + f(1 (z
- wz)
for 1z 1< 1. Let S denote a Stolz angle in V with vertex w. The integrand in (17) is bounded for ( E T and z E S and it equals 1 if ( = wand it tends to zero as z -7 w if ( 1= w. The bounded convergence theorem yields
(18)
lim(l-wz)p(z)
z->w
=
e/-L({w})
where z E S. Since e > 0 and /-L is a non-negative measure, this limit is a nonnegative real number and it is zero if and only if /-L( {w}) = O. Let
(19) If A is non-vacuous then either A is finite or A is countably infinite. ip(z) - (j W(j From (13) we obtain ( _ ) ( ). Hence what was shown above z- w 1- wz p z about lim (1 - wz)p(z)
z->w
CXl or ~(j}) dependz- w e/-L w 0 or /-L( {w }) > 0, respectively. Hence the set A defined
implies that the non-tangential limit lim ip(z) - (j is either z->w
ing on whether /-L( {w })
=
FRACTIONAL CAUCHY TRANSFORMS AND COMPOSITION
by (19) has the properties stated in the theorem.
113
We see that if w E A then
1
,( w) = ep,( {w }) > O. Also '"' _1_ ~ ,(w)
= '"' ep,( {w}) < ep,(T) = e = ~
-
Re{ 11 +- iTCP(O)} < 1+ iTcp(O) - 1-
1 1
cp(O) I. cp(O) 1
This proves (12). Finally we prove the assertion about the non-tangential limit of cp'. From (13) we obtain cp'(z)
=
;'(~zi
and hence (16) yields
ue cp' (z) = [e
h
(1 -((z)2 dp,(()
r _1__(z dp,(() + J]
iT 1 -
2·
This can be written
r((1 -
ue ~Z)2 dp,(() cp' (z) = ___i:....:7T'----'---I_-_('-z-'--_ _ _-". [e
r 1 - ~z dp,(() + J(1 _ WZ)] (z
iT 1 -
2·
Let w belong to A and let S be a Stolz angle in V with vertex w. Then both integrands in the last expression are bounded on S. The bounded convergence theorem yields
'() · I Imcp z = uewp,({w}) = (3() w
[ep,({w})]2
z->w
where z E S. 0 Theorem 13. Suppose that 0 < 0: < 1 and cp E Co. Let u E T and let A and ,( w) be defined as in Theorem 12 where 1w 1= 1. If A is non-vacuous then
(20) Proof of Theorem 13: For 1z
1< 1 let J(z)
= [
1
_ (
1 - ucp z
.
)t· The assumptlOn cp E Co
implies there exists v E M such that
J(z)
(21) for 1z
1< 1.
Let wET and let 0
=
iTr (1-1) (z
< r < 1.
dv(() 0
Then (l-r)O J(rw)
and the bounded convergence theorem yields
(22)
lim (l-r)oJ(rw)
r--+ 1-
=
r ] iTr [1 1- -r(w
lim (l-r)oJ(rw)
r--+ 1-
dv(()
= v({w})
Let the function p be defined by (13). Then (1 - r)O J(rw) Hence (18) yields
(23)
0
= [ep,({w})t
= [(1 - r)p(rw)t.
114
T.H. MACGREGOR
where e and J.l have the same meaning as in the proof of Theorem 12. From (22) and (23) we obtain
I/({W}) = [eJ.l({w})]'"
(24)
= eJ.l( ~W })
In particular, if W E A then 'Y( w)
L
and thus 'Y a (w)
= 1/( {~ }).
This gives
1 [ (w)]'" = LI/({w}). Hence
WEA'Y
wEA
1
~ b(w)]",
(25)
Ilg 0 and
the
= LOk. Let e be any real number such that e 2: 0
= -
00
for k = 1,2, .... Then
e probability measure on T such that
(31)
=
k=l
Ok
Ek
'Yk
for k
00
series LOk converges. Let 0 and let
1
=-
Ek
> 0 and'" Ek ~
k=l
0
= - :::; 1. Let
e
J.l be any
FRACTIONAL CAUCHY TRANSFORMS AND COMPOSITION
for k
=
1,2, .... For 1 z 1< 1 let
q(z) =
(32)
r _1(z__ dJ.L(().
iT 1 -
Then q is analytic in V, Req(z) > 1/2 for number and let p
1z 1
l-r,
11 - arl
1
> '211 -
ai,
Second, use induction to verify that for a sequence N
(2.5)
II (1 n=l
0 < r < 1, (bn)n~l C
N
bn ) ~ 1 -
Lb
n,
VN EN.
n=l
Third, one can verify, via a routine computation, the identity Ir - Un'12 = 1 _ (1 - r2)(1 - lan l2 ) 11 - a n rl2 11 - a n rl2
lal
< 1.
