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00
the Bergman projection PQ
Alec spent several years at Oklahoma State University. While there he collaborated with Dale Alspach and Joe Rosenblatt on projections to translationinvariant subspaces of Ll (G) [3). He also worked with David D. Ullrich on weighted averages in Hardy spaces. After leaving Oklahoma State University he went to Lamar University. Although Lamar did not offer as many research opportunities as Oklahoma State, he kept up his research interests. In particular he worked with John Cannon on free boundary value problems related to the combustion of a solid [4). Altogether Alec published 37 papers and one book. In addition to those mentioned above he has worked with his colleague Valentin Andreev as well as Kevin Madigan, Alexander Pruss, Paul Bourdon and Bill Ross. Alec took advantage of an offer to visit SUNY at Albany. This was arranged by Michael Stessin. He really enjoyed his year at SUNY and Michael and he finished two papers together (see, for example, [5)). Alec spent many weeks visiting me in my home both at Chapel Hill and in Missoula, Montana. Part of this time together was work but we both enjoyed walking in the Rockies and the Smokies. Finally, Alec, Bill Ross and I worked on material related to Cauchy transforms. In addition to two papers on this subject Alec, Bill and I worked diligently on a recent book The Cauchy Transform [6).
MATHESON NOTE
3
Let me add a few more theorems that occurred during the period when Alec and I were working closely together. One that he was realIy happy with occurs in the paper [7]. It relates to composition operators Cq, on Hardy spaces and the work by D. Sarason and work by Joel Shapiro and C. Sundberg on compactness of such operators. We were able to give a "direct function theoretic" proof of some of this material. One of the last areas we planned to consider was a study of weak type inequalities for functions from the classical Banach spaces of analytic functions. In particular we revisited a result ofD. BekolIe which stated that the Bergman projection P on functions in L 1 (]jJ), dA) satisfied a weak type estimate of the form area(z E]jJ): IPf(z)1 > t):S
C
T11l111.
BekolIe's result is valid for the Bergman space on the balI in C m as well. We have shown the folIowing (unpublished).
Theorem C. Let J.L be a finite Borel measure on ]jJ) and let P J.L denote the Bergman projection of J.L, that is, dJ.L(w) PJ.L(Z) = IIJJ (1  zw)2· Then P J.L satisfies the weak type estimate C area(z E]jJ): IPJ.L(z)I > t) :S TIIJ.LII,
1
where C is independent of J.L. This is a consequence of a more general result. Let 0 be a domain in the complex plane and let
K:OxOC be a function such that for each fixed z E 0, the function K (z, .) is bounded and harmonic in O. Then for every finite Borel measure J.L on 0, the integral
fp(z)
=
In K(z, () dJ.L(()
satisfies a weak type 1: 1 inequality ( e.g., area {z E
Ollfp (z) I > t} :S i IIJ.LII).
Lemma. For every finite Borel measure J.L on 0, there is a function fELl (0, dA) such that
IIfl11 = IIJ.LII and fp(z) =
In K(z, () f(() dA(()
for every z E O.
Let me end this note with an apology to any of Alec's coauthors and friends whose works I have slighted in this note. Those of us who worked closely with Alec (especialIy Bill Ross and Michael Stessin) can attest to Alec's ability in mathematics, his hard work, care and deep commitment to his research and his students. During these latter years when Alec had his operations for cancer and was taking heavy chemotherapy he never complained. He worked like a trooper until the end. His close friend Valentin (Andreev) found him on the floor of his home and rushed him to hospital where he spent his last days. Ross and I spoke to him by phone the last ten days of his life. Again and again he would give me a short rundown on his condition and then telI me about a mathematical result we should think about. He received the printed copy of our joint book on the morning he passed away.
4
JOSEPH A.eIMA
He is missed for his ability and his humanity. REFERENCES [I] Matheson, A. Approximation of analytic functions satisfying a Lipschitz condition. Michigan Math. J. 25 (1978), no. 3, 289298. [2] Cima,1. A.; Matheson, A. Approximation in the mean by polynomials. Rocky Mountain 1. Math. 15 (1985), no. 3, 729738. [3] Alspach, D.; Matheson, A.; Rosenblatt, J. Projections onto translationinvariant subspaces of L1(G). J. Funct. Anal. 59 (1984), no. 2, 254292. [4] Cannon, John R.; Matheson, Alec L. A free boundary value problem related to the combustion of a solid: flux boundary conditions. Quart. Appl. Math. 55 (1997), no. 4, 687705. [5] Matheson, Alec L.; Stessin, Michael!. Cauchy transforms of characteristic functions and algebras generated by inner functions. Proc. Arner. Math. Soc. 133 (2005), no. 11,33613370 (electronic) [6] Cima, Joseph A.; Matheson, Alec L.; Ross, William T. The Cauchy transform. Mathematical Surveys and Monographs, 125. American Mathematical Society, Providence, RI, 2006. [7] Cima, Joseph A.; Matheson, Alec L. Essential norms of composition operators and Aleksandrov measures. Pacific 1. Math. 179 (1997), no. 1,5964.
UNC, CHAPEL HILL, N.C. 27599
Contemporary Mathematics Volume 454, 2008
ON THE INVERSE OF AN ANALYTIC MAPPING JOSEPH A.crMA
SECTION 1.
Assume n is a domain in a separable Hilbert space JH[ and f is an analytic mapping from n c JH[ into JH[. This statement means that for each a E n, there is associated (in a continuous way) to f a bounded linear operator (the Frechet derivative) Df(a) satisfying lim Ilf(b)  f(a)  Df(a)(b b+a
Iia  bll
a)11 = 0
.
en,
If JH[ is finite dimensional, say dim JH[ = n, we may assume JH[ = and that f is an analytic mapping of a domain n c + In this case Df(a) can be identified with the Jacobian matrix of its partial derivatives and the following significant result holds. The analytic mapping f is one to one near a En if and only if f is biholomorphic on a neighborhood N(a) c n if and only if D f (a) is invertible. This theorem has no analogue if JH[ is infinite dimensional. In this case we may take JH[ to be the sequence space l2. The example below with x = (Xl, X2, X3,··· ) E lB and
en
(1)
en.
f(x) = (xi,x~,x~,x~,x5,x~,· .. )
is analytic and one to one on the ball, has nowhere dense range with D f(O) = O. In particular f has a functional inverse which is not analytic. In this paper we discuss the strongest result that we know of in the literature to produce analytic invertibility in this setting. This is a result due to Aurich [1] (with an attribution to Abt). His work begins as a study of bifurcation. His tools, which are quite appropriate for his study of the bifurcation theory, are not essential to prove the result of interest to me. The result below appears as a part of a theorem at the end of his paper and I quote only the part of that theorem that pertains to my interest. For each r E (0,1] denote the open ball with center at the origin and radius r by lBr with lBl = lB, and the boundary of lBr denoted as Sr. The derivatives are bounded operators in ,8(JH[) and I remind the reader of the class of Fredholm operators. Definition. An operator T E ,8(JH[) is a Fredholm operator if the range ofT (written R(T)), is closed and the dimension of the null space, N(T), and the dimension of the cokernel, (R(T))1., are finite. The notation (R(T))1. denotes the orthogonal complement of the closed subspace R(T). The index of T is defined as i(T) = dimN(T)  dim(R(T))1.. Date: September 25, 2007. 1991 Mathematics Subject Classification. 32K05. ©2008 American Mathematical Society
5
JOSEPH A.CIMA
6
Theorem 1 (Aurich). Assume f is an analytic and one to one mapping on lffi  t 1Hl. Assume that for each point a E lffi the operator Df(a) is Fredholm of index zero. Then it follows that f is biholomorphic on lffi. I cast the proof in a Linear Algebra setting and it is of a local nature. The proof is accessible to graduate students that have had a basic course in Functional Analysis. This type of phenomenon is in some sense typical of the behavior of analytic maps on infinitedimensional Hilbert spaces. Part of the difficulty in this setting is that the closed unit ball is not compact. In addition to this aspect of topology there is an unsatisfactory aspect to the analyticity of such mappings in the following sense. The analytic function f itself may have interesting properties but we do not know how this affects the behavior of the derivative (e.g., in the example (1) f is one to one yet its derivative at the origin is the zero linear transformation ). Conversely, the derivative may possess interesting properties but we have not been able to use these properties to recapture information about the function itself (see Section 3 below). In Theorem 1 strong conditions are imposed on both the function and the derivative. In the last section I give a short list of problems that I feel are challenging and if solved would flesh out our understanding of the interplay between the local properties of the function and conditions on the derivative. It is difficult to give interesting examples in the infinitedimensional setting. I thank Warren Wogen for his advice and interest in this work. SECTION
2.
LOCAL BEHAVIOR
The idea is to use a factorization which appears in reference [1]. We begin by stating a special result and then observing its relationship to Theorem 1 above.
Theorem 2. Suppose that f is an analytic mapping of lffi into IHl and that a E lffi. Let T = Df(a). Set M = kerT and suppose that N = (M)~ is the range ofT. If M has finite positive dimension, then f is not one to one in any neighborhood of a.
Proof. We have IHl = M
Efj
N, so relative to this decomposition,
~),
T=(g
where Tl is invertible from N onto N. We write a
= al Efj a2
E M Efj N,
and assume without loss of generality that f(a) = f(al
Efj
a2) = 0 Efj 0 EM
Efj
N.
We can find neighborhoods U(al) ~ M and V(a2) ~ N so that U Efj V is a neighborhood of a in B. With P the orthogonal projection of H onto M and Q = 1 P the orthogonal projection of H onto N we may write for x E B f(x)
Note u : B
t
M and v : B
= Pf(x) Efj Qf(x) == u(x) Efj v(x). t
N. We have
Du(a)
=
PDf (a)
=
PT(a)
=0
ANALYTIC MAPPING
7
and
Dv(a) = Q(Df(a)) = T I . By the Implicit Function Theorem for Banach spaces (Ref [2]) applied to v(a) = (OEB 0)) and Dv(a) = T I , we have neighborhoods VI(al) <;;; U <;;; M and V2(a2) <;;; V <;;; N and an analytic function h : VI > v2, h(al) = a2 and v(ml EB nr) = 0 for ml E VI, nl E V2 if and only if nl = h(mr). This implies that f(ml EB h(mr))
=
u(ml EB h(mr)) EB 0,
or
f(ml EB h(ml)) E M for ml E VI' Now for ml E VI, define the mapping u from Minto M as u(m) = u(m EB h(m)). Since Du(a) is the zero operator, the Chain Rule gives that the operator Du(al) is the zero operator on M. Applying the Inverse Mapping Theorem in finite dimensions we have distinct points ml and m2 in VI with u(ml) = u(m2)' But for mE VI, and
f(m2 EB h(m2)) = u(m2 EB h(m2)) EB 0
= u(m2) EB 0 = u(m2) EB O. Since the direct sum decomposition is unique, we have ml EB h(ml) i= m2 EB h(m2) and f(ml EB h(ml)) = f(m2 EB h(m2))' Hence, f is not one to one near a2. 0 Theorem 2 yields the proof of Theorem 1. We show that if f is analytic and one to one on E and if T = D f (a) is Fredholm on index zero at a E E then T must be invertible. It then follows by the Inverse Mapping Theorem that f is biholomorphic near a. Suppose T has index zero and M = ker T has positive dimension. If N is equal to the range T, then N.L has the same dimension as M. So choose a unitary operator U with U(M) = N.L and U(M.L) = N. If 9 = Uf, then 9 satisfies the hypothesis of Theorem 2, so 9 is not one to one in any neighborhood of a. Hence, neither if f. SECTION
3
There is an interesting example of Heath and Suffridge which does the following. They produce a mapping from the open unit ball in the nonseparable Banach space Hoc onto the space which has an open set in its range; it is one to one and yet the range of f is not an open set. In this case the Banach space is a difficult space to work in and so one might expect more aberrant behavior of analytic mappings because of the difficult structure of such nonseparable Banach spaces. I feel the following problems are worthy of study and positive results or counterexamples would be useful in the study of such mappings. As above 1HI is a separable Hilbert space. Q 1. Assume f is analytic and is a one to one mapping from E > 1HI with f (E) an open set. Is f a biholomorphic mapping? Q 2. Assume f is analytic and is a one to one mapping from E > 1HI with Df(O) = O. Show that range of f can not be locally open at O.
JOSEPH A.CIMA
8
Q 3. Assume f as above and Df(x) invertible for 0 <
Ilxll < r <
1. Show that
f is biholomorphic on lB. REFERENCES [1] Aurich, V., Bifurcation of the solutions of Holomorphic Fredholm equations and complex analytic graph theorems, Nonlinear analysis. Theory, Methods and Applications, Vol. 6, No. 6, pp 599613, 1982. [2] Dieudonne, J., Foundations of Modern Analysis, Academic Press, New York and London, 1960. [3] Heath, L.F. and Suffridge, T.J. Starlike, convex, close to convex, spirallike, and .p like maps in a commutative Banach algebra with identity, Transactions of the American Mathematics Society, Vol. 250, June 1979.
UNC, CHAPEL HILL, N.C. 27599
Contemporary Mathematics Volume 454, 2008
Isometric composition operators on the Bloch space in the polydisk Joel Cohen and Flavia Colonna In memory of Alec L. Matheson ABSTRACT. In this work, we study the holomorphic functions
f 0
1. Introduction
Let j be a complexvalued holomorphic function on the unit polydisk
6,n = {z = (Zl, ... ,zn)
E
en: IZkl < l,k = 1, .. . ,n}
and, for Z E 6,n and U E en, let (\1f)(z)u denote by n
= L:~=l
it; (Z)Uk'
For u,v E en,
_
_ ""' UkVk Hz(u, v) = ~ (1 l zkI 2)2 k=l the Bergman metric on 6, n, that is, the positive definite bilinear form which is invariant under biholomorphic transformations of 6, n. The function j is said to be Bloch if f3f = sup Qf(z) is finite, where zEt. n
Qf(Z)
=
sup uElCn\{o}
1(\1j)(z)ul. Hz(u, u)l/2
Denote by B the space of all Bloch functions on 6,n. The map I ft f3 f is a seminorm on B, and B is a Banach space, known as the Bloch space, under the norm 1I/IIs = 11(0)1 + f3f· The above definition was given by Timoney in [20] on a bounded homogeneous domain (Le. a bounded domain Dee n whose group of biholomorphic transformations acts transitively on D) with Hz(u, v) representing the Bergman metric on the domain, although the notion of Bloch function in higher dimensions was first introduced by K. T. Hahn in [14]. For excellent references on 1991 Mathematics Subject Classification. Primary: 32A18j Secondary: 47B33. Key words and phrases. Bloch functions, Bergman metric.
30D45, 32M15,
©2008 American Mathematical Society
9
10
JOEL COHEN AND FLAVIA COLONNA
the theory of Bloch functions on a bounded homogeneom; domain, see [20] and
[21]. For n = 1, the above definition of Bloch seminorm reduces to the wellknown formula f3f = sUPzEb..(I IzI2)1f'(z)l. References on the theory of Bloch functions on the unit disk include [1] and [3]. A holomorphic function 'P mapping ~n into itself induces on B the composition operator C
~ > ~
analytic, C
In Theorem 2.1 we shall list several statements that are equivalent to the condition f3
(1  IZkI2)IB'(Zk)1 =
II !IZk ZkZjZj !,
j#
nontrivial examples of symbols 'P whose corresponding operator C
k~oo
II ! 1Zk ZkZZ! = 1. J
#k
J
Sequences satisfying the above condition are called thin. In [11] it was shown that a sequence {Zk} satisfying the condition Ilzk+11 · 11m 1 IZkl
k~oo
=0
is thin. Thus, examples of thin sequences are {I  11k!} and {I  kk}. Thin Blaschke products have the property that the preimage of every a E ~ is a thin sequence and are known to be indestructible, that is, if B is a Blaschke product whose zeros form a thin sequence, then for all a E ~, La 0 B is a Blaschke product, where La(z) = t~;z'
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN fl. n
11
By Theorem 1.1 and Theorem 2.1, the zero sets Z of the symbols of the isometric composition operators that are not rotations form infinite sequences {Zk} kEN satisfying the condition
IT
lim sup kHXJ
I Zk 1
(EZ,(¥Zk
( Zk(
I = 1.
We call these almostthin sequences. Almostthin sequences form a much wider class than the thin sequences. Indeed, in [6] we proved that if C
k~oo
= lim (1 IZkI2)lg(Zk)IIB'(Zk)1 = 1. k~oo
Then from F'rostman's Theorem (see [13], Theorem 6.4) it follows that '{Jl(a) is an almost thin sequence for all a E ~, and is thin only for a set of logarithmic capacity O. For domains Dee n, D' c em, denote by H (D, D') the class consisting of the holomorphic functions from D to D'. For a domain D in en, let us denote by Aut(D) the group of biholomorphic transformations of D onto itself, which we call automorphisms of D. The automorphisms of ~ n are the transformations of the form
T(z)
=
(S1(Zr(1)),"" Sn(zr(n))), Z = (Z1, ... , zn)
where Sj E Aut(~),j = 1, ... ,n, and [18], p.167).
T
is a permutation of {l, ... ,n} (e.g. see
For '{J E H(~n, ~n), let J'{J(z) = (~(z)) Zk
E ~n,
1~j,k~n
denote the Jacobian matrix
of'{J at Z so that J'{J(z)u is the usual matrix product where u is viewed as a column vector. Then
B
sup
zEfl. n uEIC
n\
{O}
H
is finite. Indeed, in [20] (proof of Theorem 2.12) it was shown that if D1 c e nl and D2 C e n2 are bounded homogeneous domains, then there is a positive constant c such that for all '{J E H(D 1,D2), Z E D 1, and u E en
H::C~)(J'{J(z)u, J'{J(z)u) ::; cHfl (u, 71), where Hfl and H::C~) are the Bergman metrics on D1 and on D2 at Z and '{J(z), respectively. If '{J E Aut(~n), then for any choice of Z E ~n and u E en
H
Qjo
sup uEIC"\{O}
Thus
(1.1)
~n,
we have
(H
12
JOEL COHEN AND FLAVIA COLONNA
and Qfo
= 1, Hz( u, _u) = B
lul 2 (1lzI2)2'
and thus
sup (1 ;
=
zEt.

Izr)(I~~~z)l, cP z
which is no greater than 1 by the SchwarzPick lemma. REMARK 1.2. In higher dimensions, B
B2 _
Setting Wj
(1.2)
where
= Uj / (1  IZj B
Ilwll =
=
1
2 ),
sup max zEt. n Ilwll=1
VL 7=1I
sup
1
Lk=1
2:.1=1 ~ (z)Uj 12 (lI
n
LI=1
uElCn\{o}
IUl12
(1lzI12)2
we obtain
(
n
~
I",n
~(z)(1lz·12)W·1 J J
uJ=1 OZj
(1 ICPk(Z)I2)2
2) 1/2 ,
w jI2.
A bounded domain D in C n is said to be symmetric if for each a E D there exists an automorphism S of D for which a is an isolated fixed point and such that S1 = S. The polydisk is a bounded symmetric domain, since it is homogeneous and the mapping S(z) = z is its own inverse and has 0 as an isolated fixed point. Bounded symmetric domains are homogeneous (see [15], pp. 170, 301). E. Cart an [2] classified the bounded symmetric domains (up to biholomorphic transformations) into six irreducible classes (all containing 0), four of which consist of large families of domains and two others consisting of a single domain each, known as exceptional domains. A finite product of domains of the first four types is known as a Gartan classical domain. The polydisk and the unit ball are Cartan classical domains. In Theorems 2 and 3 of [5] we expressed the value CD of the supremum of the Bloch seminorms of the bounded holomorphic functions mapping a Cartan classical domain D into ~ in terms of the Bergman metric of the domain at the origin and the corresponding values of the irreducible factors of D. Specifically, we showed that for each irreducible Cartan domain D, CD
=
1
infuEoD H{? (u, u)l/2
.
Furthermore, we proved that CD is the maximum over all j = 1, ... , N of CD j , where D 1 , ... , DN are the irreducible factors of D. We also gave in Corollary 1
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN ll.n
13
the specific value of the constant CD in terms of the class and the dimension of the domain D. This result was later extended to the exceptional domains in [22]. In particular, for D = b. n, we obtain (1.3)
CD =
sup{;3J : f
E H(b., b.)} = 1.