(0,1), we have
INDESTRUCTIBLE BLASCHKE PRODUCTS
121
and so
~ (1 - r2)(1 - la n l2)
~
1- n=l ~
=
1_
f
11 _ an r 12
'
(byeq.(2.5))
(1 - r2)(1 - lanl2)
n=l
11 -
anrl11 - anrl .
Now use the inequalities in eq.(2.4) and the dominated convergence theorem to get lim IB(r)1
(2.6)
r-+l-
=
1.
To finish, we need to show that lim argB(r)
r->l-
exists. Use the identity
to get
argB(r)
(2.7) If an =
~ arg {1- 1_1~12}.
=
~ 1- anr n=l an + i!3n, where a n ,!3n E JR, some trigonometry will show that
arg
(1 _1- lanl2) anr
= sin-1 (
1-
!3n r (l - la n l2) ) . lanllan - rill - anrl
From here, one can argue that the right-hand side of eq.(2.7) converges absolutely and uniformly in r and so lim arg B(r) r->l-
exists. Combine this with eq.(2.6) to complete one direction of the proof. See [3, p. 34] for the other direction. 2.8. (1) If the zeros (an)n~l do not accumulate at (, the condition in eq.(2.3) is easily satisfied and in fact, B extends analytically to an open neighborhood of ( [18, p. 68]. (2) The zeros can accumulate at ( and eq.(2.3) can still hold. For example, let tn 1 0 satisfy Ln tn < 00 and let
REMARK
a
n
l't = -12 + -e' 2
n
Notice how these zeros lie on the circle Iz - ~I = ~, which is internally tangent to 8][Jl at ( = 1, and accumulate at ( = 1. A computation shows that
122
WILLIAM T. ROSS
and so
lanl2 I1 _ an I:;:::
1_
00
L
,,=1
00
L tn
B*(() is discontinuous at (0 if and only if (0 E E. By Fatou's theorem, the radial limit function
¢*(():= lim ¢(r(), T-l-
for a bounded analytic function ¢ on ID, exists for m-alrnost every ( E aID [8, p. 6]. If I¢*(()I = 1 for almost every (, then ¢ is called an inner function and can be factored as (2.9) Here J-L is a positive finite measure on aID with J-L ..l m. The first factor in eq.(2.9) is the Blaschke factor and is an inner function. The second term in eq.(2.9) is called the singular inner factor. By a theorem of Fatou [8, p. 4], (2.10)
lim_ ,.~I
r
Ji:J1IJi
I( 1 -
ri2012 dJ-L(() re
= (DJ-L)(e iO )
whenever DJ-L(e iO ), the symmetric derivative of J-L at eiO , exists (and we include the possibility that. (D J-L) (e iO ) = (0). By the Lebesgue differentiation theorem, D J-L exists at m-almost every eiO • Moreover, since J-L ..l m, we know that
(2.11)
DJ-L = 0 m-a.e.
and
DJ-L =
00
J-L-a.e.
See [30, p. 156 - 158] for the proofs of eq.(2.11). The first identity in eq.(2.11), along with the identity (2.12)
lexp (
-fulIJi ~ ~ ~:::dJ-L(()) I = exp ( -fulIJi I( ~~;i:12 dJ-L(()) ,
shows that the radial limits of this second factor are unimodular m-almost everywhere and hence this factor is an inner function. Furthermore, if J-L ¢ 0 (i.e., the inner function ¢ has a non-trivial singular inner factor), we can use the second identity in eq.(2.11) along with eq.(2.12) once again to obtain the following theorem of Frostman [9].
INDESTRUCTIBLE BLASCHKE PRODUCTS
123
THEOREM 2.13 (Frostman). If an inner function ¢ has a non-trivial singular inner factor, there is a point ( E 8lIJ) such that ¢* (() = o. From Remark 2.8 (4), the condition ¢*(() = 0 for some ( E 8lIJ) does not completely determine the presence of a non-trivial inner factor. Another result of Frostman (see [9, p. 107] or [3, p. 32]) completes the picture. THEOREM 2.14 (Frostman). An inner function ¢ is a Blaschke product if and only if
1
27r
lim
(2.15)
T-d -
log 1¢(reili)ldB =
0
o.