In this article, we present an overview of the many characterizations of the isometric composition operators on the Bloch space in the onedimensional case (Corollary 2.2). After giving some preliminary results on Bloch functions on the polydisk (Theorem 3.1, Theorem 3.2, and Theorem 3.3), we obtain some necessary conditions on t,p = (t,pl, ... , t,pn) for C'P to be an isometry on B. Specifically, in Theorem 4.4 we prove that if C'P is an isometry on B, then t,p(0) = 0 and the components of t,p must be linearly independent and have maximal Bloch seminorm. Moreover, in Theorem 4.5 we show that if C'P is an isometry on B, then there exist sequences {Tj,(k)hEN (with j = 1, ... , n) of automorphisms of b. n such that Z ft (t,pl(T1,(k)(z)), ... , t,pn(Tn,(k) (z))) converges to the identity of b. n . We use the results presented in section 2 to provide in Theorem 4.10 a wide class of functions t,p on b. n whose induced composition operator is an isometry on
B. We conclude the paper with a conjecture and some open questions.
2. The onedimensional case In this section, we present several characterizations of the isometric composition operators on the Bloch space of the unit disk. 2.1. For t,p E H(b., b.), the following statements are equivalent: (a) ;3'P = 1. (b) Either t,p E Aut(b.) or for every a E b. there exists a sequence {Zk} in b. such that !Zk! + 1, t,p(Zk) = a, and THEOREM
(2.1)
lim (1  !Zk!2)!t,p'(Zk)! k>oo 1 !t,p(Zk)!2
= 1.
(c) Either t,p E Aut(b.) or the zeros of t,p form an infinite sequence {ZdkEN such that lim sUPk>oo (1 !Zk!2)!t,p'(Zk)! = 1. (d) Either t,p E Aut(b.) or t,p = gB where g is a nonvanishing analytic function mapping b. into itself or a constant of modulus 1, and B is an infinite Blaschke product whose zero set Z contains a sequence {zd such that !g(Zk)! + 1 and
II
lim sup I Zk  ( I = 1. k>oo (EZ'(#k 1  Zk(
In particular, the zeros of t,p form an almostthin sequence. (e) Either t,p E Aut(b.) or there exists {SkhEN in Aut(b.) such that !Sk(O)!+ 1 and {t,p 0 S k} approaches the identity locally uniformly in b.. (f) C'P preserves the Bloch seminorm on B. (g) B'P = 1. The fraction on the lefthand side of (2.1) is also known as the hyperbolic derivative of t,p at Zk. The equivalence of (a) and (b) with the condition t,p(Zk) = a replaced by the apparently weaker condition t,p(Zk) + a was proved in [17].
14
JOEL COHEN AND FLAVIA COLONNA
PROOF. The equivalence of (a) and (b) follows from Theorem 2.7 of [6]. The equivalence of (a), (c) and (d) was shown in Theorem 3 and Corollary 1 of [7]. The equivalence of (a) and (e) was proved in Theorem 2 of [9]. Assume (e) holds and show that (f) holds. If cp E Aut(~), then by the invariance of the Bloch seminorm under right composition of automorphisms, it follows immediately that f3f o
oo 1  Icp( Wk) 12
= 1.
In the first case, cp E Aut(~). In the second case, for each kEN, let LWk(z) = l~;;;kzZ' for Z E ~. By a normality argument, we may assume (passing to a subsequence if necessary) that {cp 0 L Wk } converges to some analytic function F : ~ . ~ such that IF'(O)I = 1  IF(o)l2. By the SchwarzPick lemma, we deduce that F E Aut(~). Letting Sk = LWk 0 Fl, it follows that Sk E Aut(~), ISk(O)1 . 1, and {cp 0 Sd converges locally uniformly to the identity in ~. 0 From Theorem 1.1 and Theorem 2.1 we deduce the following result. COROLLARY 2.2. For cp E H(~, ~), C
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN t.. n
15
3. Bloch functions on the polydisk In this section, we give alternate descriptions of the Bloch seminorm and analyze some properties of Bloch functions. We begin by observing that the Bloch functions are precisely the Lipschitz maps between the polydisk under the distance p induced by the Bergman metric and the complex plane under the Euclidean distance. To see this, we recall a very useful result of [13] connecting local derivatives to Lipschitz mappings. To put this in context, we look again at the definition of Qf (z) for z E ~ n. If we consider f : ~ n >
=
sup z,wE~n,z#w
If(z)  f(w)1 p(z,w)
It is the relation between /3f and dil(f) that we are looking for. A length space is a Riemannian manifold in which the distance between two points is the infimum of the lengths of geodesics connecting the points. Examples include ~ n as well as
In the present context, we deduce THEOREM 3.1. Let f : ~ n >
/3f = sup If(z)  f(w)l. z#w p(z, w) A proof that does not use differential geometry for the special case n = 1 can be found in [23]. For a proof for Bloch functions on the unit ball, see [24]. We now prove a convergence theorem for Bloch functions on the polydisk which was proved in [8] in the onedimensional case. THEOREM 3.2. Let {In} be a sequence of Bloch functions converging locally uniformly to some holomorphic function f. If the sequence {/3 fn} is bounded, then f is a Bloch function and /3 f ~ lim inf /3f n • n>oo
That is, the function f 14 /3 f is lower semicontinuous on B with respect to the topology of uniform convergence on compact subsets of ~ n. PROOF. Since {/3fn} is bounded, B = lim infn>oo /3fn exists and is a nonnegative number. Let {nd be a sequence of positive integers such that B = limk>oo /3 fn k' Let z, w E ~n and fix E > 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 seminorm. THEOREM 3.3. Let fEB. Then {3f =
z~in II (:~ (z)(1l 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
<:t IWkl2
(L~1
IWkil u kl)2 L~=1 akl u kl 2  k=1
and equality is attained for
IUj
I=
Proof of Theorem 3.3. Fixing z E
(
Iw] I/a]
E k=l (IWk I/ a k)2
~n
ak '
)'/2' for each j = 1, ... ,n.
0
and applying Lemma 3.4, we obtain
o NOTATION 3.5. In the remainder of the paper, to avoid confusion, sequences in higher dimensions will contain a superscript rather than the conventional subscript, which will be reserved for the components. We recall a result that will be used in the next section to prove a convergence theorem for holomorphic functions from the polydisk into itself. THEOREM 3.6. (Corollary 4 of [5]) Let D = D1 x··· X DN be a Cartan classical domain with D 1, ... , DN irreducible, where least one of the factors is~. Let f E H(D,~) with {3f = CD· Then there exists a sequence {SkhEN of automorphisms of D and an integer m = 1, ... ,N such that Dm = ~ and {f 0 Sk} converges locally uniformly in D to the projection map z f> Zm.
4. Holomorphic self maps of
~n
inducing an isometry on B
The polydisk, viewed as a Riemannian manifold under the structure induced by the Bergman metric, is a length space. Thus, applying [13]' Prop. 1.8 bis as in Theorem 3.1, we obtain the following result.
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN
Ll n
17
PROPOSITION 4.1. Let
=
sup p(
p(z,w)
zo/w
Thus, Bep can be interpreted as the Bergman dilation of <po From Proposition 4.1 and the invariance of the Bergman distance under the action of Aut(Do n ), we deduce COROLLARY 4.2. The correspondence
~
Bep is invariant under
left and right composition of automorphisms of Don.
From Proposition 4.1, we also derive the lower semicontinuity of the map Bep on the space H(Do n , Don).
~
COROLLARY 4.3. Let {
are linearly independent. PROOF. Given
J
with Ilull = 1, we have L~=1 (1~1:~~2)2 2: and Uk = 0 for all k i= j. Thus
(1~1;;~2F
max IUjl = max Ilull=l Hz (u, u)1/2 Ilull=l (",n
with equality holding for IUj I = 1
IUjl IUk 12
= llz.1 2 )
1/2
J
.
L..k=l (lIZk 12)2
Consequently
lajl+ sup zELl"
lajl
(11
ajZj
1  ajZj
12)
+ 1.
Since Cep is an isometry, II'l/Jj 0
JOEL COHEN AND FLAVIA COLONNA
18
In addition, \l('l/Jj
0
cp) = \lCPj, so
1 = (3'I/Jjo
l\l('l/Jj
0 cp)(z)ul H ( )1/2 z u, u
Next, assume there exist k = 1, ... , n and constants aj E
\II{(z)=
{
Zmj
Zj
ifhij,mj, ifh=j, ifh=mj,
and set Tj,(k) = Sj,(k) 0 \Ilj E Aut(~n). Then the sequence {cpj o Tj,(k)} converges locally uniformly to the projection function Z f+ Zj. Consequently, the sequence {( CPl oT1,(k), ... ,CPn oTn,(k»)} converges locally uniformly to the identity in ~ n. 0 The converse of Theorem 4.5 is false, as the following simple example shows. EXAMPLE 4.6. For Zl, Z2 E ~, CP(Zl' Z2) = (Zl' Zl) and let Tl and T2 be the automorphisms of ~2 defined by Tl(Zl' Z2) = (Zl' Z2) and T2(Zl' Z2) = (Z2' Zl). Then (CPl 0 Tl(Zl' Z2), CP2 0 T2(Zl' Z2)) = (Zl' Z2), yet, as observed in Remark 1.2, C
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN
t,. n
19
The following lemma is straightforward and so we omit the proof. LEMMA 4.9. Given nonnegative numbers r1, ... , r n , n
L
max rkl wkl 2 = max{r1, ... , rn}. Il w ll=l k=l The following result shows that the class of functions inducing an isometric composition operator on the Bloch space is quite large. In particular, we obtain nontrivial examples of isometric composition operators on B by considering functions whose components depend on a distinct single variable and constitute almostthin Blaschke products fixing O. THEOREM 4.10. For Z E ~n and T permutation of {I, ... , n}, let 'P(z) = (h 1(zT(1)), ... ,hn (zT(n))), where for each j = 1, ... ,n, hj E H(~,~) such that hj(O) = 0, {3h j = 1. Then C
Next we show that B
=
1. Recalling (1.2) and using Lemma 4.9 we deduce
By Remark 4.7, to prove that C
~n
such that
as m + 00. By Theorem 2.1 (b), for each k = 1, ... ,n corresponding to each ak', there exists a sequence {( zk') II} in ~ such that hk (( zk') II) = ak' and lim (1 1(zk')1I12)lh~((zk')II)1 = 1 1  Ih k ((zk')I1)12 .
11+00
20
JOEL COHEN AND FLAVIA COLONNA
Let (zm)v = ((zl')v, ... , (z:)v). Then cp((zm)v) = (h 1 ((zl')v), ... , hn((z:)v)) = am and
Q
((zm)) = fo
(~I 8f (am )1 2 (1_ lamI2 )2)1/2 (1I(zr)vI2)lh~((zr)v)1 ~ 8z k
as m, 1I > 00. Thus (3fo
SUPzEAn
Qfo
>
1
1 lh k ((zr)v)i2
k
~
1. From (4.1) it follows that 0
5. Open questions We propose the following conjecture as a generalization of the onedimensional case (Theorem 1.1 and Corollary 2.2 (ii)). 1. For cp E H(f:l.n,f:l. n ), C
CONJECTURE cp(O) = 0, {3
1. If C
p(z,w) = sup{lf(z)  f(w)1 : {3f ::; I}
(5.1)
for any z, w E f:l. (see [23], Theorem 5.1.7). From Theorem 3.1 it follows immediately that sup{lf(z)  f(w)1 : {3f ::; I} ::; p(z, w) for all z, w E f:l. n . Does the opposite inequality hold for n ~ 2? Formula (5.1) also holds if the domain is the unit ball of (see [24], Theorem 3.9).
en
References [IJ J. M. Anderson, J. Clunie, Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 279(1974), 1237. [2J E. Cartan, Sur les domains bournes de l'espace de n variable complexes, Abh. Math. Sem. Univ. Hamburgh 11 (1935), 116162. [3J J. Cima, The basic properties of Bloch functions, Internat. J. Math. & Math. Sci. (2)3 (1979), 369413. [4J J. Cima, W. Wogen, On isometries of the Bloch spaces, Illinois J. Math. 24 (1980), 313316. [5J J. M. Cohen, F. Colonna, Bounded holomorphic functions on bounded symmetric domains, Trans. Amer. Math. Soc. (1)343 (1994), pp. 135156. [6] J. M. Cohen, F. Colonna, Preimages of onepoint sets of Bloch and normal functions, Mediterr. J. Math. 3(2006), 513532. [7] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc., 72 (2005), pp. 283290. [8] F. Colonna, Bloch and normal functions and their relation, Rend. Circ. Mat. Palermo, Ser. II XXXVIII (1989), pp. 161180. [9J F. Colonna, The Bloch constant of bounded analytic functions, J. London Math. Soc. (2)36 (1987), pp. 95101.
ISOMETRIC COMPOSITION OPERATORS ON THE BLOCH SPACE IN .6. n
21
[10] C. Cowen, B. MacCluer, Composition Operators on Spaces of Analytic FUnctions, Studies in Advanced Mathematics, CRC Press, Boca Raton (1995). [ll] P. Gorkin, R. Mortini, Universal Blaschke products, Math. Proc. Camb. Phil. Soc., 136 (2004),175184. [12] P. Gorkin, R. Mortini, Value distribution of interpolating Blaschke products, J. London Math. Soc., (2)72 (2005), 151168. [13] M. Gromov, Structures Metriques pour les Varietes Riemaniennes, Cedic/Fernand Nathan, Paris, 198!. [14] K. T. Hahn, Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem, Canad. J. Math. 27 (1975), 446458. [15] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New YorkLondon, 1962. [16] K. Madigan, A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., (7)347, 1995. [17] M. J. Martin, D. Vukotic, Isometries of the Bloch space among the composition operators, Bull. London Math. Soc. (to appear). [18] W. Rudin, FUnction Theory in Polydiscs, W. A. Benjamin, Inc., New York, 1969. [19] J. H. Shapiro, Composition operators and classical function theory, Springer, New York, 1993. [20] R. M. Timoney, Bloch functions in several complex variables, I, Bull. London Math. Soc. 12 (1980),241267. [21] R. M. Timoney, Bloch functions in several complex variables, II, J. Reine Angew. Math. 319 (1980), 122. [22] G. Zhang, Bloch constants of bounded symmetric domains, Trans. Amer. Math. Soc. 349(1997), 29412949. [23] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. [24] K. Zhu, Spaces of Holomorphic FUnctions in the Unit Ball, Springer, New York, 2005. [25] Z. Zhou, J. Shi, Composition operators on the Bloch space in poly disks, Complex Variables 46 (2001), 7388. [26] C. Xiong, Norm of composition operators on the Bloch space, Bull. Austral. Math. Soc. 70 (2004), pp. 293299. UNIVERSITY OF MARYLAND, COLLEGE PARK, MARYLAND
Email address: jmc(Qmath. umd. edu GEORGE MASON UNIVERSITY, FAIRFAX, VIRGINIA
Email address: f colonna(Qgmu. edu
Contemporary Mathematics Volume 454, 2008
Pluripolarity of Manifolds Oleg Eroshkin
1. Introduction
A set E c (Cn is called pluripolar if there exists a nonconstant plurisubharmonic function if; such that if; == 00 on E. Pluripolar sets form a natural category of "small" sets in complex analysis. Pluripolar sets are polar, so they have Lebesgue measure zero, but there are no simple criteria to determine pluripolarity. In this paper we discuss the conditions that ensure pluripolarity for smooth manifolds. This problem has a long history. S. Pinchuk [16], using Bishop's "gluing disks" method, proved that a generic manifold of class C 3 is nonpluripolar. Recall that the manifold M c (Cn is called generic at a point p E M, if the tangent space TpM is not contained in a proper complex subspace of (Cn. A. Sadullaev [17], using the same method proved that a subset of positive measure of a generic manifold of class C 3 is nonpluripolar. In the opposite direction E. Bedford [4] showed that a realanalytic nowhere generic manifold is pluripolar. For some applications to harmonic analysis the condition of realanalyticity is too restrictive. However, the result does not hold for merely smooth manifolds. K. Diederich and J. E. Fornaess [7] found an example of a nonpluripolar smooth curve in (C2. They construct a function f E COO [0, 1] such that the graph of this function is not pluripolar. In this example the derivatives f(k) grow very fast as k ~ 00. Recently, D. Coman, N. Levenberg and E. A. Poletsky [6] proved that curves of Gevrey class GS, s < n + 1 in (Cn are pluripolar. We generalize this result to higher dimensional manifolds. Recall that the submanifold M c (Cn is called totally real if for every p E M the tangent space TpM contains no complex line. THEOREM
If dim M
=m
1.1. Let M c (Cn be a totally real submanifold of Gevrey class GS. and ms < n, then M is pluripolar.
In fact we prove a stronger result. It follows from Theorem 2.1 in [1] that a compact set X C (Cn is pluripolar if and only if for any bounded domain D containing X and E > 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 nonconstant 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 Kdim X) with the following properties.
en
(1) O:S Kdim X :S n. (2) Kdim (3)
X=
Kdim X. m
Kdim
UXj = max{Kdim Xj : j = 1, ... ,m} . j=l
en
(4) If D c is a domain, XeD and ¢ : D then Kdim ¢(X) :S Kdim X. (5) If Kdim 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 Kdim X :S ms. REMARK 1.4. This estimate is sharp. The similar estimates hold for more general class of CRmanifolds. 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 Eentropy of traces on X of bounded holomorphic functions. The definition and basic properties of Eentropy 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 multiindices 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
o<>f=
01<>1 f· 0<>1 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 multiindex 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 centropy Let (E, p) be a totally bounded metric space. A family of sets {Cj } of diameter not greater than 2c is called an ccovering of E if E ~ UCj . Let Nc;(E) be the smallest cardinality of the ccovering. A set Y ~ E is called cdistinguishable 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 cdistinguishable set. For a nonempty totally bounded set E the natural logarithm
J(c;(E) = log Nc;(E) is called the centropy. The notion of centropy 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 centropy. 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 supmetric, 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 Ecovering of E j • Then the family
{C ll ! x C2l2 is an Ecovering of E.
X ... X
Ckl k
:
lj = 1, ... ,Nj }
o
We also need upper bounds for Eentropy of a ball in finitedimensional Coo space. Let IR~ be IR n with the supnorm:
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 Eentropy 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 simplyconnected 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,';',<''HO 10gn+1(1/E) and to calculate it explicitly. V. P. Zahariuta [20] showed how the solution of Kolmogorov problem will follow from the existence of the uniform approximation of the relative extremal plurisubharmonic function UX,D by multipole pluricomplex Green functions with logarithmic poles in X (Zahariuta conjecture). Later this conjecture was proved by Nivoche [15] for a "nice" pairs (D, X). Therefore it is established that for such pairs . :Hg(A~) 1lm";'c,<''HO 10gn+1(1/E)
C(X,D) (21l')n
PLURIPOLARITY OF MANIFOLDS
27
where C(X, D) is the relative capacity (see [5]). The pluripolarity of X is equivalent to the condition C(X, D) = 0 ([5]). If J{£(A~) = o(logn+1(~)) then X is "small" (pluripolar) and the asymptotics of an Eentropy can be used to determine how "small" X is. We will use the function
IlJ(X, D) = lim sup logJ{£(A!) 1 £>0 log log E to characterize the "dimension" of X. For a compact subset X c en we define the Kolmogorov dimension Kdim X = IlJ(X, D), where D is a bounded domain containing X. A. N. Kolmogorov proposed in [12] to use IlJ(X, D) as a functional dimension of space of holomorphic functions on X. The idea to use IlJ(X, D) to characterize the "size" of compact X seems to be new. We proceed to prove that IlJ(X, D) is independent of the bounded domain containing X. (4.1)
LEMMA 4.1. Let D c en be a bounded domain. If Xl, X 2 , ... , Xk are compact subsets of D, then PROOF. Let X = UX j . Embeddings Xj > X generate a natural isometric embedding C(X) > C(X l ) x C(X 2) x ... X C(Xk)' Restriction of this embedding on A~ gives an isometric embedding A~ > A~, X A~2 X ... X A~k' By Lemma 3.2 k
J{£(A~) ::; LJ{£(A~j) ::; kmaxJ{£(A~j) ,
(4.2)
j=l
o
and the result follows. For a point a E en and R > 0, we denote R with center at a ~(a,R)
~(a,
R) the open polydisk of radius
= {z = (Zl,Z2, ... ,Zn): IZj ajl < R,
j
= 1, ... ,n} .
The following well known result follows directly from Cauchy's formula. LEMMA 4.2. Let Rand r be real numbers, R > r > O. Let a E en and f be a bounded analytic function on the poly disk ~(a, R). Then for any positive integer k there exists a polynomial Pk of degree k such that (4.3)
sup
wE~(a,r)
1 (r)k If(w)  Pk(w)1 ::; ~ Ii 
r
sup
zE~(a,R)
If(z)l·
LEMMA 4.3. Let Rand r be real numbers, R > r > O. If a polynomial P of degree k satisfies the inequality IP(w)1 ::; A for every w E ~(a,r), then for z E ~(a, R) we have (4.4) PROOF. Let Q(A) = P((A(Z  a) + a). The inequality follows from the appli0 cation of maximum modulus principle to Q(A)/Ak. THEOREM 4.4. Let X be a compact in en. If D l , D2 c en are bounded domains containing X, then IlJ(X, Dd = IlJ(X, D 2 ).
OLEG EROSHKIN
Without loss of generality we may assume that Dl c D 2. Then A~l :J A~2 and IJ1(X, DJ) ~ IJ1(X, D2)' We establish the special case of two polydisks first. Suppose Dl = ~(a, r) and D2 = ~(a, R), where R > r > O. Choose r' > 0 such that r > r' and X c ~(a, r'). Let {II, 12, ... , fN} C A~l be a maximal cdistinguishable set with N = Me:(A~l). By Lemma 4.2 there exist a positive integer k and polynomials {PI, P2, ... , PN } of degree k such that PROOF.
sup IfJ(z)  pj(z)1 < c/3
for j
=
1,2, ... ,N
zEX
i,
and k :::; L log where L depends only on Rand r'. Then polynomials {PI, P2, ... , PN} are c/3distinguishable. By Lemma 4.3, polynomials
qj =
(~)k Pj
are bounded on D2 by 1. There exist positive constants c,.x, which depend only on R, rand r', but not on c, such that polynomials {ql, Q2, ... , qN} are bdistinguishable (as points in A ~2), where b = cc A • Hence No(A~2) :::; Mo(A~2) :::; NE(A~l) ,
(4.5)
and IJ1(X, DJ) = IJ1(X, D2)' If Dl = ~(a, r) and D2 is an arbitrary bounded domain containing D 1 , then there exists R > 0, such that D3 = ~(a, R) :J D 2 . In this case the theorem follows from the inequalities
IJ1(X, Dd
~
IJ1(X, D 2) ~ IJ1(X, D 3 )
=
IJ1(X, D 1 )
.
Now consider the general case. Let polydisks ~1' ~2"'" ~s C Dl form an open cover of X. There exist compact sets X 1 ,X2,oo.Xs such that Xj C ~j for j = 1,2, ... , sand UXj = X. Then
IJ1(Xj, D 1 )
=
IJ1(Xj, ~j) = IJ1(Xj, D 2) ,
for j = 1,2, ... , s.
o
The theorem follows now from Lemma 4.1. EXAMPLE 4.5. Let X =
~(O,
r). Let R > rand D
= ~(O, R). Kolmogorov
[12] (see also [14]) showed that (4.6)
J(E(A~)=C(n,r,R) ( log~1) n+ 1 +0
( (
log~1) n loglog~1 )
Therefore IJ1(X, D) = nand Kdim X = n. THEOREM 4.6. Let X be a compact subset of en. The K olmogorov dimension Kdim X satisfies the following properties. (1) 0:::; Kdim X :::; n. (2) Kdim{z}=O. (3) Kdim X = Kdim X. (4) IfY c X then Kdim Y:::; Kdim X. (5) If {Xj} is a finite family of compact subsets of en, then m
Kdim
UXj = max{Kdim Xj : j = 1, ... , m} . j=1
(6) If D c en is a domain, XeD and ¢ : D then Kdim ¢(X) :::; Kdim X.
+
ek
is a holomorphic map,
PLURIPOLARITY OF MANIFOLDS (7) If Kdim X
29
< n, then X is pluripolar.
REMARK 4.7. Property (5) does not hold for countable unions. There exists a countable compact set X such that n = Kdim X (see Example 4.10). Such set X also provides a counterexample to the converse of (7). PROOF. Properties (4) and (6) follow immediately from the definition. Property (2) follows from Lemma 3.3. Lemma 4.1 implies (5). The inequality Kdim X ::::: immediately follows from the definition. From (4.6) follows that Kolmogorov dimension of a closed polydisk equal n. Therefore (4) implies the second part of (1). To show (3) consider a Runge domain D containing X. Let W = X. Then A~ and A{.?r are isometric, hence \lI(X, D) = \lI(X, D) and (3) follows. The property (7) follows from the following theorem and Lemma 4.11. 0
°
THEOREM 4.8. Let X be a compact subset of en, such that Kdim X = s < n. Then for every 1 < h < ~ and every N > No = No(h) there exists a nonconstant polynomial P E Z[ZI,Z2, ... ,Zn], degP::::: N with coefficients bounded by exp(N h ), such that sup {IP(z)1 : z E X} < exp( _N h ) .
(4.7)
PROOF. The result follows from Dirichlet principle. Let D = 6.(0, R) be a polydisk containing X. Assume that R > 1. Let t: = ~ exp(  2N h  N log R n log N). Choose t such that s < t < ~. For large enough N there exists an t:covering of A~ with cardinality::::: exp {(log ~)t+I}. Let T = [exp(n h )] (integer part of exp(n h )). There are M
= T(N~n)
polynomials of degree at most N with coefficients in {I, 2, ... T}. Let {PI,P21 ... PM} be a list of all such polynomials. Clearly the polynomial q J 
1 NnRN exp(Nh)
p. J
belongs to A~. By our choice of t, h(t + 1) < n + h, therefore there are more polynomials qj than cardinality of the t:covering, so there are two polynomials, let say ql and q2 such that Iql(Z)  q2(z)1 ::::: 2t:
Then P
=
for every z E X.
o
PI  P2 satisfies (4.7).
Let PN be the set of all polynomials (with complex coefficients) on en of the degree::::: N, whose supremum on a unit polydisk 6.(0,1) is at least 1. COROLLARY 4.9. If Kdim X = s < n, then for every 1 N> No = No(h) there exists polynomial P E PN, such that (4.8)
<
h
<
~
and every
sup {IP(z)1 : z E X} < exp( _N h ) .
Corollary 4.9 may be used to bound Kolmogorov dimension from below.
OLEC EROSHKIN
30
EXAMPLE 4.10. Given 0 < r < 1 and a positive integer N there exists a finite set Xr,N C ~(O,r), such that for any polynomial P E PN, the following inequality holds 1
(4.9)
max IP(z)12 _r N 2
ZEXr,N
.
For example, if E = ~tJ r N , then a maximal Edistinguishable subset of ~(O, r) satisfies condition (4.9). Let 00
x=
U
X1/k,k'
k=2
Then X is compact and for any N > 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 Kdim X = n. To finish the proof of Theorem 4.6 we need the following wellknown 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 mdimensional 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, realvalued 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 multiindex 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 = NaN. 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!)Sl 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 supnorm 1R~ the ball B of a radius eN (N!)SI. By Lemma 3.3,
t
1C,,(B) ::; 2 (N ;
m)
log
(eN (~!)8I + 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), Kdim X = W(X, D) ::; mt. The only restriction imposed on t so far was t > s. Hence Kdim 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:!)81 .
= (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, 407417. [2] K. I. Babenko, On the entropy of a class of analytic functions, Nauchn. Dokl. Vyssh. Shkol. Ser. Fiz.Mat. Nauk (1958), no. 2, 916. [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 LelongHenri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, Lecture Notes in Math., vol. 919, Springer, Berlin, 1982, pp. 294323.
PLURIPOLARITY OF MANIFOLDS
33
[5] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 12, 140. [6] D. Coman, N. Levenberg, and E. A. Poletsky, Quasianalyticity and pluripolarity, J. Amer. Math. Soc. 18 (2005), no. 2, 239252. [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, 931936. [8] V. D. Erohin, Asymptotic theory of the centropy of analytic functions, Dokl. Akad. Nauk SSSR 120 (1958), 949952. [9] ___ , Best linear approximation of functions analytically continuable from a given continuum to a given region, Uspehi Mat. Nauk 23 (1968), no. 1 (139),91132. [10] S. D. Fisher and C. A. Micchelli, The nwidth of sets of analytic functions, Duke Math. J. 47 (1980), no. 4, 78980l. [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, centropy and ccapacity of sets in functional space, Amer. Math. Soc. Trans!. (2) 17 (1961), 277364. [13] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 6972. [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, 413450. [16] S. I. Pincuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki 15 (1974), 205212. [17] A. Sadullaev, A boundary uniqueness theorem in en, Mat. Sb. (N.S.) 101(143) (1976), no. 4, 568583, 639. [18] M. Waldschmidt, Diophantine approximation on linear algebraic groups, SpringerVerlag, Berlin, 2000. [19] H. Widom, Rational approximation and ndimensional diameter, J. Approximation Theory 5 (1972), 34336l. [20] V. P. Zahariuta, Spaces of analytic functions and maximal plurisubharmonic functions, Doc. Sci. Thesis, RostovonDon, 1984. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW HAMPSHIRE, DURHAM,
03824 Email address: oleg. eroshkinlDunh. edu
NEW HAMPSHIRE
Contemporary Mathematics Volume 454, 2008
On a question of Brezis and Korevaar concerning a class of squaresummable 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 Bn3zisNirenberg 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 BrezisNirenberg Result We shall proceed by a number of lemmas. LEMMA
u( 0) :=
2.1.
Under
the
hypothesis
the
(1),
21rperiodic
function
2::%"=00 akeikIJ is welldefined and unimodular for almost all 0 E [0, 21r).
PROOF. By the RieszFischer 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
Lt
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 0
21r 121r 1U(O)  U(tp) 12 '0'
e'  e''P
0
00
= 47r 2
L
dO dtp
Ikllbkl 2 .
k=oo
Unfortunately, it is in general false that ample is 1/(0) := eiO since
2::;:"=00 Ikllbkl2 < 00;
21r 121r 1 0  tp 12 iO i dO dtp 1o 0 e e'P
an easy counterex
= 00.
The next result we shall need must be well known (in fact a stronger result can be found in [I, pp. 2122]) and we therefore state it without proof: LEMMA 2.2. There exists a sequence {UN}r of continuously differentiable unimodular functions UN(O) := 2::;:"=00 ak(N)e ikO such that
00
R. FOURNIER AND L. SALINAS
38
and 00
:L
Ikllak(NW <
00,
N 2: 1.
k=oo Also related to Lemma 2.2 is an older result due to Krein [9] stating in particular that the class H 1 / 2 (see the definition below) is a Banach algebra closed under composition with certain analytic functions. We shall now prove that the result of Bn§zis and Nirenberg holds for sequences {an}~=_oo corresponding to smooth enough functions u(O) := L:~=oo akeikfJ. LEMMA 2.3. Let u(O) := L:~=oo akeikfJ where {ad satisfies (1) and (2) and with a continuous derivative u' (0) over [0,2rr]. Then L:~=oo klak 12 is an integer. r 27r  
Let 1> = 1/2rri Jo u(t)u'(t) dt. By using the functions ur(O) as in the proof of Lemma 2.1, we obtain PROOF.
00
1>
=
:L
klakl 2.
k=oo Let us also define A(O) := J~ u(t)u' (t) dt and cp(O) := u(O)e)'(fJ). Then by Lemma 2.1
cp'(O) = (1 lu(OW)u'(O)e)'(fJ) == 0, We have that cp(O) == cp(O) = u(O) = u(2rr) and follows.
0 E [0,2rr].
e)'(27r)
=
e27ricl>
= 1. The claim D
We may now prove the general Bn§zisNirenberg result. Our arguments become clearer when expressed within the frame of certain Sobolev spaces (although we shall not use any of the deeper results concerning these spaces). Let Hl/2 denote the set of all 2rrperiodic Fourier series L:~=oo CkeikfJ which satisfy L:~=oo IkliCk 12 < 00. It is known [11, pp. 246247] that Hl/2 can be turned into a Banach space when endowed with the norm
For example Lemma 2.2 amounts to the fact that the subspace of smooth functions in Hl/2 is dense in Hl/2. Let us consider a sequence {an}~=_oo satisfying (1), (2) and u(O) := L:~=oo ane infJ ; let also {UN }N=l' UN(O) := L:%,:oo ak(N)e ikfJ , be the sequence of smooth functions given by Lemma 2.2. Because limN~oo IluN  ull = 0, we have that IluN II is bounded above and by Lemma 2.3, the sequence of integers L:~=oo klak(N)1 2, N = 1,2, ... is therefore also bounded above; we may therefore assume that for N large enough, L:%,:oo klak(N)j2 := I is a fixed integer depending only on u. Further 00
00
00
k=oo
k=oo
00
k=oo
00
00
k=oo
k=oo
k=oo
QUESTION OF BREZIS AND KOREVAAR
39
where
and
12 k~OOkak(N)(akak(N))1
2k~oolkllak(N)llakak(N)1
:::;
(J:~klla.(N)I') 'I' (J:~kllak a,(N)I') 'I'
:5 2
:::; 211uNliliu  uN11 :::; 2(llull + l)llu  uNII· We clearly have 00
00
k=oo
k=oo 3. Proof of Theorem 1.1
We write W(z):= L:%"=oakzk for z E lDl:={z Ilzl <1}. Because L:%':o lakl 2 < 00, W is welldefined and in fact belongs to the Hardy space H2(lDl) (see [7] for a standard reference concerning Hardy spaces). By Lemma 2.1, (with ak = 0 for k < 0), the radial limits (which are known to exist a.e. [0,27l"))
W(e iO ):= lim W(re iO ) r>l O
satisfy IW(eiO)1 = 1 almost everywhere, i.e., W is an inner function. If W is neither a constant nor a finite Blaschke product, it is known that W takes on all values in the unit disc lDl infinitely many times, except possibly for an exceptional set of planar measure zero (this is due to Frostman [5, p. 35]). In other words, 00
= lim r+1
Je r
} x2+y2
~r2
IW'(x
+ iy)1 2 dxdy
This of course contradicts the hypotheses (2), again because of the Abel continuity theorem. A more refined argument leading to the same result can be found in [6, pp. 6061]. We may therefore assume that W is a constant of modulus 1 or else a finite Blaschke product, J
W(z)
= ;.., II z  aj j=1
1  a·z J
,
Iaj I < 1, j = 1,2, ... , J , 1(1 = 1
R. FOURNIER AND L. SALINAS
40
3
2
1
1
FIGURE
2
3
1
and then
by the argument principle. We end this paper by a remark: any unimodular function u( (}) of the type considered here is a limit of the type
u({}) = lim Bl,j(e iO)B2,j(eiIJ),
a.e. [7r,7r),
JOO
where {Bi,j} and {B2,j} are two sequences of Blaschke products. This is a result essentially due to Douglas and Rudin [7, p. 153] and one may wonder particularly in view of Theorem 1.1, if
(6) where Bl and B2 are Blaschke products (or more generally inner functions). We show that this is not true in general; consider for example a real trigonometric polynomial UIR as displayed in Fig. 1 and perturb UIR ((}) to a real function U({}) 7 6
3
2
1
FIGURE 2
1
2
3
QUESTION OF BREZIS AND KOREVAAR
41
whose graph is shown in Fig. 2. Clearly URI. E Hl/2 and since
IU(O)  U(
:s IUIR(O) 
UIR(
0,
we also have U E Hl/2. Let u(O) := eiU(!)). By the formula of Devinatz, u belongs to H 1 / 2 and if (6) holds for u, the function Bd B2 (i.e., a quotient of bounded analytic functions) has constant radial limits almost everywhere on an arc of the unit circle; it is well known [5, p. 41] that this can occur only if Bl is a constant multiple of B2 and this is ruled out by the definition of U.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13.
A. Boutet de MonvelBerthier, V. Georgescu, and R. Purice, A boundary value problem related to the GinzburgLandau model, Comm. Math. Phys. 142 (1991), no. 1, 123. H. Brezis, New questions related to the topological degree, The Unity of Mathematics, Progr. Math., vo!' 244, Birkhauser Boston, Boston, MA, 2006, pp. 137154. H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197263. ___ , Degree theory and BMO. II. Compact manifolds with boundaries, Selecta Math. (N.S.) 2 (1996), no. 3, 309368. E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, Cambridge, 1966. R.A. Hibschweiler and Thomas H. MacGregor, Fractional Cauchy Transform, Chapman and Hall/CRC, USA, 2006. P. Koosis, Introduction to Hp spaces, Cambridge Univ. Press, Cambridge, 1998. J. Korevaar, On a question of Brezis and Nirenberg concerning the degree of circle maps, Selecta Math. (N.S.) 5 (1999), no. 1, 107122. M. G. Krein, On some new Banach algebras and WienerLevy type theorems for Fourier series and integrals, Amer. Math. Soc. Trans!. 93 (1970), 177199. E. Landau and D. Gaier, Darstellung und Begriindung einiger neuerer Ergebnisse der F'unktionentheorie (German) [Presentation and explanation of some more recent results in function theory], 3rd ed., SpringerVerlag, Berlin, 1986. R. C. McOwen, Partial differential equations, Prentice Hall, Upper Saddle River, 2003. D. J. Newman and H. S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249255. B. Simon, Orthogonal polynomials on the unit circle. I. Classical theory, Amer. Math. Soc. Colloq. Pub!., vol 54, Part 1, Amer. Math. Soc., Providence, R.I., 2005.
DEPARTEMENT DE MATHEMATIQUES ET CENTRE DE RECHERCHES MATHEMATIQUES, UNIVERSITE DE MONTREAL, C.P. 6128, Succ. CENTREVILLE, MONTREAL, QUEBEC H3C 3J7, CANADA Email address: fournier~dms. wnontreal. ca DEPARTAMENTO DE INFORMATICA, UNIVERSIDAD TECNICA FEDERICO SANTA MARIA, VALPARAiso, CHILE Email address:lsalinas~inf.utfsm.cl
Contemporary Mathematics Volume 454, 2008
Approximating z in Hardy and Bergman Norms Zdeilka Guadarrama and Dmitry Khavinson ABSTRACT. We consider the problem of finding the best analytic approximation in Smirnov and Bergman norm to general monomials of the type znzm. We show that in the case of approximation to z in the annulus (and the disk) the best approximation is the same for all values of p. Moreover, the best approximations to z in Smirnov and Bergman spaces characterize disks and annuli.
1. Introduction
Throughout this paper, G denotes a bounded domain in C with boundary r consisting of n simple closed analytic curves. R( G) will stand for the uniform closure of the algebra of rational functions in G with poles outside of C. Let ds be the arclength measure on the boundary of G. Recall that a function belongs to the Smirnov class lEp(G) for 1 :::; p < 00, if it is analytic in G and there exists a sequence of finitely connected domains {Gn}~=l' G 1 C G 2 C G 3 c ...
I
=
with rectifiable boundaries r n so that U G n n=l
= G, and a constant M > 0 such that
1
1IIIIEp
:=
s~p [lnll(Z)lPdS] P :::; M
<
00.
For a nice and concise introduction to
Smirnov spaces see [6], also cf. [14].
da
We let denote area measure in G. The Bergman space Ap(G) for 1 :::; p < is the set of analytic functions I(z) in G, with finite norm IIIII Ap = IIIII Lp(da,G)
00
=
1
[fcll(Z)lPda] P (cf.
[7]).
D. Khavinson, in [10], [12], [13], [15], [16] posed the question of " how far" z is from being approximable by rational functions that are analytic in G. In particular, the following concept was introduced in [15], also cf. [4].