Again, for the sake of giving the reader a feel for how all these ideas are related, and since this result will be used later, we outline a proof. We follow [3, p. 32]. Indeed, suppose ¢ = B, a Blaschke product. Let Bn be the product of the first n terms of B and, given t > 0, choose a large n so that
Thus,
o ~ T~rr-1a1D log IB(r()ldm(() = =
laID
lim_ T-+1
~
r 10giBB (r()i dm(()-
lim_ T-d
r
laID
n
i (roi
log BB
lim_ T-d
r 10gIBn(r()ldm(()
laID
dm(()
n
10g(1 - c).
The last inequality comes from the sub-mean value property applied to the subharmonic function log IBI Bnl [12, p. 36]. It follows that eq.(2.15) holds for ¢ = B. Now suppose that ¢ is inner and eq.(2.15) holds. Factor ¢ = Beg, where
g(Z):=-
r (+zd/-L(()
lalD(-z
and notice, using the fact that Rg is non-positive and harmonic along with the mean value property for harmonic functions, that if Rg has a zero in lIJ), then Rg == 0 on lIJ) and consequently /-L == o. Use the mean value property again to see that
r
laID
log 1¢(r()ldm(()
=
r
laID
log IB(r()ldm(()
+ Rg(O).
As r ----> 1-, the integral on the right-hand side approaches zero since B is a Blaschke product (see above) and the integral on the left-hand side approaches zero by assumption. This means that Rg(O) = 0 and so, by what we said before, /-L == 0 and so ¢ = B is a Blaschke product. This completes the proof. The linear fractional maps
Ta(Z)
z-a 1- az
:= - _ - ,
lal < 1,
WILLIAM T. ROSS
124
are automorphisms of IDl (the complete set of automorphisms of IDl is {(Ta : ( E 81Dl, a E 1Dl}) and also satisfy Ta(81Dl) = 81Dl. So certainly the Frostman shifts
<Pa
Ta
:=
1
~
r (r
J K JalTh
r (lim r
JK 1'->1- JalTh
(dominated convergence theorem)
log 11 - w~(r~) I dm(()) da(w) w - ¢ r( log 11 -
w~(r~) Idm(()) da(w)
w - ¢
r(
(Fubini's theorem) (Fatou's lemma)
> 0 (byeq.(2.19)) which is a contradiction.
o
Let us make a few remarks about the limits of Theorem 2.18. REMARK 2.20. (1) Frostman [9, p. 113] showed that if E is relatively closed in J])) and has logarithmic capacity zero, then there is an inner function ¢ with £(¢) = E (see also [3, p. 37] and the next two comments). (2) Recall from Proposition 2.17 and Theorem 2.18 that £(¢) is an Fa set of logarithmic capacity zero. The authors in [22] showed that if E c J])) is of type Fa and has logarithmic capacity zero, then there is an inner function ¢ such that £(1;) = E.
WILLIAM
126
T.
ROSS
(3) Suppose that E is a closed subset of lDl, 0 rf. E, and E has logarithmic capacity zero. We claim that there is a Blaschke product B such that Ba := Ta 0 B is a Blaschke product whenever a E lDl \ E and Ba is a singular inner function whenever a E E. To see this, let B be the universal covering map from lDl onto lDl \ E [7, p. 125]. Notice that B*() E im u E. First note that B is inner. Indeed, suppose that IB* () I < 1 for ( E A and m(A) > O. Then B*(A) c E and, since E has logarithmic capacity zero, we see that B == 0 [3, p. 37] which is a contradiction. Second, note that Ba is a Blaschke product for all a E lDl \ E. Indeed, Ba maps lDl onto lDl \ Ta(E) and 0 rf. Ta(E). Moreover, B~() E 8lDl U Ta(E) and so B~() can never be zero. An application of Theorem 2.13 completes the proof. Third, Ba is a singular inner function whenever a E E. To see this, note that B maps lDl onto lDl \ E and so a rf. B(lDl) which means the inner function Ba has no zeros. Thus Ba must be a singular inner function. (4) If one is willing to work even harder in the previous example, one can find an interpolating Blaschke product B such that Ba is an interpolating Blaschke product for all a E lDl \ E while Ba is a singular inner function whenever a E E [14, Theoerm 1.1]. In fact, the above proof is part of this one.