2000 Mathematics Subject Classification. 30ElO, 41A99. Key words and phrases. Analytic approximation, Hardy spaces, Bergman spaces. This work was supported in part by NSF Grant DMS013900S. 43
44
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
DEFINITION
1.1. The analytic content A( G) oj a given domain G is defined as
A(G):=
inC
gER(G)
liz 
g(z)111L (ds) = 00
inC
gER(G)
liz 
g(z)111L (da)' 00
It turns out that A( G) can be bounded above and below by basic quantities depending on the geometry of the domain G, specifically, its area and perimeter. If we let A (G) denote the area of G and P (G) the perimeter of its boundary, the following inequality holds: 2A(G) < A(G) < VA(G) P(G) 1f
(1)
The upper bound is due to Alexander [3], and the lower bound is due to D. Khavinson [13]. We will refer to this inequality from now on as the AK inequality. It follows immediately from (1) that A (G) ::::: P:~G), which is the isoperimetric inequality. Moreover, when we notice that both inequalities in (1) are sharp, since they become equalities when the domain is a disk, we obtain the isoperimetric theorem (cf. [10]) .
The question of what are the extremal domains for the lower bound of (1) still remains open. A few equivalent formulations for the equation A( G) = in terms of geometry and potential theory can be found in [12] and [15]. The reader may consult the survey [4] which focuses on extremal domains for the left inequality in (1). The following conjecture [4], [14] remains open.
2:/:;/
CONJECTURE 1.2. For a fixed A (G), the only extremal domains Jor which the lower bound in (1) becomes an equality are the disks oj radius A(G), and annuli {z:r< Izl
For an extensive discussion about different forms and various ramifications of this conjecture we refer the reader to [4].
If we denote by EHG) the unit ball in El(G). We can write A(G):=
inC gER(G)
liz 
g(z)lllLoo(ds.f')
=
sup
fEIE; (G)
IIrzJ(z)dzl,
and there exist extremal functions g* (z) and 1* (z) for which the infimum and the supremum above are attained [19]. If the domain is a disk centered at the origin, the best rational approximation to z in G is the zero function. In the case of the annulus centered at the origin, the best approximation is g*(z) = ~r [10]. The main focus of this paper is to extend the concept of analytic content to the context of Smirnov and Bergman spaces for p ~ 1. The paper is organized as follows. In the next section we define the Smirnov panalytic content of a domain and show that the AK inequality extends to the Ep case yielding bounds for the Smirnov panalytic content in terms of the area
APPROXIMATING
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IN HARDY AND BERGMAN NORMS
45
and perimeter of the domain. In section 3 we find the best approximation to any monomial znzm in the Smirnov pnorm of the annulus and the disk. For disks and annuli, the best approximation to z turns out to be the same rational function for all p. We prove a converse for this result in the case of the disk, and for p = 1 in the case of the annulus. In section 4 we consider the Bergman panalytic content of a domain and explore similar questions, now for the case of the Bergman space pnorm. We conclude with some remarks and open questions.
2. Smirnov panalytic content DEFINITION AlE
(G):= p
2.1. The Smirnov panalytic content of a domain G is defined by liz  g(z)IIIL (ds r)'
inf gElEp(G)
P
,
The following general result summarizes the study of extremal problems in Smirnov classes (d. [19], Theorem 4.3).
COROLLARY
hold: (i) inf
gElEp(G)
2.2. Letp ~ 1, ~+i
Ilw(z)  g(z)IIIL (ds r) = p,
=
1, and letw(z)
sup
fElE~(G)
E
lLp(G) then the following
IJrw(z)f(z)dzl·
(ii) There exist extremal functions g*(z) E lEp(G) and J*(z) E lEq(G) for which the infimum and the supremum are attained in (i). (iii) g*(z) E lEp(G) and J*(z) E lEq(G) are extremal if and only if, almost everywhere on r, i8
(2)
J*(z)(w(z)  g*(z))dz
=
A:1Iw(Z)  g*(z)IPds, lEp
where b is a real constant and AlE p = Ilw(z)  g*(z)IIILp(ds,r)' We will refer to this last equality as the extremality condition in Smirnov spaces. For p > 1 the extremal functions g* (z) and J* (z) are unique, the latter up to a factor of eiO:. For p = 1 the extremal function J* (z) is unique up to a factor of eio:. If the domain G is simply connected, then g* (z) is also unique. If the domain G is n connected, n > 1, then the extremal function g* (z) is unique provided that f* (z) has more than n  2 zeros in G or that on a certain set T C r, meas(T) > 0, 1f*(z)1 < 1. Otherwise, the extremal function g*(z) may fail to be unique. (cf. Part 3, Theorem 3.2, in [19], and Theorem 3.6 (ii) below).
In our first Theorem we show that following the same strategy used in [10] we can find bounds for AlE p (G) in terms of the perimeter and the area of the domain G, obtaining the AK inequality as a limiting case when p approaches infinity.
46
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
THEOREM 2.3. Let p > 1, ~ + ~ = 1. Let G be a multiply connected domain in C bounded by n simple closed analytic curves, as before, A( G) denotes the area of G and P( G) the perimeter. Then 2A(G) < A (G) ylP(G)  IEp
(3)
lfp = 1, then 2A(G)
~
Al(G)
~
JA~C)
< 
J
A (G)
P(G)~
7r
P(G).
PROOF. We first address the lower bound in (3) for p > 1.
By Jensen's inequality, since p > 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
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IN HARDY AND BERGMAN NORMS
47
Holder's inequality yields
Let Fc(z)
=
~ J [w~zd(T,
z E C. Now, the AhlforBeurling 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
'
= <;
=
(ll~J[w~zd(TIP dS)*
(l (A~G))' d'r VA~G) p(G)i.
The theorem is proved.
o
3. Characterization of disks and annuli in terms of approximations to in lEp norm
z
PROPOSITION 3.1. Let p ~ 1 and let G = {z E C: Izl < r}. Then: (i) The best approximation in lEp(G) to a general monomial oJ the type w(z) = znzm for m > 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 (pl)(n+m)+k+l ei(k+mn)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 izmn1(znzm)dz = r 2m d(), Hence J*(z) = J(z)
Ir  izmnl+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(nrn)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)pl] . zP dz This last equation implies that for each contour S(z)pl is analytic throughout the
(5)
domain, except at the origin. We will now consider a few cases. CASE 1. p = 1
APPROXIMATING
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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:1arglzIP2ds = 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)pl 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)pl = 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
= constlzlkIds,
which in turn yields
(7)
[J*(Z)]2 Z2k
d
= const dz
[S(z)2kI] .
(6) implies that S(Z)2k1 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)2k1 is the same for every boundary component. Then z2k1 = S(z)2k1 = ~~F~~ + g(z) for g(z) analytic in G. Multiplying through by Z2k1 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
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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 = constlzlp2 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 * nm r2m+q(nm)+~ +R2m+q("m)+~ equal to g (z) = cz where c(n, m,p) = q(nmH!l q(nmH.'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(nm)+~ + Rq(nm)+~ :=
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
n7n
enTTL
zz"zzm__ :zn
m
IP
~~. Then
p(nm)+q(k+l) 1
2m ,P)127r ei(k+mn)O dB
r r2m 
C
C
0
= 0, unless k = n  m.
and 27r
 clP + rP(nm)+1 Ir2m  clP ) ( RP(nm)+1 IR2m R2m  c r2m  c
if
RP(nm)+1 IR 2m R2m 
clP C
P _ _ rP(nm)+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 )Pl
= (!!.)p(n ml+l
R2m  c
r
and after some algebra we obtain that r2m+q(nm)+~
c=
rq(nml+~
Therefore, f (z) annihilates IEp (G). Now let 1*(z) = IIIII I so that 111*11 Lq(d.)
+ R2m+q(nm)+~ + Rq(nm)+'f;
Lq(d.)
= 1 and let g*(z) = cz n m . Then,
Iznzm  cznml P ''_""7 1
Ilzn zm
cznmll~
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)nm(R2m_r2m) q(nmH!l q(nmH!l p
= 1 and n 
+R
m
q(n=H~+
R2=+q(n=H'l. q(nmH!l P,
p+R
p
2"'" [(r~R%(1nm))P "
53
we have
+ (R~r%(1nm))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 ksheeted covering of the unit disk. It is not a constant since znzmdz = 0 unless n = 1. So j*(z) maps G onto a ksheeted 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 panalytic 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 ksheeted 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 realvalued 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 panalytic content of a domain G is
AA(G):= P
inf
gEAp(G)
Ilzg(z)lllL(d17G)' p ,
By the HahnBanach 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<7,G). When p = 1 the conditions above become e ili J*(z)(w(z)  g*(z)) = Iw(z)  g*(z)1 a.e. in G. (iv) For p > 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 panalytic content of ][)) as follows: AAp (][)))
=
fo27r foT Iteit 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
APPROXIMATING z IN HARDY AND BERGMAN NORMS
57
In particular, if n = 0 and m = 1, for p 2: 1, the Bergman panalytic content of G in Ap is AAp (G) = lR t 1 
p
It 2

c1 P  1 sgn(t 2 .
2n lR
c)dt _
f
PROOF. Conslder f(z) 
n miP )
 cz 1.G ( iznzm zn z m  cz n m
=
Ic  t21P C ~ dt
where c(O, 1, p) is such that
O.
z n z 1Tt_ cz nm.[P
znzm_czn
zk da = =
1 0
m
,
21T JR iznzm  cznmi P zktdtd() r znz m  cz n  m
It2m  cl J Rt~Cnm)+k+l 2m t  c r
P
(1 21T e·Ck+Cnm))(Jd() . ) dt 0
= 0, unless k = n  m.
If we choose
f(z)
E
c = c(p), so that lRtPcnm)+1 It2m  c1 P  1 sgn(t2m 
c)dt = 0, then
Ann(Ap(G)).
Defining J* (z)
=
II f II f
we can check that necessary and sufficient (Corol
Lq(d",G)
lary 4.2 (iii)) conditions for extremality hold and the result follows. To compute A~ p (G):
o THEOREM 4.5. Let G be a bounded domain with analytic boundary. The best analytic approximation to z in AP (G) is g* (z) = 0 if and only if G is a disk. PROOF. If G is a disk, the best rational approximation to z in Ap(G) is g*(z) =
oby Remark 4.3.
Now suppose 0 is the best approximation to z in AP(G). First assume p > 1. In this case Corollary 4.2 (iii) can be written as izi P = APiJ*i q, so f(z) =
EF
annihilates AP(G), J*(z) u E w~,q(G). Hence,
=
dCWllq' and by Khavin's Lemma J*(z) = ~~ for some au
izi P
 =const. Oz z Integrating with respect to z we obtain u(z) =
J~dZ
=
const
Jzh~ldZ
=
constizi P + h(z),
where h(z) is analytic. Since u(z) E w~,q(G) and izi Pis real analytic near r, it is easy to see that for any sequence of domains G j , UG j = G with rectifiable boundaries r j , iiuiilLqcrj,ds) are bounded, so h(z) E lEq(G). Now, u(z) = 0 a.e. on r, hence h(z) = izi Pa.e. on r. Since r is analytic, and h(z) E lEq(G) for q 2: 1 and has bounded boundary
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
58
values, h(z) is bounded in G. But h(z) has real boundary values on r a.e., hence h(z) is constant. Thus IzlP is constant a.e. on r, so G is a disk centered at the origin. The case p = 1 and q = 00 requires only small modifications that are left to the reader. D
Along the same lines we also have: THEOREM 4.6. Let G be a finitely connected domain with analytic boundary, p ?': 1. G is an annulus centered at the origin if and only if the best analytic approximation to z in Al (G) is g* (z) = ;.
PROOF. If G is an annulus we have already seen in Proposition 4.4 that the best rational approximation to z in Ap (G), p ?': 1, is g* (z) = ;. To prove the converse, suppose the best approximation to z in AP (G) is g* (z) = ;. Once again for the sake of clarity we focus on the case p > 1, the remaining case only requires small modifications that are left to the reader. Corollary 4.2 (iii) yields that J*(z) = rzz:}'r E Ann(AP(G)), and Khavin's Lemma yields that J*(z) = ~~ for some u E w~,qz(G). Denoting Izl by r we have
ou
Izz  clP zzc
Oz Integrating we have that
(10)
J
JIr2  c1 =J J
Ou =dZ = uZ
r P
2
z :e. :e.
Z2Z2
P 1
z
sign(r2  c)dZ
(r2  C)pl d r P  2 sign(r2  c) dZ log IzzldZ
=2
(r2  C)pl 2 sign(r2  c)dlogr. r P
Since 0 rt. G, this integral is bounded away from zero and yields a realvalued function F(r) for all r > O. So u(z) = F(r) + h(z) with h(z) analytic. As in the proof of Theorem 4.5, h(z) extends across the boundary and hence belongs to lHlOO(G). Now, because u(z) E w~,q(G), u(z) = F(r) + h(z) = 0 a.e. on the boundary of G. So h(z) is real valued almost everywhere on the boundary. Hence it has to be real inside the domain as well and therefore constant. Now note that u = F(r) + const and u = 0 on r. Moreover from (10) it readily follows that F'(r) = 0 only at one point ro = .jC, where F' changes sign. Hence F may take the same value only twice, so r consists of two components and on each one the value of F(r) is the same, i.e. r consists of two concentric arcs centered at the origin. Since G is bounded and 0 rt. G (~ is analytic in G!), G must be an annulus. D
APPROXIMATING
z
IN HARDY AND BERGMAN NORMS
59
5. Final Remarks For the Bergman norm, assuming that G is a multiply connected domain with analytic boundary, we were able to prove that for all p 2: 1 the domain is an annulus whenever the best approximation to z is ~ (and that the domain is a disk, whenever the best approximation to z is a constant function). Our proof relies on the assumption of analyticity of the boundary. However, it is easy to see from the proof that this assumption can be relaxed and we only need assume that the domain G is Smirnov. We do not know whether the result holds for domains with arbitrary rectifiable boundaries. The Smirnov norms case turns out to be more difficult. One of the reasons is that knowing the best approximation in Bergman norm determines the extremal function in the dual problem throughout the domain (although in a vast set Ann(Ap)). In the lEp setting, it only determines the extremal function in the dual problem, although analytic in the domain, on parts of the boundary where z does not coincide with its best approximation, which unfortunately could happen a priori. In the Bergman setting this can never happen because two real analytic functions can never coincide on a set of positive area measure without being identical. If a constant is the best approximation to z in lEp, we showed in Theorem 3.2, for multiply connected domains and under the assumption of analyticity of the boundary, that the domain is a disk. We were able to reach the same conclusion for Jordan domains with rectifiable boundaries satisfying the Smirnov condition, but only when the domain is simply connected. We think it should be possible to generalize Theorem 3.2 to multiply connected domains with weaker regularity conditions imposed on the boundary.
The following question seems natural in connection with Remark 3.4 (and Thm. 3.2). Let G be a finitely connected domain containing the origin and assume that Q: E R, on the boundary r is a representing measure at the origin for analytic functions in G, say, continuous in C.
(*) the measure constlzl<>ds,
Does condition (*) alone imply that G must be simply connected? If so, (cf. Remark 3.4) then for Q: > 2, G must be a disk centered at the origin. Perhaps, condition (*) implies that G is simply connected only for specific values of Q:, what are these values and what happens in the remaining cases? Under a less restrictive regularity assumption, say assuming the boundary of G merely rectifiable, even for Q: = 0, there exist highly nonregular, nonSmirnov domains, so called pseudocircles, for which (*) still holds (cf. [6], Ch. 10). When the domain is an annulus we found the best lEpapproximation to any monomial znzm explicitly for all p > 1 and for p = 1 when n  m = 1. Yet, when p = 1 and n  m =1= 1, the extremal function f* in the dual problem is a trascendental function, hence we can only conclude that the best approximation to znzm is not a monomial. It would be worthwhile to study the best approximation of such monomials in lEl of the annulus in greater detail.
60
ZDENKA GUADARRAMA AND DMITRY KHAVINSON
In Theorem 3.10 we show for p = 1 that the domain is an annulus whenever the best approximation to z is:;. In the proof we use Lemma 3.9 which, via the argument principle, shows that the boundary of the domain consists of two boundary components. However, the hypothesis of the lemma assumes that the boundary of the domain is analytic, and that z =I :; on every boundary component of the domain. If z = :; on some component, then that boundary contour is a circle but already our argument that the boundary has two components fails since the argument principle can only estimate the change in the argument of f* on the remaining components. It should be possible to coach the proof of Theorem 3.10 to include the case when z = :; on a boundary component and to extend it to all p ~ 1, but we have not been able to do it.
References
[1] Adams R. A., Sobolev Spaces, Academic Press, New York,1975. [2] Ahlfors L., Beurling A., Conformal invariants and function theoretic null sets, Acta Math., 83 (1950) 101129. [3] Alexander H., Projections of polynomial hulls, J. Funct. Anal. 3 (1973), 1319. [4] Beneteau C., Khavinson D., The isoperimetric inequality via approximation theory and free boundary problems, Comput. Methods Funct. Theory, 6 (2006), No.2, 253274. [5] Davis P., The Schwarz Function and its Applications, Carus Math. Monographs 17, Math. Assoc. of America, 1974. [6] Duren P., Theory of lHIP Spaces, Academic Press, New York, 1980. [7] Duren P., Schuster, A., Bergman Spaces, American Mathematical Society, Providence, Rhode Island, 2004. [8] Ebenfelt P., Khavinson D. and H. S., Shapiro, A free boundary problem related to singlelayer potentials, Ann. Acad. Scie. Fenn. Math., 27 (2002), 2146. [9] Evans L., Partial Differential Equations, American mathematical Society, Providence, Rhode Island, 1998. [10] Gamelin T., Khavinson D., The isoperimetric inequality and rational approximation, Amer. Math. Monthly 96 (1989), 1830. [11] Gustafsson B., Khavinson D., Approximation by harmonic vector fields, Houston J. Math. 20 (1994), no.l, 7592. [12] Khavinson D., An isoperimetric problem, Linear and Complex Analysis, Problem Book 3, Part II, V.P. Khavin and N.K. Nikolsky, eds., Lecture Notes Math., SpringerVerlag, 1574 (1994), 133135. [13] Khavinson D., Annihilating measures of the algebra R(X), J. Funct. Anal. 58 (1984), 175193. [14] Khavinson D., Smirnov classes of analytic functions in multiply connected domains, Appendix to the English translation of Foundations of the Theory of Extremal Problems for Bounded Analytic Functions and various generalizations of them, by S. Ya. Khavinson, Amer. Math. Soc. Transl. (2), 129 (1986), 5761. [15] Khavinson D., Symmetry and uniform approximation by analytic functions, Proc. Amer. Math. Soc. 101 (1987), 475483.
APPROXIMATING
z
IN HARDY AND BERGMAN NORMS
61
[16] Khavinson D., Luecking D., On an extremal problem in the theory of rational approximation, J. Approx. Th. 50 (1987), 127132. [17] Khavinson D., McCarthy J. and Shapiro H. S., Best approximation in the mean by analytic and harmonic functions, Indiana University Mathematics Journal, (4),49 (2000), 14811513. [18] Khavinson D., Stessin M., Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math J., 46 (1997), 933974. [19] Khavinson S. Ya., Foundations of the Theory of Extremal Problems for Bounded Analytic Functions and various generalizations of them, Amer. Math. Soc. Transl. (2), 129 (1986), 156. [20] Shapiro H. S., The Schwarz Function and its generalization to higher dimensions, University of Arkansas Lecture Notes in Mathematical Sciences, 9 Wiley, (1992). DEPARTMENT OF MATHEMATICS AND PHYSICS, ROCKHURST UNIVERSITY, KANSAS CITY, MIS
64110 Email address: guadarrama«!rockhurst. edu
SOURI,
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF SOUTH FLORIDA, TAMPA, FLORIDA
Email address: dkhavins«!cas. usf . edu
33620
Contemporary Mathematics Volume 454, 2008
A General View of Multipliers and Composition Operators II Don Hadwin and Eric Nordgren
1. Introduction
In this paper we continue our investigation of multiplier pairs begun in [5]. We give an alternative view of a multiplier pair in terms of an algebra of operators with a separating cyclic vector. A case of particular importance is LOO[O, 1], and we are led to study symmetric norms in this context and then unitarily invariant norms on type I h factor von Neumann algebras. In addition we start an investigation of multiplier pairs of tensor products. 2. Preliminaries We call a pair (X, Y) a multiplier pair provided X is a Banach space, Y is a Hausdorff topological vector space, X C Y, and the inclusion map is continuous. Moreover, suppose we have a bilinear map (multiplication) from m : X x X > Y, with the notation m (u, v) = u . v such that (1) m is separately continuous, (2) The sets Co = {x EX: x . X c X} and no = {x EX: X . x C X} are dense in X, (3) There is an e E X such that, for every x E X, X· e = e· x = x. (4) There are dense subsets E C Co, Fe X, G c no such that,
(u·v)·w=u·(v·w) whenever u E E,v E F,w E G. If x E X we define Lx and Rx from X into Y by
Lxw
=x
. wand Rxw
= w . x,
where the domain of Lx is Dom (Lx) = {w EX: X· wE X} and the domain of Rx is Dom (Rx) = {w EX: W· x E X}. We define C = {Lx: x E Co} and R={Rx : x E no}. THEOREM 2.1. The following are true. (1) The multiplication· is jointly continuous from X x X to Y. 2000 Mathematics Subject Classification. Primary 46B28. ©2008 American Mathematical Society
63
64
DON HADWIN AND ERIC NORDGREN
(2) For every x E X, Lx, Rx are densely defined closed operators on X. (3) Lx is bounded on no if and only if x E Co, and Rx is bounded on Co if and only if x E no. (4) C, ncB (X). (5) If u, v E Co or v, W E no or u E Co, W E no, then
(u· v) . W =
U·
(v· w).
(6) C' = nand n' = C. (7) LvLw = L v.w if v, WECo and RvRw
= R w.v
if v, W E no.
It was proved in [5] that it is possible to choose Y so that Y is a Banach space and ball Y is the closed convex hull K = co ((ball X) . (ball X)) if and only if K contains no lines (Le., lib for some nonzero vector x). In this case (X, Y) is called a natural multiplier pair and Y is called the cospace of X. A cospace exists whenever Y can be chosen so that the continuous linear functionals on Y separate the points ofY. In the multiplier pair setting a notion of composition operator was defined that coincides with the usual notion in all of the classical cases. Suppose Ct : C  C is a unital algebra homomorphism. Since C and Co are isomorphic ( x 1+ Lx), Ct induces a unital algebra homomorphism (} : Co  Co defined by
L&(x) =
Ct
(Lx) .
If (} is bounded (closable) on Co, we denote its continuous extension (closure) on X by CO/., We call CO/. the (left) composition operator induced by Ct. Similarly, if f3 : n  n is a unital algebra homomorphism, we can define a unital algebra homomorphism /3 : no  no by
RI3(x) = f3 (Rx ). Note that the multiplicativity of /3 follows from
RI3(x.y) = f3 (Rx.y) = f3 (RyRx) = f3 (Ry) f3 (Rx) = RI3(y)RI3(x) = R 13 (x).I3(y)" We denote the continuous extension (closure) of /3 on X by Cfj, and we call Cfj the (right) composition operator induced by f3. 3. Completing Norms on U,o [0,1] There is another way to view multiplier pairs. Suppose X is a Banach space, and A a unital weakoperator closed algebra of operators on X having a separating cyclic vector e. The identification of A with Ae induces a multiplication ·:AexAe Ae defined by (Ae) . (Be) = (AB) e. We get a multiplier pair when we can extend this multiplication to all of X (allowing the product to be contained in a larger space Y. From this point of view, we can actually eliminate X. We can define a new norm v on the Banach algebra A by v (A) =
IIAell.
In this way, we can view X as the completion of A with respect to the norm v. Note that we have v (AB) :::; IIAII v (B).
A GENERAL VIEW OF MULTIPLIERS AND COMPOSITION OPERATORS II
65
In this section we begin an investigation of norms on Loo [0,1] so that the pointwise multiplication on L 00 [0, 1] extends to a multiplication on the completion that gives rise to a multiplier pair. If Loo [0,1] is contained in the multipliers on the completion of Loo [0,1], then, with an equivalent norm [6], we have Ilfgll ~ Ilfll oo Ilgll and 11111 = 1 always hold. Let XliII denote the completion of Loo [0,1] with respect to the norm 1111. Define AIIII,'1]IIII: [0,1]> (0,00) by AIIII (t)
= inf {llxeli : JL (E) = t},
'1]1111 (t) = sup {llxEII : JL (E) = t},
where JL is Lebesgue measure. We will restrict ourselves to norms for which the completions can be realized as measurable functions on [0, 1]. Let Y be the topological vector space of all complex JLmeasurable functions on [0,1] with the topology of convergence in measure. PROPOSITION 3.1. Suppose 1111 is a norm on Loo [0,1] such that 11111 = 1, Ilfgll ~ Ilflloo Ilgll and AIIII (t) > 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 IIcauchy 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 IIcauchy. 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 IIcauchy 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 JL (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 JL) is a symmetric norm if
(1) 11111 = 1 (2) Illflll = Ilfll for every f E LOO [0,1] (3) Ilf 0
(1) (2) (3) (4)
3.2. Suppose 1111 is a symmetric norm on LOO [0, 1]. Then
Ilfgll ~ 1I11111glioo for every f, 9 E LOO [0,1] IIfl11 ~ Ilfll ~ Ilflloo for every f E LOO [0, 1] 1111 is equivalent to 111100 if and only if 1111 is not continuous. If Un} is a II IICauchy sequence in Loo [0, 1] and fn ; 0 in measure, then Ilfnll ; O.
(5) The (pointwise) multiplication extends uniquely to a bilinear map from Xliii x Xliii to Y that makes (Xliii, Y) into a multiplier pair.
(6) Xliii has a cospace only if . 11m sup
t
t>o+
IIX[o,tJ II
2<
00.
(7) If XliII has a cospace, the cospace norm (normalized so 1 has norm 1) is a symmetric norm.
(8) If 1111 is continuous, then on U E Loo [0,1] : Ilflloo ~ I} the IIIItopology coincides with the topology of convergence in measure.
(9) In the multiplier pair (Xliii, Y) we have Co = no = Loo [0,1] and, lor every
IE
Loo [0,1] , IILfl1 = IIRfl1 = Ilfll oo '
Proof. (1) It follows from Ilfll = Illflll that multiplication by a function 9 with Igl = 1 is an isometry on Loo [0,1] with respect to 1111. However, every hE Loo [0,1] with Ilhlloo ~ 1 can be written as the average of two functions gl and g2 with Ig11 = Ig21 = 1. Thus (1) is true. (2) It follows from (1) and 11111 = 1 that Ilfll ~ Ilflloo for every f E LOO [0,1]. For the other inequality, suppose I = 2:;;'=1 (}:kXE k is a simple function with {E 1 , .•. , Em} a measurable partition of [0,1]. Let [x] denote the greatest integer function of x. let n E N, and let Sk = [nJL (Ek)] for 1 ~ k ~ m. Choose a measurable partition {F1 , ... , Fn} of [0,1] so that each Fj has measure ~ and, for each k, 1 ~ k ~ m, Ek contains exactly Sk of the Fj's. Next choose a measurepreserving isomorpism
Ilgnll
~ ~L
J=o
II1I1
0
= 11/11·
A GENERAL VIEW OF MULTIPLIERS AND COMPOSITION OPERATORS II
67
However,
JIf dIL11
Ilgn 
I
S Ilgn 
JIf
I dILlloo
s ~ Inkl (IL (Ek)  [nIL ~Ek)]) 1+ :
Ilflloo
>0 as n > 00. Hence, IIfl11 S Ilfll when f is a simple function. However, the inequality Ilfll S Ilflloo and the 111100density of the simple functions in L oo [0, 1] yields IIfl11 S Ilfll for every f E LOO [0,1]. (3) The only if part is obvious. Suppose limt>o+ IIX[o,tJiI = L > O. It follows from invariance that IlxE11 ~ L whenever IL (E) > O. If r < Ilfll oo ' then E = {x : If (x) I ~ r} has positive measure and Ilfll ~ IlxE Ifill ~ r IlxEli ~ rL.
It follows that Ilfll ~ L Ilfll oo ' (4) If 1111 is equivalent to 111100 the assertion is obvious. Hence we can assume that 1111 is continuous. In this case we can apply part (2) of Proposition 3.1. (5) If f E L oo [0,1] and E > 0 and E = {x E [0,1] : Ifm (x)1 ~ E}, then Ilfll ~ IlfxEII ~
E
IlxE11 ~
E
IIxEll 1 =
Ell
(E).
Hence the inclusion map from LOO [0,1] to Y is continuous with 1111 on L oo [0,1] and convergence in measure on Y. By (4), the inclusion map extends to a continuous injective linear map from Xliii into Y. Thus the statement (5) is clear. (6) Suppose lim sup II t 112 = 00. Clearly, limt>o+ IIX[o,t) II = O. Choose a t>O+
X[O.t)
decreasing sequence {tn} in (0,1/4] with tn > 0 such that II
tn
112 >
00.
Let
X[o.t n )
kn
= [t~] 1
k n
Then
1
k n 1
L j=O
IIX[o,tn)11
1
1
2 (X[jtn,(j+l)tn)X[o,!)) E co (ball (Xliii) . ball (Xliii)) .
However,
kn
kn
L J=O
1 II
X[O,tnl
1 112 (X[jtn,(j+l)tnlX[O,!))
= tnk n II
tn X[o,tn)
11 2X [0,!),
and, since tnkn > 1, we see that co (ball (Xliii) . ball (XliII)) contains the line ~X[O,!). Hence there is no cospace.
(7) Let K = co (ball (Xliii) . ball (Xliii))' Clearly K is convex absorbing and balaced in Loo [0,1] and the existence of a cospace says that K contains no line, so that the Minkowski functional IIIIK is a norm on Loo [0,1]. It is clear from the fact that 1111 is a symmetric norm that f E K if and ony if If I E K if and only if f 0 'P E K for every measurepreserving Borel isomorphism 'P : [0, 1] > [0, 1]. It follows that IlfilK = IllflllK = Ilf 0 'PIIK for every f E L OO [0,1] and every measurepreserving Borel isomorphism 'P on [0,1]. Moreover, we have Ilfgll K S IIfll1l911 for all f,g E L OO [0,1]. Hence 11111K S 1.
68
DON HADWIN AND ERIC NORDGREN
(8) The fact that 11111 :::; IIII implies that convergence in IIII implies convergence in measure. Suppose IIII is continuous and Un} is in the unit ball L 00 [0, 1] and In + 1 in measure. Suppose E > O. The continuity of II II allows us to choose 8 > 0 such that /L (E) < 8 implies IlxE11 < E/4, and convergence in measure implies that there is an nn such that /L ({x: 11 (x)  In (x)1 ~ E/2}) < 8 whenever n ~ no. It follows that, for n ~ no, E
E
111  lnll < "2 + 4111  lnlloo :::; E. (9) Suppose
1 E Co.
Then L f is bounded on Xliii so
Illgll :::; IILfllllgl1 for every g E LOO [0,1]. If 1111100 > 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 RussoDye theorem tells us that the closed unit ball of M is the normclosed 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 measurepreserving 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 <€
T)  P (TA  T) P
whenever .A ~ .Ao. Let MT denote the completion of M with respect to the topology of convergence in measure. It is a fact [10J that the multiplication on M extends to a multiplication on MT that is jointly continuous. One class of examples of unitarily invariant norms are the pnorms, II lip' for 1 :::: p < 00 defined by 1
IITllp = T
((T*T)P/2) P .
The norm 11112 arises naturally in the GNS construction. In [5J we showed that the completion X of M with respect to 11111 is a subset of Y and that (X, MT) is a multiplier pair with Co = Ro = M. Here we show that this result extends to all unitarily invariant norms on M. If v is a unitarily invariant norm on M, let My denote the completion with respect to v. We call a unitarily invariant norm v continuous if the corresponding symmetric norm on Loo [0, 1J is continuous, equivalently, if lim IIPII = 0 T(P)~O
as P ranges over the projections in M. THEOREM
4.1. Suppose v is a unitarily invariant norm on a I h factor M.
Then
(1) v (ST) :::: v (S) IITII and v (ST) :::: v (T) IISII for every S, T EM (2) IITlll :::: v (T) :::: IITII for every T E M (3) IIII is equivalent to 111100 if and only iflimt~o+ >0 (4) The (pointwise) multiplication extends uniquely to a bilinear map from My x My to MT that makes (My, MT) into a multiplier pair. (5) My has a cospace only if
IIX[o,tJlI
lim sup p=p2=p', T(P)~O+
(P~ < 00. IIPII
T
(6) If My has a cospace, the cospace norm (normalized at I) is a unitarilyinvariant norm on M. (7) If v is continuous, then on ballM the vtopology, the *strong operator topology and the topology of convergence in measure coincide. (8) In the multiplier pair (My, MT) we have Co = Ro = M and, for every A E M, we have IILAII = IIRAII = IIAII.
Proof. (1) We already proved this above. (2) This follows from part (2) of Theorem 3.2 and the description of V07r above. (4) This is the result of Nelson mentioned above. (3), (5), (6), (7) These follow from their analogues (or their proofs) in Theorem 3.2.
DON HADWIN AND ERIC NORDGREN
70
(8) It follows from (1) that IILTII, IIRTII ::; IITII for every T E M and it follows from [6] that IILTII = IIRTII = IITII for every T E M. If v is equivalent to IIII on M, the desired assertion is obvious. Hence we can assume that v is continuous. Suppose TELa. It follows from IIAIII ::; v (A) and the definition of convergence in measure that, for every n E N there is a projection Pn EM with T (1  Pn ) < lin with TPn E M. It is clear that IITPnl1 = IILTPnll ::; IILTII. Since v is continuous v(l Pn ) ~ 0, and since LT is continuous, v(T(l Pn )) ~ O. Hence TPn converges to T in vnorm, but it follows from (7) that T E M and IITII ::; IILTII .• REMARK 4.2. In an early version of [7] the proof of proposition 7.5 included a proof that if {an} is a normbounded sequence in M and Ilanll p ~ 0 for some 0< p < 1, then IIanl1 2 ~ O. However, Ilanll p ~ 0 easily implies an ~ 0 in measure, and by part (7) of the preceding theorem that v (an) ~ 0 for any continuous unitarily invariant norm v on M.
5. Tensor Products We now want to consider tensor products of multiplier pairs. Suppose (Xi, Yi) are multiplier pairs for i = 1,2. Let X = X l 0X 2 and Y = YI 0 Y2 be the algebraic tensor products. We can define a multiplication· on X with values in Y by
0 bd . (a2 0 b2) = (al . a2) 0 (b l . b2). question that interests us is, if 1IIIx is a tensor norm on X, Iia 0 bll x = Ilallllbll , (al
The first
i.e.,
then can we extend the multiplication on X to the completion X of X that creates a multiplier pair? An important special case is when (Xi, Yi) are natural multiplier pairs and 1IIIx is a tensor norm on X, when does there exist a tensor norm lilly on Y so that the multiplication extends so that (X, Y) is a natural multiplier pair? One way to show that the latter question has a negative answer is to show that
K
= co ((ballX)
. (ball X))
contains a line, although this does not answer the first question. If K does not contain any lines, then we have
Ilu· vll y ::; Ilull x Ilvll x for all u, vEX. The only difficulty is that, although the inclusion map from X to Y is injective and satisfies
Ilxll y ::; 1I10111x Ilxllx ' it is not clear that the inclusion from X to Y is still injective. Of course all of these problems go away when Xl and X 2 are finitedimensional, and since e m0e n is the space Mmxn (e) of mxn matrices, different multiplications on em and en yield multiplications on Mmxn (e) ; in particular, coordinatewise multiplication on em and en yields the Shur (entrywise) product on Mmxn (C). These ideas lead to a myriad of finitedimensional examples. Of all the tensor norms on X there is a unique smallest one 1IIImin and a unique largest one 1lllmax. Hence, if 1IIIx is a tensor norm on X, then ballllil max X C ballllllxX C ballllllm;nX, so if co ((ballllllm;n X) . (ballllllm;nX)) contains no lines,
A GENERAL VIEW OF MULTIPLIERS AND COMPOSITION OPERATORS II
71
then co ((ballllli x X). (ballllli x X)) contains no lines for every tensor norm 1111x. Hence the 1IIImin is a natural first case to check. Similarly, if co ((ballllli x
X)· (ballllll x X))
contains no lines for some tensor norm 1111x, then co ((balllili max contains no lines. Here is an interesting example.
X) . (balllili max X))
PROPOSITION 5.1. If Xl = X 2 = L2 [0,1] with pointwise (a.e.) multiplication, then co ((ballllllm;n X) . (ballllllrn;n X)) contains a line.
Proof. Note that Xl (9min X 2 is the set of compact operators on L2 [0, 1] with the operator norm. Suppose n E N and let Fk = VnX[~,~] for 0 ~ k < n. Then IlFkll = 1 for each k and IIFi (9 Fjllmin = 1 for 1 ~ i,j ~ n. For each integer s with let
o~ s < n
lijl=8 mod n Since IIlIminis the operator norm, IIT811min = 1. Moreover,
L
L
(Fi . Fi ) (9 (Fj . Fj ) = n 2
lijl=8 mod
X[*,'~l](9 X[*,4?]
lijl=8 mod
n
n
is in co ((ballllllm;n X) . (ballllllm;n X)). Since co ((ballllllm;n X) . (ballllllm;n X)) contains 0 and is absolutely convex, it follows that co ((ballllllm;n X) . (ballllllm;n X)) contains the line IR (1 (9 1) . • COROLLARY 5.2. Suppose G is a countable discrete abelian group and Xl X 2 = £2 (G) with the convolution product. Then Xl (9min X 2 has a cospace if and only if G is finite.
Proof. If {; is the dual group of G, with Haar measure
Jl,
then Xl and X 2 are
isometrically isomorphic to L2 ( (;, Jl) with pointwise (a.e.) multiplication. If G is infinite, then Jl is continuous and we are in the situation of the preceding theorem .
•
For the projective tensor product things work better. This has to do with the definition of 1IIImax on V (9 W, i.e., Ilsll If we let
tk =
= inf
{t
Ilvkllllwkll : s
Ilvkllllwkll and replace ballllil max (V (9 W)
Vk (9wk
=
t
Vk
(9 Wk}'
with II~kll (9 lI:kll' we see that
= co{v (9 w:
Ilvkll
=
Ilwkll
=
I}.
The following lemma, which is obvious from the definition of a cospace, shows the relationship between multiplier pairs and projective tensor products. In particular, it gives a representation of the cospace when it exists.
DON HADWIN AND ERIC NORDGREN
72
LEMMA 5.3. Suppose (X, Y) is a multiplier pair and T : X 0 X  Y is the linear mappin9 defined by T(a0b)=a·b.
Then
(1) T (balllili max (X 0 X)) = co ((ball X) . (ball X)) (2) If Y is a cospace for X and T : X 0 max X  Y is the bounded extension of T, then the induced map from (X 0 max X) / ker T to Y is an isometric isomorphism.
Suppose Vj and Wj are Banach spaces and Tj : Xj  Yj is an injective bounded operator for j = 1,2. Then the operator Tl 0 max T2 : Xl 0 max X 2  Yl 0 max Y2 defined by (Tl 0 max T 2) (VI 0 V2) = TIVI 0 T2V2 may not be injective [1, p. 49]. However, if either VI or V 2 has the approximation property, then Tl 0 max T2 must be injective. 5.4. Suppose (Xl, Yl ) and (X2' Y2 ) are natural multiplier pairs, with Ti : Xi  Yi the inclusion maps. If Tl 0 max T2 : Xl 0 max X 2  Yl 0 max Y2 is injective, then (Xl 0 max X 2, Yl 0 max Y2) is a natural multiplier pair. THEOREM
Proof. The closed unit ball of Yj is the closed convex hull of {a· b: a,b E Xj, Iiall = Ilbll = I}
and the closed unit ball of Yl 0 Y2 is the closed convex hull of {Yl 0 Y2 : Yl E Yl , Y2 E Y2, IlyIiI
= IIY211 = I},
so the closed unit ball of Yl 0 max Y2 is the closed convex hull of
E = {(al' a2) 0 (bl · b2): al,a2 E X l ,a2,b2 E X 2, Ilajll = Ilbjll = I}
= {(al 0 bl ) . (a2 0
b2) : aI, a2 E Xl, a2, b2 E X 2, Ilaj II
= Ilbj II =
I} .