3. Indestructible Blaschke products From Frostman's theorem (Theorem 2.18), we know that the exceptional set of an inner function ¢ is small. A Blaschke product B is indestructible if qB) = 0. This next technical result from [21] helps show that indestructible Blaschke products actually exist.
q ¢)
PROPOSITION 3.1. If B is a Blaschke product such that B*() is never equal to a E lDl \ {O}, then B is indestructible. PROOF. Suppose that for some a E lDl \ {O}, Ba = Ta 0 B has a non-trivial singular inner factor. By Theorem 2.13, there is a ( E 8lDl such that B~() = O. However, for 0 < r < 1, 1 IBa(r()1 ~ "2IB(r() - al
and so, taking limits as r assumption.
~
1-, we see that B* ()
=
a, which contradicts our D
COROLLARY 3.2. If B is a Blaschke product whose zeros (an)n;;d satisfy (3.3)
~ l-Ian l ~ .,-----'------'- < n=l I( - ani
00
for every ( E 8lDl, then B is indestructible.
PROOF. By Theorem 2.2, tion 3.1.
IB*()I
= 1 for every ( E 1['. Now apply Proposi-
D
Certainly any finite Blaschke product satisfies eq.(3.3). The infinite Blaschke product in Remark 2.8 (2) also satisfies eq.(3.3) and thus is indestructible. Let us say a few words about the origins of the concept of indestructibility. The following idea was explored by Heins [15, 16] for analytic functions on Riemann surfaces but, for the sake of simplicity, we outline this idea when the Riemann
INDESTRUCTIBLE BLASCHKE PRODUCTS
127
surface is the unit disk. Our discussion has not only historical value, but will be useful when we discuss a fascinating example of Morse later on. If f : ]]J) ----. ]]J) is analytic and a E ]]J), the function z f---> -log Ifa(z)l, where fa = Ta 0 f, is superharmonic on ]]J) (i.e., log Ifal is subharmonic on ]]J)). Using the classical inner-outer factorization theorem [8, Ch. 2], one can show that
(3.4)
-log Ifa(z)1
L
=
-n(w) log ITw(z)1
+ 1La(z),
f(w)=a where n(w) is the multiplicity of the zero of f(z) - a at z = w, and Ua is a nonnegative harmonic function on]]J). The focus of Heins' work is the residual term Ua. His first observation is that Ua is the greatest harmonic minorant of -log Ifal. Moreover, since U a is a non-negative harmonic function on ]]J), Herglotz's theorem [8, p. 2] yields a positive measure /-la on 8]]J) such that
the Poisson integral of /-la. Heins proves that if /-la = Va + (ja is the Lebesgue decomposition of /-la, where Va « m and (ja -.l m, then the m-almost everywhere defined function 1- a qa(() := log a - f*(()
1 !*(()1
is integrable on 8]]J) and (3.5) In the general setting, and the actual focus of his work, Heins examines the residual term U a in Lindelof's theorem
G s , (f(z), a) =
L
n(w)Gs2 (z, w)
+ ua(z),
f(w)=a where Sl and S2 are Riemann surfaces with positive ideal boundary, f is a conformal map from Sl to S2, and GSj is the Green's function for Sj. To study the residual term U a in this general setting, Herglotz's theorem and the Lebesgue decomposition theorem are replaced by an old decomposition theorem of Parreau [27, TMoreme 12] (see also [17, p. 7]). When Sl = S2 = ]]J), observe that
I
z- a . G s (z, a) = -log --_1 1 - az
I
We state this next theorem in the special case of the disk but refer the reader to Heins' paper where an analog of this theorem holds for Riemann surfaces. THEOREM 3.6 (Heins). The functions Ua and PVa satisfy the following proper-
ties. (1) Either Pva(z) = 0 for all (a, z) E ]]J) x ]]J) or Pva(z) 1:- 0 for all (a, z) E ]]J) x]]J). (2) The set {a E ]]J) : Ua - PVa > O} is an Fa set of logarithmic capacity zero. PROOF. Observe that if a E ]]J) is fixed and Pva(z) = 0 for some z E ]]J), we can use the fact that PVa is a non-negative harmonic function along with the mean
128
WILLIAM T, ROSS
value property of harmonic functions to argue that Plla == 0, Thus, from eq.(3.5), we have, this particular a,
. Plla(r() = log 0= hm r->l-
l 1 - Cif!*(()1 (() , *
a-
a.e. (E alI)).