On the other hand the closed unit ball of X = Xl 0 max X 2 is the closed convex hull of {al . a2 : al E Xl, a2 E X 2, IlaIil = IIa211 = I}, so the closed convex hull of (ball X) . (ball X) is also the closed convex hull of E defined above. _ COROLLARY 5.5. If (Xl, Yl ) and (X2' Y2 ) are natural multiplier pairs, and if either Xl or X 2 has the approximation property, then (Xl 0 max X 2, Yl 0 max Y2 ) is a natural multiplier pair.
Another class of tensor product examples comes from complex analysis. The Hardy space H2 consists of all analytic functions in the open unit disc Jl)l with power series having square summable Taylor coefficients. The product of H2 functions in the unit ball is in the unit ball of HI, and every HI function in the unit ball can be factored into a product of two functions in the unit ball of H2. As was pointed out in [5], HI is the cospace of H2. If the tensor product H2 0 H2 is given the Hilbert space norm, then its completion H 202H2 may be identified with H2(Jl)l2). The multiplication of elements of the tensor product defined by (h 091)(12092) = (hh)0(9192) corresponds under this
A GENERAL VIEW OF MULTIPLIERS AND COMPOSITION OPERATORS II
73
identification to ordinary multiplication of functions, and thus the cospace is identified with a subspace of HI (]j))2) but the situation is more complex than that of the disk. A result of Rosay [12] (see also Rudin [12] and Miles [9]) shows that functions in ball HI (]j))2) can not always be factored as products of functions in ball H2(]j)) 2). However a result of 1. J. Lin [8] asserts that every element! in HI (]j))2) has a representation! = 2:~=lFnGn with Fn,Gn E H2(]j))2) and 2:~=lllFnIIIIGnll ~ ell!ll l . Since Ilflll ~ 2:~=1 IlFnllllGnll, the cospace is Hl(]j))2) and the cospace norm is equivalent to the Hl(]j))2) norm.
References [1] Defant, Andreas and Floret, Klaus, Tensor norms and ideals, NorthHolland, Amsterdam, 1993. MR1209438 (94e:46130) [2] Ding, Huiru and Hadwin, Don, Approximate equivalence in von Neumann algebras, Sci. China Ser. A 48 (2005), no. 2, 239247. MR2158602 (2006c:46050) [3] Fack, Thierry Sur la notion de valeur caracteristique, J. Operator Theory 7 (1982), no. 2, 307333. MR0658616 (84m:470l2) [4] Fack, Thierry and Kosaki, Hideki, Generalized snumbers of rmeasurable operators, Pacific J. Math. 123 (1986), no. 2,269300. MR0840845 (87h:46122) [5] Hadwin, D. and Nordgren, E., A general view of multipliers and composition operators, Linear Algebra Appl. 383 (2004), 187211. MR2073904 (2005g:47123) [6] Hadwin, Don and Orhon, Mehmet, A noncommutative theory of Bade functionals, Glasgow Math. J. 33 (1991), no. 1, 7381. MR1089956 (91m:46085) [7] Haagerup, U. and Schultz, H., Invariant subspaces for operators in a general IIIfactor, preprint. [8] Lin, Ing Jer and Russo, Bernard, Applications of factorization in the Hardy spaces of the polydisk, Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), 331349, Lecture Notes in Pure and Appl. Math., 175, Dekker, New York, 1996. MR1358170 (96g:460l8) [9] Miles, Joseph, A factorization theorem in HI(U 3 }, Proc. Amer. Math. Soc.52 (1975), 319322. MR0374459 (51 #10659) [10] Nelson, Notes on noncommutative integration, J. Functional Analysis, 15 (1974), 103116. MR0355628 (50 #8102) [11] Rosay, JeanPierre, Sur la nonfactorisation des elements de l'espace de Hardy HI(U 2 ), Illinois J. Math. 19(1975}, 479482. MR0377098 (51 #13272) [12] Rudin, Walter, Function theory in polydiscs, Benjamin, New York, 1969. MR0255841 (41 #501) DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW HAMPSHIRE, DURHAM,
NH 03824 Email address, D. Hadwin: [email protected]
Email address, E. Nordgren: ean4lmath. unh. edu
Contemporary Mathematics Volume 454, 2008
A General View of BMO and VMO Don Hadwin and Hassan Yousefi Dedicated to the memory of Paul Halmos ABSTRACT. We formulate a very general setting in which the spaces BMO and VMO can be defined, and we prove several results in this general setting. We prove general versions of the JohnNirenberg theorem and characterizations of VMO. One main result is that VMO is never complemented in BMO.
1. Introduction
In this paper we construct a general setting in which functions of bounded mean oscillation BMO and vanishing mean oscillation V MO can be studied. One benefit of developing this general theory is that, when we throwaway all unnecessary structure, we are forced to identify the" real reasons" that a theorem is true. For example, we prove a version of Sarason's characterization of V MO as the closure of the uniformly continuous functions in BMO that uses very simple ideas and doesn't depend on the clever, but specialized, use of a convolution argument. Another benefit is that proving a theorem in the general setting gives a proof in all the different realizations. For example we prove that V MO is not complemented in BMO, and one realization yields the conclusion that if Jt is a continuous probability measure on a compact metric space X, then C(X) is not complemented in L oo (Jt). Another benefit is that the general setting provides a framework in which certain questions and concepts (e.g., products, homomorphisms) become natural that would not arise in a specific instance. In this setting we also prove a version of the JohnNirenberg theorem. The notion of functions of bounded mean oscillation (BMO) made its first appearance in [6] where F. John studied BMO on ]Rn with Lebesgue measure. The celebrated JohnNirenberg inequality was proved in the next article in the same issue of Comm. Pure Appl. Math. [7]. Later a great deal of work was done on BMO on the circle with Haar measure. 2000 Mathematics Subject Classification. Primary 32A37, 46E30; Secondary 46B20. Key words and phrases. BMO, VMO, JohnNirenberg Theorem, BMO triple, complemented subspace. This paper is in final form and no version of it will be submitted for publication elsewhere. ©200B American Mathematical Society
75
76
DON HADWIN AND HASSAN YOUSEFI
Roughly speaking, functions of Bounded Mean Oscillation are the ones that on average, are not too far from the local averages of the function. To be more precise, an integrable function I defined on jRn is said to be in BMO if
Ilfll* =supI(I/I(I)I) < 00, where the supremum is taken over all cubes I that are Cartesian products of subintervals of the coordinate axes, III is Lebesgue measure of the cube I, and I (I) = 1;1 I I is the average of the function on I. For BMO on the circle, we let I range over all arcs and let III denote the Haar measure (normalized arc length)
J
of!.
A martingale version of BMO can be found in [8], and [1] studies BMO on spaces of homogeneous type. Later D. Sarason [9] introduced a subspace of BMO functions called V MO functions. In his paper [9], he gave several characterizations of this subspace, including that it is the closure in BMO of the set of uniformly continuous functions in BMO. Roughly speaking, a function in BMO is said to have Vanishing Mean Oscillation, if its mean oscillation is locally small, in a uniform sense. More precisely, a BMO function I is in V MO if I (II  I (1)1) tends to zero as III tends to zero. We refer the reader to [10] and [3] for beautiful accounts of the spaces of BMO and VMO. 2. BMO Triples We call a triple (X, JL, I) a BMO triple if X is a complete separable metric space with no isolated points, JL is a nonatomic regular Borel measure on X whose support is X (i.e., if U =I 0 is open in X, then JL (U) > 0), and I is a collection of Borel subsets of X such that:
(1) 0 < JL(I) <
00 for every I E I, (2) for every nonempty open set U C X there is an I E I such that leU, (3) there is a countable subset {In}n~l of I such that
(4) for every I and J in I, there are I = h, h ... ,In each j, 1 ::; j < n, either I j C IHI or Ij+! C I j .
= J in I such that, for
REMARK 2.1. If in (3) above we have h C he··· and if we have that, whenever I, J E I and JL (I n J) > 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 ofrealvalued BMO(I, /l) functions forms a lattice. In other words, if the realvalued 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 Ml
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 Llnorm to a function 9 E L1 (I). We have:
< <
I(lgn  9  I(gn  g)!) I(lgn  gm  I(gn  gm)!)
+ I(lgm 
Ilgn  gmIIBMO(I,/Ll
+ 2I(lgm  gl)
Ilgn  gmIIBMO(I,/Ll
+ fL(I)
9  I(gm  g)l)
2
Suppose above):
E:
Ilgm  glll,1'
> 0 is given. There exists N such that m, n E:
I(lgn  9  I(gn  g)l) < "2
2
+ fL(I)
~
N implies that (from
Ilgm  glll,1'
By letting m  00 it follows that 1(1 gn  9  I(gn  g) I) ~ ~ for every I. Therefore gn converges to 9 in BMO(I, fL) norm. Since X ~ I, then In = gn in BMO(I, fL) and so In is convergent to g. If X E I, we have X (lin  Iml)
~
X (lin  1m  X Un  1m)!)
~
Il/n  ImIIBMO(I,/Ll
+ IX Un 
Im)1
thus In converges in L 1norm to a function I E L1 (X) . The proof of convergence of gn can be applied to show that In converges to I in BMO(I, fL) norm. 0 In the proof of the next corollary we have used the ideas of the proof of the preceding proposition. COROLLARY 1. Let (X, fL,I) and (X, fL,.:J) be two BMO triples and BMO(I, fL) c BMO(.:J, fL)· Then there exists M > 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 ClosedGraph 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 JL (X) = 00. Choose I' E I and l E 3 such that JL (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, JL). 0 Let C u (X) denote the set of uniformly continuous functions on X. If I E I, define the measure JLI as the restriction of JL to the O'algebra of Borel subsets of I. LEMMA
2.4. If (X, JL, I) is a BMO triple, then:
(1) Cu(X) n BMO(I,JL) c VMO(I,JL) and VMO(I,JL) is a closed linear subspace of BMO(I,JL). (2) There is a countable collection of continuous linear functionals on BMO(I, JL) that separates the points of BMO(I, JL). (3) For every f E Loo(p),
IlfIIBMo(I,/L)
~
311fll00 .
In particular, the inclusion map from Loo(JL) to BMO(I, JL) is continuous. (1) The inclusion Cu(X) n BMO(I, JL) c V MO(I, JL) is easily proved. For each I E I, we define TI : BMO(I, JL) * U (JLI) by 1
PROOF.
TJ (f) = JL(I) (f  I(f)) II.
Then IITJ II ~ 1 and f
E
V MO(I, JL) if and only if
/L(I)+1!:!(I)>0 IITI (f) II
= O.
It easily follows that V MO(I, JL) is a closed linear subspace of BMO(I, JL). (2) For every In in the definition of BMO triple there exist continuous linear functionals {¢n,d k2 1 on U (JLIJ that separate the points of Ll (JLIJ. Define 'l/J n k : BMO(I,JL) * C by
'l/Jn,k(f) = ¢n,k
(JL(~n) (f 
In(f)) lIn) .
Note that
I'l/Jn,k
(f)1 ~ II¢n,kll
1
JL(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, JL). (3) This is obvious.
o If X is the unit circle, JL 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 CTfinite 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 realvalued 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:<:;11:<:;211' 00 and that IIRe(g)IIBMo(I,/L) ::::; IlgIIBMo(I,/L) for every 9 E BMO(I, JL). 3) First we will show that V MO(I, JL) C C(X). Suppose f E BMO(I, JL)\C (X) is real valued. By a theorem of D. Hadwin [5], there exists a point a E X such that f cannot be continuous at x = a by redefining the function on any set of measure zero. For every n ?: 1 define the monotone sequences Mn and mn to be the essential
f)11
BMO AND VMO
81
supremum and essential infimum of the function f on the open disk centered at a and radius ~, B (a;~) , respectively. The sequence Mn  mn does not converge to zero, otherwise f would be continuous at x = a by redefining it at x = a to be lim Mn. Therefore there exists € > 0 such that, without loss of generality, n>(X)
Mn  mn ~ € for every n ~ 1. Choose Borel subsets In,l and I n,2 of B (a;~) , of equal measure such that f(t2)  f(tl) > i for every h E In,l and t2 E I n,2. Thus I n ,2(f)  In,l (f) ~ i· By letting In = In,l U In,2 we will have: 1 1 "2In,2 (If  In(f)I) ~ "2IIn,2 (f  In(f))1 1
1
"2 IIn,2 (f)  In (f) I = "2 IIn,2 (f)  In,l (f) I >

4
Therefore f t/: V MO(I, /1,)We have shown that
Cu(X) n BMO(I, JL) C V MO(I, JL) C C(X). To finish the proof suppose that 9 E V MO(I, JL)\Cu(X), Thus there exists a continuous function f on X in BMO(I, JL) such that f = 9 almost everywhere. There exist x n , Yn E X and € > 0 such that d (x n , Yn) 70 but If (xn)  f (Yn)1 ~ € for every n ~ 1. The continuity of f at Xn and Yn implies that there are Borel sets Xn E In,l and Yn E I n ,2 in small neighborhood of Xn and Yn such that JL (In,l) = JL (In,2) 7 O. If we let In = In,l U I n ,2 and do a similar calculations as above, then we arrive to the contradiction that f t/: V MO(I, JL). D
3. Characterizations of VMO Here is a natural question that arises from the classical case and the preceding proposition: Is V MO(I, JL) the 11·IIBMo(I,/L)closure of Cu(X) n BMO(I, JL)? We will give an affirmative answer to the above question in some special cases. The proof of the next lemma is a small modification of [9, Lemma 2]. We present the proof here for the sake of completeness. LEMMA 3.1. Suppose there exists M ~ 1 for which for every n In C I that partitions X and satisfies in the following conditions:
(1) JL (J) + diam (J) ::; ~ for every J E In, (2) VII, h E In if h U 12 C B (x;~) for some x hUh c I and
E
~
X, then 31
1 there exists
E
I such that
JL (I) ::; M JL (h) and diam (I) ::; M diam (h) for k = 1,2 (3) If I E I and ~ :( JL (I) +diam (I) , then there are finitely many h, ... , h E In such that I C h U· .. U h a.e. and JL (h U· .. U h) ::; MJL (I). Then fn defined by fn (x) =
L: JEJn
for every function f E VMO(I,JL).
J (f) XJ (x) converges to f in the BMOnorm
DON HADWIN AND HASSAN YOUSEFI
82
PROOF. For given c > 0 there exists J > 0 such that I (II  1(1)1) < c whenever I E I and JL (1) + diam (I) < J. Suppose n E N and ~ < We claim that . 1 \/x, y E X If d (x, y) < , then lIn (x)  In (y)1 < 2Mc. n To see this, assume that x E hand y E 12 where h,I2 E In. Since II U 12 C B (x; ~) , there exists I E I that satisfies in the condition (2) of the lemma. Thus we have
t.
lIn (x)  In (y)1
< lIn (x)  1(1)1 + II (I)  In (y)1 < h (II  1(1)1) + 12 (II  1(1)1)
<
JL(I) 1(1/ 1(1)1)+ JL(I) 1(1/1(1)1) JL(h) JL(I2)
< 2Mc. To estimate II I  In II BMO(I,I") , suppose I E I. If JL (I)
I (II  In  1(1  In)!)
~
I (II  I(I)!)
< c + 2Mc If JL (I) + diam (1) :2: ~ there are h"" the lemma. Then
I (II  In  1(1  In)!)
~
+ diam (1) < 1., n then
+ JL (~)2111/n (x)
(y)1 dJLdJL
3Mc.
,h
E
In that satisfy in condition (3) of 2
< 21(11  In I) ~ JL (I) 2 JL (1)
 In
t; j
t; j II L
Ii
2
L
Ii
II  Idf)1 < JL (I)
Inl
t; L
JL (Ii) c
< 2Mc. Therefore for every I E I we have I that III  InIIBMo(I'I") ~ 9Mc. EXAMPLE
convex. Let I
(II  In  1(1  In)!)
~
3M c which implies 0
3.2. Suppose A
! 0 is any open subset of jRm which is bounded and
= {v + aA : v E
jRm, a E jR+} and for each n E Z+ let
In =
{v + n(JL (A) +ldiam (A))A: vE jRm}.
Then we can show that I and In satisfy the conditions of the previous lemma. To see this, let I E I. To simplify the calculations we assume that JL (A)+diam (A) = 1. Suppose A' and l' are the largest cubes contained in A and I, respectively. If we need N many cubes of the form v + A'to cover A, then N many cubes of the form v + I' would cover I and N many cubes of the form J~,v = v + ~A' c In,v = v
1
+ nA E In
would cover any element of In· Let
f3
=
JL (I) JL (A)
=
JL (I') JL (A') . Then
BMO AND VMO
83
~ ~ ~ IL (In,v) + diam (In,v) implies that IL (1) ~ IL (1) IL (In,v) , and so {3nm = (J ) ~ 1. Each v + [' can be covered by 2m (l{3nm J + 1) IL n,v J~,v's, so [ can be covered by at most 2m N (l{3n m J + 1) In,v's. Therefore:
Note that IL (1)
+ diam (1)
IL (In,v's) IL (I)
~
2m N ( l{3n m J + 1) IL (~A) (31L(A)
~
2mN (1
~
2m3N.
+ (3~m)
The next lemma states a condition under which we can approximate a function by a continuous function. LEMMA 3.3. Suppose that Y is a Hausdorff para compact topological space, c > 0, f : Y + C, and U is an open cover of Y such that, for every U E U and every x, y E U we have If (x)  f (y)1 < c. Then there is a continuous function 9 : Y + C such that, for every x E Y, If(x)  g(x)1 < c.
PROOF. Let {g>. : A E A} be a partition of unity subordinate to U, and for each A E A choose x>. E supp (g>.). We define 9 : Y + C by
g(x)
=
L
f(x>.)g>. (x).
>'EA
Suppose x E Y and let D = {A E A : g>. (x) > O}. From the definition of a partition of unity there is a U E U, an open set V such that x EVe U and E g>. (y) = 1 for every y E V and
E
>'ED
g>. (y) = 0 for every y E Y\U. Hence, for every A E D,
>'ED
x>. E U. Thus, for every A E D we have If (x)  f (x>.)1 < c. It follows that 9 is continuous on V (and by the generality of x, on Y) and that If (x)  9 (x)1 =
IL
I
(f (x)  f (x>.)) g>.(x) < c.
>'ED
o COROLLARY 2. Suppose (Y, d) is a compact metric space, f : Y + C, c > 0, and for every x E Y there is a Ox > 0 such that whenever y E Y and d (x, y) < Ox we have If (x)  f (y) I < c. Then there is a uniformly continuous function 9 : Y + C such that If(x)g(x)l
for every x E Y. The preceding corollary uses compactness to guarantee that the continuous function 9 is actually uniformly continuous. If Y c jRn, we can obtain uniform continuity with a weaker hypothesis.
84
DON HADWIN AND HASSAN YOUSEFI
PROPOSITION 3. Suppose E is a subset of JRn, a function such that, for every x, y E E,
Ilx  Ylloo
< 15 * l
E
> 0,15 > 0 and
Then there is a uniformly continuous function F : E l
<
>
>
C is
E.
C such that
E
for every x E E.
fdt) =
= 15/4.
!
Let w
PROOF.
For each integer k E Z we define
ift«2k1)w
0 t (t  (2k l)w) 1  t (t  (2k + 1) w)
if (2k l)w::; t < 2kw if 2kw ::; t ::; (2k + 1) w if (2k + 1) w < t ::; (2k + 2) w if (2k + 2) w < t
o
It is easily seen that
(1) Ilk (s)  lk (t)1 ::; tis  tl for every k E Z and all s, t E JR, and (2) L lk (t) = 1 for every t E R kEZ
If K, = (kl"'" kn ) E zn, we define fK : JRn
>
C by
n
fK (tl,"" t n )
=
II lk j (tj). j=1
It is easy to show,
3. for all s, t E JRn with lis  tll oo
< 1,
that
n
IfK (s)  fK (t)1
=
II [(lk j (Sj) j=1
n
fkj (tj))
+ lk j (tj)]

II lk j (tj) j=1
4.ifxEJRn, ball II 1100 (x, w) C
U
Supp UK) c ball II 1100 (x, 3w),
xEsupP(j,,)
5. LKEZn fK (t) = 1 for every t E JR n . Define A = {K, E zn : ::IXK E E with dist (x K, Supp UK)) < 3w}, and we define F : JRn > C by
Suppose x E E. It follows from (4) above, for each K, E zn with x E SUPP UK) , that Ilx  xKlloo < 3w < 15, which implies by hypothesis that l
IF (x) 
=
BMO AND VMO
85
To show that F is uniformly continuous on E, suppose x, y E E and Ilx  Ylloo < min (1, w) . Let D be the set of all i'L such that x or Y is in supp (f",). It follows from (4) that, whenever i'L E D,
which implies
lep (x)  ep (x",)1 < c. Hence, by (5)
(Y)I = I L
IF (x)  F
[ep (x",)  ep (x)]
[1", (x)  f", (y)] ,
",ED
and, by (3) and the fact that the cardinality of D is at most 2 (3 n ) ,
IF (x) 
F
(Y)I ~ 2 (3 n ) c (~) nils  tll oo '
Hence F is uniformly continuous on E.