Whence it follows that I!*(()I = 1 almost everywhere, i.e., f is inner. The fact that f is inner along with the fact that Tb maps alI)) to alI)) for each b E II)) shows that lim Pllb(e) = 0
r---+l-
a.e. (E alI)).
Thus, from eq.(2.1O), we see that for each b E
II)) ,
0= lim Pllb(e) = Dllb(() r---.l-
a.e. (E alI))
and so lib ~ m. But since lib « m it must be the case that lib == o. Thus we have shown part (1) of the theorem. To avoid some technicalities, and to keep our focus on Blaschke products, let us prove part (2) of the theorem in the special case when Plla == 0 for some (equivalently all) a. Note that f is inner. If U a has a zero in II)) , then, as argued before using the mean value property of harmonic functions, U a == o. Recall from our earlier discussion that U a = m( -log Ifal), where m denotes the greatest harmonic minorant. If we factor fa = bg as the product of a Blaschke product b and a singular inner function g, one can argue that
m( -log Ifal) = m( -log Ibl)
+ m( -log Igl).
It follows from Theorem 2.14 and a technical fact about greatest harmonic mino-
rants [12, p. 38], that m( -log Ibl) == O. But since we are assuming U a == 0, we have m( -log Igl) == o. However, 9 has no zeros in II)) and so -log Igl is a non-negative harmonic function on
II))
and thus
0== m( -log Igl) = -log Igl· Hence 9 == eic , c E lR, equivalently, fa is a Blaschke product. Thus we have shown U a == 0 =? fa is a Blaschke product. If fa is a Blaschke product, then, as pointed out before, U a = m( -log Ifa I) == o. It follows that (3.7)
{a
ElI)):
u a > O}
= £(1).
Now use Proposition 2.17 and Theorem 2.18.
D
In summary, u a == 0 if and only if fa is a Blaschke product. Moreover, U a == 0 for every a E II)) if and only if f is an indestructible Blaschke product. Heins did not coin the term 'indestructible' in his work. McLaughlin [21] was the first to use this term and to explore the properties of these products. 4. Zeros of indestructible Blaschke products
McLaughlin [21] determined a characterization of the indestructible Blaschke products in terms of their level sets. Suppose ¢ is inner and a E II)) \ {¢(O)}. Let (Wj k~ 1 be the solutions to ¢( z) - a = 0 and factor
¢-a
¢a = 1 _ Ci¢ = b· s,
INDESTRUCTIBLE BLASCHKE PRODUCTS
129
where b is a Blaschke product whose zeros are (w j k;~ 1 and s is a singular inner function. Taking absolute values of both sides of the above equation and evaluating at Z = 0, we get
11¢~0~;(;) I~ (fl, IWjl) 1,(0)1·
As discussed in the proof of Theorem 2.14, notice that Is(O)1 = 1 if and only if sis a unimodular constant, i.e., ¢a is a Blaschke product. In other words, for a =I=- ¢(O), ¢a is a Blaschke product if and only if
rr
- a I I1¢(O) - a¢(O) = j=1 IWjl . oo
What happens when a
= ¢(O)?
Let
¢(z) - ¢(O)
=
bnz n + bn+ 1 z n+ 1
be the Taylor series of ¢ - ¢(O) about z = 0 and let of ¢(z) - ¢(O). As before, write
+ ...
(Zj)j~1
be the non-zero zeros
~ ¢ - ¢(O) = b . s zn 1 - ¢(O)¢ , where s is a singular inner function and b is the Blaschke product whose zeros are (Zj)j~1' Again, take absolute values of both sides of the above expression and evaluate at z = 0 to get 1
Moreover, ¢
=
_llb;ioJl' ~
(fl,
IZj
I) 1,(0) I·
¢o is a Blaschke product if and only if
Ibnl
00
1 -1¢(0)12
= }1l zjl.