D
We can now prove a generalization of Sarason's theorem [9].
3.4. If X is a compact space or X is any subset ofJRn and I satisfies the conditions of Lemma 3.1, then VMO(I,JL) = Cu (X) n BMO (I,JL)II·IIBMO(I.I'). THEOREM
PROOF. Suppose f E V MO(I, JL), X is compact, and c > O. Then by Lemma 3.1 there exists fn as in Lemma 3.1 such that Iii  fnIIBMO(I,/l) < ~. It was shown in the proof of Lemma 3.1 that Ifn (x)  fn (Y)I < ~ whenever d (x, y) is small enough. Lemma 3.3 can be applied to find a continuous function hn such that llin  hnll oo ~ ~ and so Ilfn  hnIIBMO(I,/l) ~ ~. Therefore Ilf  hnIIBMO(I,/l) < c. If X c JRn, then Lemma 3.1 and the previous proposition will imply that V MO(I, JL) = Cu (X) n BMO (I, JL)II·IIBMO(I,I'). D
REMARK
(1) If A and I are as in Example 3.2, then by the previous
3.5.
theorem V MO(I, JL) = C u (X) n BMO (I, JL)II·IIBMO(I.I'). As a special case that A is a ball or a cube in JRn, then we get the Sarason's theorem [9]. (2) If X is a circle with I the set of open arcs or X is an interval with I the open subintervals, and if JL is a finite continuous (i.e., JL{(x)} = 0 for every x) measure whose support is X, then the hypotheses of Lemma 3.1, so V MO(I, JL)
= Cu (X) n BMO (I, JL)II·II BMo (I.I').
The next theorem gives another characterization of V MO(I, JL). Following Sarason's notation [9], for a positive measurable function f and a > 0 we let: Na
(f)
sup
=
/l(I)+diam(I):<:;a
I
(f)
I
(II) and
No
(f)
lim Na (f) .
=
atO
The Schwarz's inequality implies that Na (f) :::: 1 for every a the next theorem we refer the reader to [9]. THEOREM
if and only if f
E BMO(I, JL) be a real valued function. Then No V MO(I, JL).
3.6. Let f E
> O. For the proof of
(e f ) = 1
DON HADWIN AND HASSAN YOUSEFI
86
It is clear that if Ie .1, then BMO (.1, JL) c BMO (I, JL) and V MO (.1, JL) V MO (I, JL) . The next result is a generalization of this.
c
PROPOSITION 4. Let (X, JL, I) and (X, JL,.1) be two BMO triples and M ~ l. If for every J E .1 there exists an I E I such that J C I and JL (I) :::; M JL (J), then
(1) IlfIIBMo(J.,L):::; 2M IlfII BMo (I,I') , (2) if also diam(I):::; Mdiam(J) , then VMO(I,JL) c VMO(.1,JL)' PROOF. Suppose that f E BMO(I, JL) and J E .1. Choose I E I that contains J and JL (I) :::; M JL (J) . The proof will be apparent by the fact that:
J
(I
f  J(f) I)
< J (I f  I(f) 1+ 1I (f)  J(f) I) :::; 2J (I f  I(f) I) :::; 2M I (I f  I (f) I) .
o EXAMPLE 3.7. Let X = ~n and let JL be Lebesgue measure. If we let I be the collection of all disks in ~n and .1 be the collection of all cubes in ~n, then by Proposition 4 and Corollary 1 BMO(I, JL) = BMO(.1, JL), V MO(I, JL) = V MO(.1, JL). More generally, if A and K are as in Example 3.2, then BMO(I, JL) = BMO(K, JL) and V MO(I, JL) = V MO(K, JL).
4. JohnNirenberg Theorem In this part we will prove a version of JohnNirenberg theorem for any function f E BMO(I, JL) whenever I has certain properties. We let essdiam(J) denote the essential diameter of J (i.e., the diameter modulo sets of measure 0). Note: when we talk of a partition, we always mean modulo sets of measure zero. A key idea in this section is the existence of certain partitions of sets. Suppose JL is a Borel measure on a metric space Y. A sequence {An} of measurable partitions of Y is called a null sequence of partitions if
(1) An+1 is a refinement of An for every n ~ 1. (2) If {In } is a sequence and I n E An for every n JL(Jn)
~
1, then
+ essdiam(Jn ) > O.
LEMMA 4.1. Suppose JL is a Borel measure on a metric space Y, and suppose {An} is a null sequence of partitions of Y Then the following are true:
< U n21An U {F : JL(F) = O} > contains all Borel sets in Y. (2) If JL(Y) < 00, (X > 0, f is a nonnegative and integrable function such that, for every n and every JEAn we have J(f) :::; (x, then we can conclude that f(x) :::; (X almost everywhere.
(1) F =aalg
PROOF. (1) Suppose E is a closed subset of Y with positive measure. Since An is a partition of Y for every n, we can find Ii', 12, ... E An such that JL(Ij n E) > 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 = aa 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 Mdivisible 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 JohnNirenberg 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) Icp(t)  J(cp)1 :::; a, a.e. on J\ Uj I?),
(2) 11;1) (cp)  J(cp) I < aM, (3) L JL(I?)) :::; iJL(J)· On each I?) we again apply Lemma 4.4 to Icp  I?) (cp) I to obtain 1;2) as before such that each Iy) is contained in some I?). Then, a.e. on J\ Uj IY), we will have:
Icp(t)  J(cp)1
:::;
Icp(t)  I?)(cp) I + II?) (cp)  J(cp)1
< a + aM < 2aM. Suppose Iy) is contained in Ik 1 ), then
II?) (cp)  J(cp) I <
IIY) (cp)  Ik 1 )(cp) I + IIk 1 ) (cp)  J(cp) I
< aM + aM = 2aM. We also have:
LJL(I?)):::;
~a LJL(I?)):::; (~)2JL(J). a
Continue this process inductively. At stage n we get intervals It) such that
(1) Icp(t)  J(cp)1 :::; anM, a.e. on J\ Uj It),
(2) LJL(It)) :::; (i)nJL(J). If anM :::; A < a (n
+ 1) M, n
~
1, then
JL{tEJ:lcp(t)J(cp)I>A}
< LJL(I;n)) < (~tJL(J):::; eCAJL(J), a
BMO AND VMO
for c = o~ In 0:. Thus inequality in the theorem holds for o:M :=:; A. If 0 then obviously
89
< A < o:M,
J.L{t E J: 1'P(t)  J('P) I > A}:=:; J.L(J) < eoMcec>J.L(J). Therefore the inequality holds for all A.
o
REMARK 4.6. (1) If every I E I satisfies in the conditions of Lemma 4.4, then the previous theorem can be applied to show that for every 'P E BMO(I, J.L) and p > 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 JohnNirenberg 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} :=:; Cec>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 4divisible. 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 JohnNirenberg 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 2divisible; therefore the JohnNiremberg theorem holds in BMO(I, J.L). (6) If in our JohnNirenberg 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 11 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), 569645. [2] J. Doob, What is a Martingale?, Amer. Math. Monthly 78 (1971),451463. [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), 305326. [5] D. Hadwin, Continuity Modulo Sets of Measure Zero, Mathematica Balkanica, Vol. 3 (1989), 430433. [6] F. John, Rotation and Strain, Comm. Pure Appl. Math. 14 (1961), 391413. [7] F. John and L. Nirenberg, On Function of Bounded Mean Oscillation, Comm. Pure Appl. Math. 14 (1961), 415426. [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), 391405. [10] D. Sarason, Function Theory on the Unit Circle, Virginia Poly. Inst. and State Univ., Blacksburg 1978. MATHEMATICS DEPARTMENT, UNIVERSITY OF NEW HAMPSHIRE
Email address: don
Email address: hyousef i
Contemporary Mathematics Volume 4154, 2008
Order Bounded Weighted Composition Operators R. A. Hibschweiler ABSTRACT. Let X be a Banach space of functions analytic in the unit disc and let m denote normalized Lebesgue measure on the circle. The operator T: X + Lq(m) is said to be orderbounded if there exists hE Lq(m) such that I (Tf)(e i8 ) I:':: h(e i8 ) a.e. [mJ for all II f Ilx:':: 1. Let lIt i 0 E Lq(m) and let be an analytic selfmap of the disc. The weighted composition operator WW, is defined by WW,(f) = 1It(f 0 is studied on the weighted Bergman spaces and on more general Banach spaces of analytic functions with restricted growth. Connections are exposed between bounded ness, compactness and order bounded ness on the weighted Bergman spaces.
1. Introduction
For p :::: 1, the Hardy space HP is the Banach space of functions analytic in the disc such that
I Recall that if
f
E
f II~p=
1
27r
sup
1211" I f(re ilJ ) IP
O:Sr
HP, then the boundary function j*(e ilJ ) =
dO <
00.
0
lim
r+l
f*
defined by
f(re ilJ )
exists a.e. with respect to Lebesgue measure m. Let D denote the open unit disc. Throughout this work, will denote an analytic function mapping D into itself. As is well known, the composition operator defined by Cip (f) = f 0 is bounded on HP for each p :::: 1. The focus of this work is on order bounded weighted composition operators acting on the Hardy spaces and the weighted Bergman spaces. The general definition of order bounded operators is given next. DEFINITION 1.1. Let X be a Banach space of functions analytic in D and let q > O. Let J1. be a positive measure on the unit circle. The operator T : X t LQ(J1.) is said to be order bounded if there exists hE LQ(J1.), h :::: 0, so that the inequality
I T(f)(e ilJ ) I ~ h(e ilJ ) 1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. weighted composition operators, weighted Bergman spaces. ©2008 American Mathematical Society
93
R. A. HIBSCHWEILER
94
holds a.e. with respect to 11, for all
JE
X with
II J IIx
~
1.
In particular, let p ::::: 1 and (3 > 0 and let <1> be an analytic selfmap of D such that <1>* E Lp{3(m). H. Hunziker [5] characterized the selfmaps <1> for which the composition operator Gcp : HP > L{3P(m) is order bounded. In this context, Gcp(J) is understood to mean the boundary function (f 0 <1» * . THEOREM 1.2 (Hunziker). Let (3 > O. The Jollowing are equivalent.
(1) Gcp : HP > L{3P(m) is order bounded Jor some p ::::: 1. (2) 1/(1 I <1>* I) E L{3(m). Since the condition (2) in Hunziker's theorem is independent of p, it is clear that if Gcp : HP > L{3P(m) is order bounded for some p ::::: 1, then it is order bounded for all such p. Assume p > 0 and q > O. Let 0 "# \[I E Lq(m). The weighted composition operator Wq"cp is defined by Wq"cp(f) = \[I (f 0 <1» where J is analytic in the disc. The focus of this work is on weighted composition operators on the Hardy spaces, and more generally, on the weighted Bergman spaces. Definition 1.1 implies that if Wq"cp : HP > Lq(m) is order bounded, then Wq"cp : HP > Lq(m) is bounded. The converse is false, as shown by Hunziker's theorem and the simple example p = q, \[I = 1, and <1>(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) <
00.
1, the appropriate definition for A~ is the Hardy space HP
[14]. In contrast with the Hardy spaces, the Bergman spaces include functions that have no boundary values. See, for example, [3]. Thus a discussion of order bounded weighted composition operators on the Bergman spaces will require the assumption that I <1>*(e i8 ) 1< 1 a. e. with respect to m. Hunziker's theorem indicates that this condition is necessary for Gcp : HP > L{3P(m) to be order bounded. Let zED and let EAf) = J(z) for J E A~. The following lemma is well known, and so only a sketch of the proof will be given here. LEMMA 1.3. Fix Q ::::: 1 and p ::::: 1. There are positive constants G1 and G2 depending only on Q and p such that G1 (1
IZ
1)(<>+2)/p ~
I Ez I
~ G2 (1
IZ
1)(<>+2)/p.
PROOF. For Q = 1, see [2] and Proposition 1.1 [6]. For used the subharmonicity of I J IP to establish
I J(z) I ~
G2
I
J IIA~ (11
Q
> 1, Smith [14]
Z 1)(<>+2)/p
where G2 depends only on Q and p, and J is any function in A~. This yields the second inequality in the lemma. For the remaining inequality, let
(1
JAw)
I Z 12 )(<>+2)/p
= (1 _ zw)2(o+2)/p , I w I < 1.
95
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
Then II fz IIA;: ~ 1 [14], and thus there are constants Kl and K2 depending only on a and p such that Kl :::; II fz IIA~:::; K 2. Therefore
II
E z
II > I fz(z) I : : :  I fz IIA~
K12 (1 I z 12)(Q+2)/p.
o
The proof is complete. In what follows, C denotes a generic positive constant.
THEOREM 1.4. Let a ::::: 1, q > 0 and W E Lq(m). Let If> be an analytic selfmap of the disc such that I If> (e ili ) I < 1 a.e [m]. For fixed p, 1 :::; p < 00, the following are equivalent. (1) W W ,4> : A~ > Lq(m) is order bounded. (2) W/(lllf>* I)(Q+2)/p E Lq(m).
Suppose that w/(lllf>* 1)(Q+2)/p E Lq(m). Since I If> * (e ili ) 1< 1 a.e. [m], Lemma 1.3 provides a constant C depending only on a and p such that PROOF.
a.e. [m] for
I f(If>*(e ili )) I :::; C(l I If> * (e ili ) 1)(Q+2)/p all f with II f IIA~:::; 1. Let h(e ili ) = C I W(e ili ) I (1 I If> * (e ili ) 1)(Q+2)/P.
Then hE Lq(m), by hypothesis, and the previous inequality implies that
I w(e ili ) II
f(If>*(e ili ))
I :::; h(e ili )
a.e. [m]. Thus WW,4> : A~ > Lq(m) is order bounded, as required. Next suppose that Ww,4> : A~ > Lq(m) is order bounded. Thus there exists hE Lq(m), h ::::: 0, with
I h(e ili ) I : : : I W(e ili ) II f(If>*(e ili )) I a.e. [m] for every f with II f IIA~:::; 1. By Lemma 1.3, the inequality h(e ili ) ::::: I w(e ili ) I sup{1 E4>'(e )(f) I: I f IIA~:::; 1} = I w(e ili ) I I E4>'(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 2DY~(n+l)"Il
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 HilbertSchmidt if and only if (Xl n=O
In [13], J. H. Shapiro and P. Taylor proved that Cq, : H2 + H2 is HilbertSchmidt 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 HilbertSchmidt 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 selfmap 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 HilbertSchmidt. 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)"11 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 HilbertSchmidt. 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 <
00
00
n=O {=}
W wk,4> : D,  H2 is HilbertSchmidt.
o 3. Weighted Banach Spaces
In this section a connection is drawn between ;3order 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 nonincreasing, 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. MontesRodriguez [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)
= (lzw)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)2klr2m+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<
1.
n=l
In [7], Jarchow and Riedl showed that f E H;:' in the case 0 < k :s:; 1. However, their argument remains valid for all k > O. Let rn(t) denote the Rademacher functions given by
ro(t) = {
~1,
if 0 :s:; t  [t] < 1/2, if 1/2 :s:; t  [t] < 1.
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
For n
~
= ro(2nt).
1, let rn(t)
99
For 0 :::; t :::; 1, 00
n=l
Since Jarchow and Riedl's estimate on II f Ilv depends solely on the magnitude of the coefficients of f and since I rn(t) I = 1 for all nand t, there is a constant C such that II It Ilv :::; C for 0 :::; t :::; 1. It follows that I Ww,q,(It) IILq(m):::; KC for 0:::; t :::; 1. Thus
(KC)q
~
127r I w(e ill ) Iq I ~ 2kn r n (t) <1>*(e ill )2n Iq dm for 0 :::; t
:::; 1.
Since the previous estimate holds for all 0 :::; t :::; 1, integration with respect to t and Fubini's theorem yield
(KC)q
~
t
(27r I w(e ill ) Iq io
I f 2kn
io
rn(t) <1>*(e ill )2n
Iq
dt dm.
0
Khinchine's inequality provides a lower bound for the inner integral in the previous expression, and yields
~ (27r I w( eill ) Iq (f 22nk I <1>* (e ill ) 12"+1 )q/2 dm
C1
io
0
where C 1 is a positive constant depending only on q. Bringing all the constants together, Lemma 3.4 now implies that there is a positive constant Co such that
Co
~
(27r,
I w(e'll ) Iq
io
I <1>*(e ill ) Iq
(1 I <1>*(eili) 12)kq dm.
An easy argument now yields W/(v 0 <1>*) E £q(m). Finally suppose that w/(v 0 <1>*) E £q(m) and let f E H;:' with Since I <1>* (e ill ) I < 1 a.e [m], Lemma 3.3 implies that the inequality
I Ww,q,(f)(e II I :::; I W(e''II ) III Eq,'(e l
)
iO )
I II f Ilv :::; I W(e''II ) I
I
f
Ilv:::;
1.
1
v(<1>*(e ill ))
holds a.e. [m]. Thus the function h =1 W I (1 I <1>* I)k serves as the dominating function in Definition 1.1, and Ww,q, : H;:' > £q(m) is order bounded. The proof is complete. 0 Let wp(r) = (1  r)l/p. Lemma 1.3 implies that HP c H:;:'. Let (3 > O. p In [7], Jarchow and Riedl proved that Cq, : HP > £p{3(m) is order bounded for some (for all) p ~ 1 if and only if Cq, : Hw p > £P{3(m) is order bounded for some (for all) p ~ 1. The next corollaries present related facts for Ww,q, in the context of the weighted Bergman spaces. Here the appropriate weight will be v<>,p(r) = (1  r)(<>+2)/p, where p ~ 1 and a ~ 1. Lemma 1.3 shows that for each such p, A~ c H;:'G,p . Corollary 3.6 follows from Theorem 1.4 and Theorem 3.5. COROLLARY
3.6. Fix P
~
1, a
~
1 and let (3 > O. The following are equiva
lent. (1) Ww,q, : A~ > £P{3(m) is order bounded. (2) Ww,q, : H::':"p > £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 HilbertSchmidt [13]. By Theorem 3.1 [13], it follows that 1/(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 <1> is a selfmap of D with 1<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<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 selfmaps <1> 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 <1> a selfmap of D, W =I= <1>(0) and 'Y > 0, N,(w) =
I) log(l/
1
z I) )'
where the sum extends over all z with <1>(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 'bigoh' condition is replaced by'littleoh'. 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
17 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
17 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 17 1. = 0 ( (1 1wi)' ) as 1w 17 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 selfmap 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,
az 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 selfmap of the disc. The following are equivalent. (1) Ww,
+2)q/P) as I a lt 1. Compactness is characterized by the analogous littleoh condition.
t Ah is bounded. (1 I a 12)(<>+2)/p (1 _ az)2(<>+2)/p ,I z I < 1.