Combining these observations, we have shown the following theorem. THEOREM 4.1 (McLaughlin). Using the notation above, a Blaschke product B is indestructible if and only if
rr
- a I I1B(O) _ aB(O) = j=1 IWjl , oo
Va
=I=-
B(O),
and
Though the above theorem gives necessary and sufficient conditions (in terms of the level sets of B) to be indestructible, characterizing indestructibility just in terms of the zeros of B seems almost impossible. Consider the following theorem of Morse [23]. THEOREM 4.2 (Morse). There is a Blaschke product B for which £(B) but such that if c is any zero of B, then £ (B / Tc) = 0.
=I=-
0
130
WILLIAM T. ROSS
In other words, there are 'destructible' Blaschke products which become indestructible when one of their zeros are removed. We will not give all of the technical details here since they are done thoroughly in Morse's paper. However, since they do relate directly to the earlier work of Heins, from the previous section, we will give an outline of Morse's theorem. Suppose B is a Blaschke product such that the set {( E 8]]J) :
IB*(()I < I}
is at most countable. For a E ]]J), let U
a := m( -log IBa I),
be the greatest harmonic minorant of the non-negative superharmonic function -log IBal. This function is the residual function covered in the previous section (see eq.(3.4)). Since U a is a non-negative harmonic function on ]]J), Herglotz's theorem says that
a = PJ..La, the Poisson integral of a measure J..La on 8]]J). Moreover, since log IB~(()I = 0 for mU
almost every (, it follows that (see eq.(2.10)),
u~ (() =
0 m-almost everywhere. By Fatou's theorem
U~(() =
(DJ..La)(()
at every point where (DJ..La)(() exists (and we count the possibility that (DJ..La)(() might be equal to +(0). We see two things from this. First, (DJ..La)(() = 0 for malmost every ( and so, by the Lebesgue decomposition theorem, J..La ..1 m. Second, since we are assuming that {( E 8]]J) : IB* (() I < 1} is at most countable, we can use the facts that {( E
8]]J): (DJ..La)(() = +oo} = {(:
u~(()
= +oo}
C {(:
IB*(()I < I}
and {( : (DJ..La)(() = +oo} is a carrier for J..La (since J..La ..1 m) [30, p. 158] to see that J..La is a discrete measure. It might be the case that J..La == 0, i.e., Ba is a Blaschke product (see eq.(3.7)). If we make the further assumption that not only is {( : IB* (() I < I} at most countable but B is also destructible, i.e., Ba is not a Blaschke product for some a E ]]J), we see (see eq.(3.7)) that U a > 0 and so, for this particular a, the discrete measure J..La above is not identically zero. Define Q(B) to be the union of the carriers of the measures {J..La : U a = PJ..La > O}. Notice that (4.3)
Q(B)
C {( E
8]]J): IB*(()I < I},
and hence is at most countable, and that Q(B) is contained in the accumulation points of the zeros of B. We also see in this case that B is destructible if and only if Q(B) =f 0. A technical theorem of Morse [23, Proposition 3.2] says that if ( E Q(B), then there is an inner function g, a point a E ]]J), and a f3 > 0 such that
Ba(z) = g(z) exp ( -f3~ ~:) . Morse says in this case that B is exponentially destructible at (. It follows from here that for some 0: > 0 (4.4)
INDESTRUCTIBLE BLASCHKE PRODUCTS
131
An argument using this growth estimate (see [23, Proposition 3.4]) shows that if c is any zero of B, then
Q(B) n Q(B/Tc) = 0.
(4.5)
Morse gives a treatment of exponentially destructible Blaschke products beyond what we cover here. We are now ready to discuss Morse's example. Choose a E lIJJ \ {O} and define
(4.6)
B(z)
Ta (exp ( -
:=
~ ~;) )
.
One can see that B is an inner function, B* (() exists for every ( E olIJJ, and
IB*(()I
=
{I,
lal,
~f (E olIJJ \ {I};
If ( = 1.
By Theorem 2.13, B is a Blaschke product. It is also the case, by direct computation, that the zeros of B can only accumulate at ( = 1. Finally, notice from eq.( 4.3) and the identity
z) ,
B_a(z) = exp ( -1-+1-z that
Q(B)={l} and so B is destructible, in fact exponentially destructible at 1. We claim that if c is a zero of B, then B / Tc (B with the zero at c divided out) is indestructible. Indeed, since
{(: I(B/Tc)*(()1 < I} = {(: IB*(()I < I} = {I} we can apply eq.(4.3) to get
Q(B/TC) C {I}. However, from eq.(4.5) we see that Q(B/Tc) = 0 which means, from our discussion above, that B / Tc is indestructible. 5. Classes of indestructible Blaschke products So far, we have discussed conditions on a Blaschke product that make it indestructible. We now examine a refinement of this question. Suppose that 13 is a particular class of Blaschke products and B E 13. What extra assumptions are required of B so that Ba E 13 for all a E lIJJ? We focus on the class (and certain sub-classes) of e, the Carles on-Newman Blaschke products. These are Blaschke products B whose zeros (an)n~l satisfy the so-called 'conformal invariant' version of the Blaschke condition 00
L(1-lan l)
O. (i) ForO.. The sequence {nd of positive integers is called a lacunary sequence. We show that a necessary and sufficient condition for membership in Fg can be obtained for functions with lacunary Taylor series. Some results along this line have been obtained by Blasco and Galbis [2].
e
THEOREM C ([2], Theorem 2.3, 2.5). Let {nd be a lacunary sequence. Then Jar any sequences {ak},
/nJ
L ak znk E Fi ~ L lakl Vz;;; nl < 00
00
k=l
k=l
1
00,
We will extend the result of Blasco and Galbis to Fg, for a > 0 and 1 '5. P < 00. This will give explicit examples of functions in Fg. As a side note, we give a new proof for an analogous result concerning functions with lacunary Taylor series in the Bergman spaces AP. Another way that we can refine Theorems A and B is to make use of the mixed norm sequence spaces: let 0 < p, q < 00, and let {nd be a lacunary sequence. For each positive integer k, let Ik = [nk, nk+d n N. A sequence of complex numbers {an} is said to belong to ep,q if
(1) In the case of p =
00
or q =
00,
the respective norms are
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES
137
We can view ep,q as the vector-valued space Lq(N, v; ep) consisting offunctions from N to ep where v is the counting measure. We note that ep,p is the usual ep space, but in general ep,q and ep are different spaces. For example, the sequence whose sum on h equals 1/ k is in 1 ,q for any q > 1, but not in 1 • Also, since all sequences in ep and ep,q are bounded, it can be shown that the following inclusions hold for p < q:
e
e
The ep,q spaces are dependent on the choice of the lacunary series {nk}, but this will not affect the statements of our theorems. These spaces were introduced by Kellogg [8] in his improvements for the classical Hausdorff-Young Theorem. We will similarly improve Theorems A by giving necessary or sufficient conditions in terms of these ep,q spaces. For 0 < p ~ 00, and a space of analytic functions X, we say that a sequence of complex numbers {An} is a coefficient multiplier from X to p if for every f(z) = E:=o anz n in X, we have {Anan} E p ; we use the usual notation (X, p ) for the space of all coefficient multipliers from X to ep . Coefficient multipliers in HP and AP have been well investigated. For example, a characterization of coefficient multipliers from the Hardy space HI to e1 is found in Theorem 6.8 in [5]. We also have the following result characterizing (Al,e 1 ) due to Blasco (see Theorem 5.1,
e
e
e
[1]):
e1
THEOREM D. A sequence of complex numbers {An} is a multiplier from Al to if and only if {nAn} is in e1 ,00.
We will prove a sufficient condition for {An} to be a multiplier from F~ to e1 and show that it is in some sense the best possible. In the converse direction, we will prove a necessary condition that, curiously, differs from the sufficient condition by a factor of .;n. In the proof of theorems, we shall abuse notations and use c and C to represent positive constants that may change from step to step in the proof. The author would like to thank Oscar Blasco and Petr Honzik for helpful discussions, and Martin Buntinas for the reference [8].
2. Lacunary Taylor series 2.1. Fock spaces. A version of the following lemma is found in [2] and is needed for the proof of the generalization of Theorem C for FJ;. We remark that the lemma is similar to Lemma 3 in [10], but in this case the domain of integration is over a finite interval instead of [0, 00 ) .
LEMMA 2.1. Let p, 0: > 0, and let {nd be a lacunary sequence. Then for every k, E
c(nk!)2 o:nk
~
E
n-~+! ~J -"'-rnkPe-¥r2rdr~C(nk!)2 n-~+!, k
V!'ff
o:nk
k
where c and C are constants independent of k. PROOF. The second inequality follows directly from Lemma 3 in [10]. For the first inequality, we note that since {nk} is lacunary,
nk+lP AnkP nkP ylnkP > >+ -2 2 2 - 2
JAMES TUNG
138
when k is sufficiently large. We observe that for a > 0 and x E JR, the function x I-t xae- x is decreasing on [a, 00); together with the estimate X ---4
00
:':.1 c ( - 2 ) ¥ J¥+vf¥ -
¥
ap
~ c (:p) ¥ >c ( - 2 )
-
n~p
(n;p
2
e udu
+ In;p) n~p e-¥-vf¥
(n- kP ) ¥ 2
ap
U
e _:':.1 C (nk!) 2 n-i+~. -
a nk
k
o 2.2. Let 1 ::; p ::; 2, {nk} be a lacunary sequence, and J(z) be a Junction in Fg. Then
THEOREM
L:~=o anz n
Jor some constant C independent oj J. The finiteness of the sum is already a sufficient condition for membership in Fg (see Theorem B, part (i)). We thus obtain a characterization for a function with lacunary Taylor series to belong to Fg, 1 ::; p ::; 2. PROOF. Let gration, we have
J E Fg.
Note that we have
By Holder's inequality and breaking the domain of inte-
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES
139
for every n, by a calculation involving Cauchy's formula for an. Continuing with the calculation, we have
o
the last line follows from Lemma 2.1.
We shall prove a sufficiency condition for membership in F[:, when p 2: 2 using Theorem 2.2 and a duality argument. We first prove the following lemma.
= 2:~=1 akzk be an entire junction, and sn(z) = be its Taylor polynomials. If IlsnllF~ is bounded above for
LEMMA 2.3. Let p > 0, j(z)
2:;=1 ak zk , n = 1,2, ... all n, then j E F[:,.
PROOF. Let p > 0. On the disk Izl ::; R, the function f is continuous and the polynomials Sn converge to f uniformly. Thus {sn} is uniformly bounded on Izl ::; R. We can apply the dominated convergence theorem to conclude
1
If(zW e-¥l zI2 dA(z) = lim
Izl:SR
n->oo
::; sup n
The result follows by letting R
----> 00.
1
I
ISn(zW e-¥l zI2 dA(z)
Izl:SR
ISn (zW e-¥l zI2 dA(z) ::;
c.
IC
o
THEOREM 2.4. Let 2 ::; p < 00, and let {nd be a lacunary sequence. If {ad is a sequence of numbers such that
PROOF. Let p' = p/(p - 1) be the conjugate index of p. For each positive integer N, let SN(Z) = 2::=1 akznk. Then SN E F[:,. Since the dual space of F[:, is Fg', up to an equivalence of norms, we have
for some constant C, where the supremum is taken over all functions 9 in Fg' with 11911 FP' ::; 1. a
JAMES TUNG
140
Let g(z) = L:~=o cnz n be an entire function in F[!,' with Holder's inequality and Theorem 2.2,
I[
IlglI F {
::;
1. Then by
sN(z)g(z)e-crlzI2 dA(z)1
=C
11 t 00
akCnk r2nk e-crr2 r drl ::;
o k=1
t
lakCnk I
k=1
"c (t, lakl' (:::) "C 119I1 F,'
c
I
n;!+l) l
:~:
(t, Ic.f (:~:) >' n;~+l f
(~Iakl' (:~:) I n;'+l
r'
where C is a constant independent of N. Applying the hypothesis and taking the supremum yield IlsN11 ::; c, and the theorem follows by Lemma 2.3. 0 We summarize the results of Theorems B, 2.2 and 2.4 as follows: THEOREM (Summary). Let 1 ::; p < 00, and let {nd be a lacunary sequence. A necessary and sufficient condition Jar the Junction J(z) = L:~=1 akznk to belong to F[!, is
f
(:~:) ~ n~~+!
lakl P < k=1 Furthermore, iJ {nk} is an arbitrary sequence, then (i) Jar 1 ::; p ::; 2, the sufficiency part holds; (ii) Jar 2 ::; p < 00, the necessity part holds.
00.
2.2. Bergman spaces. For 0 < P < 00, the Bergman space AP consists of those J analytic on the unit disk ][)) such that
where dA(z) is the Lebesgue area measure. The following theorem concerning functions in AP with lacunary Taylor series was proved by Buckley, Koskela and Vukotic [3]. THEOREM E. Let 1 ::; p < 00, and let {nd be a lacunary sequence. A necessary and sufficient condition Jar the Junction J(z) = L:~1 akznk to belong to AP is 00
L
lakl Pnk -1