PROOF. First assume that Ww,
fa(z) = Since
I
1 [14],
IIA~::::::
fa
For a E D, let
there is a constant C with
e > (1 I a 12)(<>+2)q/p 
1
11 
I w(z) Iq dA/3(z) a(z) 12(<>+2)q/p
for all a E D. The first estimate in Lemma 4.5 now yields the result. Assume that the second condition holds, that is, there exists ro, 0 < ro < 1, and a constant e such that
1
I w(z) Iq dA/3 < e (1 I W
12)(<>+2)q/p for
By the closed graph theorem, it is enough to show that f E A~. A standard estimate yields a positive constant
II
I wi>
II Ww,
roo
IIA~ < 00
for all
e such that
II~q S jDl W(z) Iq (1  I ~ Z ) 12)2jD(
Ww,
11
Since w E D((z)) {:} z E l(D(w)), Fubini's theorem and Lemma 4.5 (Part 2) yield
I
W
w,
By Lemma 1.3,
II
Il q <
(f)
Ah 
q
f( ) w
D
I f(w) I sell
IIAh sell
Ww,
ejl
Iq
f IIA~ (1
q_p f IIA~
j I f(w) D
I w(z) Iq dA/3(z) dA( ) (1 I w 12)2 W . I W 1)(<>+2)/p and it follows that p J
J
Since W E Ah and f E A~,
(
Jwl'5.ro
I f(w) IP J
1
< (
2)1(<>+2) q /P jDl
1  ro
dA/3(z) dA (w) (1 I W 12)(<>+2)q/p <>
f(w)
IP jDl w(z) Iq
dA/3(z) dA(w) <
00.
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
103
The hypothesis implies that
1
I f(w) IP
1'o
JiI>l(D(w)) I w(Z) Iq dA{3(z) dA (w) < C (1 I w 12)(a+2)q/p a
II f
II~~
.
Thus I WW,iI>(f) IIAij < 00, and WW,iI> : A~ + A~ is bounded. The proof of the statement about compactness involves similar estimates and the standard criterion for compactness. The details are omitted. 0 In the case q < p, the characterization will make use of work of D. Luecking. In [8]' Luecking used Khinchine's inequality and other techniques to prove the following result on the Bergman space AP = A{;. The result is stated here in the more general setting of the spaces A~. See [8] for the ideas involved in the proof for Q: =I O. The restriction Q: > 1 is needed for the construction of suitable test functions to be used in the proof of the last theorem. THEOREM 4.7 (Luecking). Let 0 < q < p and let Q: > 1. Let f,L be a positive measure on the disc, and let L(z) = f,L(D(z)) (1 I z 12 )(a+2). The following are equivalent. (1) There is a constant C such that
(1
1f
Iq df,L)l/q
:::; C
II
f
IIA~
for all f
E
A~.
(2) L E LP/(pq)(A a ). THEOREM 4.8. Let 1 :::; q < p and let
Q:
> 1. Let W E A~. For I z 1< 1, define
I w(w) Iq dA{3(w) (1 I Z 12)a+2
J~l(D(z))
=
L(z)
The following are equivalent. (1) WW,iI> : A~ + A~ is bounded. (2) Ww,iI> : A~ + A~ is compact. (3) L E LP/(pq)(A a ). PROOF. First suppose that L E LP/(pq)(A a ), II fn IIA~:::; C and fn + 0 uniformly on compact subsets of the disc. To prove (2) it will be enough to show that I Ww,iI>(fn) IIA~ + 0 as n + 00. An argument as in the proof of Theorem 4.6 yields
II Ww,iI>(fn) II~q :::; Il
C
j
D
I fn(w) Iq
L(w) dAa(w).
Let f > 0 be given. The hypothesis (3) implies that there exists r, 0 < r < 1, such that
1
Since
I
fn liAR:::;
L(z)p/(pq) dAa < fP/(Pq).
r
1
:::; II
fn
II~~
(1
I fn(w) Iq
L(w) dAa(w)
r
r
L(w)p/(pq) dAn )(pq)/p :::; Cqf for all n.
104
R. A. HIBSCHWEILER
Since In
>
1
0 uniformly on {z:1 z
Izl::;r
I In(w) Iq
I ::; r}
L(w) dAa ::;
E
and since W E Ah,
(1  r 2 )(a+2)
I
W II~q (3
L
dA,,(z)
for all large n. Thus II WIlt,
(LI
w(z)
Iq I (f 0
<1»(z)
Iq dAj3(z) )ljq ::; C I I IIAl: .
Let v be the measure defined by dv(z) = I w(z) Iq dAj3(z). Then v is a positive measure on D and the previous expression can be rewritten as
Luecking's result finishes the proof.
o
The restriction a > 1 in Theorem 4.8 can not be removed. To see this, let W = 1, <1>(z) = z, and consider the operator WIlt,
ORDER BOUNDED WEIGHTED COMPOSITION OPERATORS
105
DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW HAMPSHIRE, DURHAM, NEW HAMPSHIRE, 03824
Email address: rah2(Qunh.edu
Contemporary Mathematics Volume 454, 2008
Fractional Cauchy Transforms and Composition T.R. MacGregor
1. Introduction
Let D = {z E C : I z I < I} and let T = {z E C : I z I = I}. Let M denote the set of complexvalued Borel measures on T. For each positive real number 0:, let :Fa. denote the set of functions I such that there exists 11 E M for which (1)
for
(2)
I z 1< 1.
The power function in (1) is the principal branch. If I E :Fa., let
IIIIIT = inf 111111 '"
J1EM
where 111111 denotes the total variation of 11 and 11 in (2) varies over all members of M for which (1) holds. This defines a norm on :Fa. and :Fa. is a Banach space with respect to this norm. A function given by (1) is called a fractional Cauchy transform of order 0:. When 0: = 1 this gives the Cauchy transform of a measure supported on T. The book [10] is an introduction to the research on the families :Fa.. For 0: > 0 let Ca. denote the set of functions 'P : D  D such that the composition 1 0 'P E :Fa. for every I E :Fa.. If 'P E Ca. then the mapping I ft 1 0 'P is a continuous linear operator on :Fa.. Let 11'Plle" denote the norm of this operator. The identity function belongs to :Fa. for every 0: > 0 and hence Ca. C :Fa. In particular, each member of Ca. is analytic in D. This paper concerns the problem of describing the functions which belong to Ca.. A survey of known facts about this problem is given and some new results are presented. In [4] Cima and Matheson study composition operators on the family of Cauchy transforms where the emphasis is on questions relating to the compactness and the weak compactness of these operators. 2000 Mathematics Subject Classification. 30E20. ©2008 American Mathematical Society
107
T.H. MACGREGOR
108
2. Basic facts about C" Theorem 1. If ip : D > 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 onetoone 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 <
00.
In [6; see Lemma 2, p. 160] it was shown that Ba C Fa. This gives an analytic way of showing membership in Fa. Theorem 5. If 0 < a < 1, ip E Ba and sup 1ip(z) 1< 1 then ip E Ca. Izl
FRACTIONAL CAUCHY TRANSFORMS AND COMPOSITION
109
Below we present two results about membership in Bo:. Combined with Theorem 5 this yields sufficient conditions for membership in Co: when sup I y?(z) 1< 1. Izl
Theorem 6. Suppose that j
HI and let F(B)
E
=
lim j(re ilJ ) and D(B, A) =
rj.l
F(B + A)  2F(B) + F(B  A) for suitable values of B and A in [71",71"]. If 0 <
Q
<1
and
(6) then j E Bo:. A proof of Theorem 6 follows from the result shown in [10; see p. 139141]: If j E HI and 0 < Q < 2 then there is a constant A such that
11 I j'(re
(7)
I (1 
ilJ )
rt 1dr :::: A
17r I DA~~:) IdA.
We see from (7) and Tonelli's theorem that (6) implies (5). 00
Theorem 7. Let 0
< < 1 and suppose that Q
y?(z)
= I,>nzn(1 z 1< 1) and n=O
00
2:(n + 1)10: 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  (lr)'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 10:
n=l
I
an
1::::1
a1
I
n=2
20: A 00 ~ 10: (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)(31dBdr :::: B
7r
t (1  r)(30:1dr <
Jo
00.
Therefore Y? E B(3 for (3 > 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
1z a J.t E M such that
F(b(z)
=
J
F(oz)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 ft 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<
1 M' For k 1
Suppose that
I(
111
= 1, 2, ... let Ak (a) denote the binomial coeffi00
1< 1).
(1 _ z)o = LAk(a)zk(1 z k=O 1= 1 and for n = 1,2, ... let n
Pn(z) = LAk(a)[b(f(zW(1 z k=O
1< 1).
Then Pn E :F0 and n
IIPnIIF" <
LAk(a) I b Ik k=O
IlfkllF"
k=O
~
k
< L..,Ak(a) I b I M
1
k
k=O
Since I b 1<
_
(1 I b I M)o = P
=
~, P < 00.
From f E M 0 it follows that f E Hoo and IlfllHoo ::; M [9; see p. 380]. (The last inequality and Theorem 1 give another proof of this theorem when a :::: 1.) Hence
I b(f (z) I::; I b I M < 1
I z I< 1.
for
00
Therefore Pn(z)
>
LAk(a)[b(f(z)jk uniformly in V as n k=O
> 00.
Since
IIPnllFo ::; P
00
for n
= 1,2, ... and Pn(z)
>
LAk(a)[b(f(z)jk for each z in V, it follows that k=O
00
00
LAk(a)[b(f(z)]k belongs to:Fo and IILAk(a)[b(f(z)]kIIFo ::; P [10; see Lemma k~
k~
1
7.9, p. 146]. We have shown that [1  (b f( z )] 0 E:Fo and for all ((I ( 1= 1). This implies that bf E Co. 0
II [1 
1
(bf (z )] 0
IIF < P ,,
5. Angular derivatives Our study of Co relates to results of Julia about the angular derivative of a bounded analytic function. Theorem 12. Suppose that the function rp : V > V is analytic. Let IJ E T. Then for each wET the nontangential limit
f3(w) == lim rp(z) 
(11)
z>w
exists or equals
00.
Let A = {w
E
Z 
T : f3(w)
IJ
W
=I
oo} and let ,(w) = ~f3(w). If IJ
wE
A than ,(w) is a positive real number and the nontangential limit limrp'(z) z>w
112
T.H. MACGREGOR
exists and equals (J(w). If A is nonvacuous 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 HerglotzRiesz 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(lwz)p(z)
z>w
=
e/L({w})
where z E S. Since e > 0 and /L is a nonnegative measure, this limit is a nonnegative real number and it is zero if and only if /L( {w}) = O. Let
(19) If A is nonvacuous 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 nontangential 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 nontangential 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 nonvacuous 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 (11) (z
< r < 1.
dv(() 0
Then (lr)O J(rw)
and the bounded convergence theorem yields
(22)
lim (lr)oJ(rw)
r+ 1
=
r ] iTr [1 1 r(w
lim (lr)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
We have 9 E Fa and
0
:::;
III/II· 1
for every 9 E Fa. If g(z) = (1 _ az)a then
(26) Inequality (25) holds for all 1/ E M which represent
o
f
in Fa. Thus (26) yields (20).
Next we show that there is an analytic function II' : V . V for which the nontangential limits of
T. Theorem 14. Let 0' E T. Suppose that {wd(k = 1,2, ... ) is a sequence of distinct points on T and {'Yd(k = 1,2, ... ) is a sequence of positive real numbers such that
1
00
L<00. 'Yk
(27)
k=l
Then there is an analytic function
1,2, ... the
(28) exists and equals
0',
the nontangential limit
13k =
(29) exists, and wkf3k
= 'Yk.
l' ip(z)O' 1m Z>Wk
Z 
Wk
Also the nontangential limit
0'
(30) exists and equals
13k
for k = 1,2, ....
Proof of Theorem 14: Let Ok 00
k=l
.
1,2, .... By assumptlOn Ok
> 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<
115
+
eq
=
1; e
+ ie.
.
1z 1<
1 and q(O)
Then p is analytic in V, Rep(z) > 1/2 for
1+e
1 and p(O) = b + te where b = 2' For 1z
ip(z) = a
(33)
Then ip is analytic in V and 1ip(z) Let wET. Then (33) implies
wip(z)a a zw
1<
1. Let e be any real
=
1<
1 let
(1 ptZ))'
1 for
1 (1wz)p(z)
1z 1< 1. 1
(1  wz) [eq(z)
+ lZe + ie]
If S is a Stolz angle in V with vertex Wk then the argument given in the proof of Theorem 12 and (31) yield
lim (1  WkZ)q(Z) =
Ek
for k = 1,2, ...
Z+Wk
where z E S. Hence lim Wk ip(z)  a = _1_ = ~ = 'Yk Z~Wk
a
z  Wk
eEk
15k
where z E S. This proves the assertions about (29). This implies the claims about (28). The assertions about (30) follow by the argument given in the proof of Theorem 12. 0 The argument for Theorem 14 shows that ip is obtained from the probability measure J.L which is only subject to the condition (31). This provides a variety of 00
functions
ip
having mass
when e > 15. If e = 15 then J.L is the measure supported on Ek
k=l
at Wk. If, in addition, we let e = 0, this yields
ip (Z ) =
a
s(z)  1 s(z)+1
By choosing the sequence
U{wd
L: 15k 1 + WkZ . oo
where
s(z) =
k=l
bd such that
1WkZ
1
L: a = 00
(27) holds and
00
for a given
0:
k=l 'Yk
where 0 < 0: < 1, we obtain further examples of analytic functions which ip rf Ca·
ip :
V . V for
References [1] L.V. Ahlfors, Conformal Invariants, McGrawHill, New York, 1973. [2] P. Bourdon and J.A. Cima, On integrals of CauchyStieltjes type, Houston J. Math 14 (1988), 465474. [3] C. Caratheodory, Theory of Functions of a Complex Variable, Vol. 2, Chelsea, New York, 1954. [4] J.A. Cima and A. Matheson, Cauchy transforms and composition operators, Illinois J. Math 42 (1998), 5869. [5] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
116
T.H. MACGREGOR
[6] D.J. Hallenbeck, T.H. MacGregor and K. Samotij, Fractional Cauchy transforms, inner functions and multipliers, Proc. London Math. Soc. (3) 72 (1996), 157187. [7] RA. Hibschweiler, Composition operators on spaces of Cauchy transforms, Contemporary Math 213 (1998), 5763. [8] RA. Hibschweiler and T.H. MacGregor, Closure properties of families of CauchyStieltjes transforms, Proc. Amer. Math. Soc. 105 (1989), 615621. [9] RA. Hibschweiler and T.H. MacGregor, Multipliers of families of CauchyStieltjes transforms, Trans. Amer. Math. Soc. 331 (1992), 377394. [10] RA. Hibschweiler and T.H. MacGregor, Fractional Cauchy Transforms, Chapman and Hall/CRC, Boca Raton, 2006.
Department of Mathematics Bowdoin College Brunswick, Maine 04011
Contemporary Mathematics Volume 454, 2008
Abstract Alec L. Matheson* (matheson\Omath.lamar. edu), Department of Mathematics, Lamar University, Beaumont, TX 77710. Continuous functions in starinvariant subspaces. This talk will examine the continuous functions in the starinvariant subspaces of HP. The existence of such functions was shown by Alexsandrov and further discussed by Dyakonov. This talk will discuss further properties of these continuous functions. (Received February 21, 2006)
117
Contemporary Mathematics Volume 454, 2008
Indestructible Blaschke products William T. Ross In memory of Alec L. Matheson
1. Introduction
Consider the following set of linear fractional maps za Ta(Z) :=  _  , lal < 1. 1 az Each Ta is an automorphism of the open unit disk ][J) := {z E C Ta(em) = em. For an inner function ¢, the Prostman shifts
¢a(z) := Ta
0
Izi <
I} and
¢(z)  a ¢(z) = 1 _ a¢(z)
are certainly inner functions. A celebrated theorem of Frostman [9] says that ¢a is actually a Blaschke product for every lal < 1 with the possible exception of a set of logarithmic capacity zero. In this survey paper, we explore the class of Blaschke products for which this exceptional set is empty. These Blaschke products are called indestructible and have some intriguing properties. 2. Frostman's theorems If (an)n~l is a sequence of points in ][J), pEN U {O}, and 'Y E JR, a necessary and sufficient condition that the infinite product 00
B(z) = eir zP defines an analytic function on
][J)
II
_
an an  z anz n=l
lan l1 
is that the series 00
converges. Such sequences (an )n~l are called Blaschke sequences and the product B is called a Blaschke product. The function B is analytic on ][J), has zeros precisely at the origin and the an's (repeated according to their multiplicity), and satisfies 2000 Mathematics Subject Classification. Primary 30D50; Secondary 30D35. Key words and phrases. Blaschke products, Frostman shifts, inner functions. ©2008 American Mathematical Society
119
120
WILLIAM T. ROSS
IB(z)1 < 1 for all z E lD. Furthermore, by a wellknown theorem of Fatou [3, Ch. 2] [8, Ch. 2], the radial limit function B*(():= lim B(r() T'fl
exists and satisfies IB*(()I = 1 for almost every ( E 8lD, with respect to (normalized) Lebesgue measure m on 8lD. REMARK 2.1. (1) In what follows, we use the notation B* (() to denote the radial limit value of B at ( whenever it exists (whether or not it is unimodular) . (2) This paper will cover a selection of results about Blaschke products. All the basic properties of Blaschke products, and more, are covered in [3, 5, 6, 8, 12, 18, 26]. For a particular point ( E 8lD, there is the following refinement of Fatou's theorem [10] (see also [3, p. 33]). THEOREM 2.2 (Frostman). A necessary and sufficient condition that a Blaschke product B, with zeros (an)n~l' and all its subproducts have radial limits of modulus one at ( E 8lD is that
~ lIa n l
(2.3)
L n=l
I( 
ani
< 00.
The Frostman theorems (like Theorem 2.2 above and Theorem 2.13, Theorem 2.14, and Theorem 2.18 below) are not always standard material for many complex analysts and so, for the sake of completeness and to give the reader a sense of how all these ideas are related, we will outline parts of the proofs of his theorems. In our discussion below, we will only use one direction of Theorem 2.2 so we prove this one direction and point the reader to [3, p. 34] for the proof of the other. Suppose, for fixed ( E 8lD, the condition in eq.(2.3) holds. We wish to show that B*(():= lim B(r() r+l
exists and IB*(()I = 1. The proof of the same result for any subproduct will follow in a similar way. Without loss of generality, we can assume ( = 1. First check the following inequalities (2.4)
11arl>lr,
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
= sin1 (
1
!3n r (l  la n l2) ) . lanllan  rill  anrl
From here, one can argue that the righthand 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 <
00.
n=1
Notice how this infinite Blaschke product B with zeros (a n )n;;'1 satisfies IB*(()I = 1 for every ( E aID. (3) With more work, one can even arrange the zeros of B t.o satisfy the much stronger condit.ion sup
~ lIanl ~
(Ei:J1IJi n=1
I( 
an
I
<
00.
We will get to this in the last. section.
(4) So far, we have examined when B*(() exists and has modulus one. Frostman [9] showed that the Blaschke product with zeros an = In 2 sat.isfies B*(I) = O. (5) If B*(() exists for every ( E aID, then results in [1, 5] say that. if E is the set of accumulat.ion point.s of (an)n;;'l, then (a) E is a closed nowhere dense subset of aID, (b) the function ( > B*(() is discontinuous at (0 if and only if (0 E E. By Fatou's theorem, the radial limit function
¢*(():= lim ¢(r(), Tl
for a bounded analytic function ¢ on ID, exists for malrnost 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 JL is a positive finite measure on aID with JL ..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 dJL(() re
= (DJL)(e iO )
whenever DJL(e iO ), the symmetric derivative of JL at eiO , exists (and we include the possibility that. (D JL) (e iO ) = (0). By the Lebesgue differentiation theorem, D JL exists at malmost every eiO • Moreover, since JL ..l m, we know that
(2.11)
DJL = 0 ma.e.
and
DJL =
00
JLa.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 ~ ~ ~:::dJL(()) I = exp ( fulIJi I( ~~;i:12 dJL(()) ,
shows that the radial limits of this second factor are unimodular malmost everywhere and hence this factor is an inner function. Furthermore, if JL ¢ 0 (i.e., the inner function ¢ has a nontrivial 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 nontrivial 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 nontrivial 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)
Td 
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~rr1a1D log IB(r()ldm(() = =
laID
lim_ T+1
~
r 10giBB (r()i dm(()
lim_ Td
r
laID
n
i (roi
log BB
lim_ Td
r 10gIBn(r()ldm(()
laID
dm(()
n
10g(1  c).
The last inequality comes from the submean 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 nonpositive 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 righthand side approaches zero since B is a Blaschke product (see above) and the integral on the lefthand 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)
za 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
:=
0
lal < 1,
are all inner functions. However, some of them might not be Blaschke products even if
T1/2
(exp ( 
~ ~ :))
turns out to be a Blaschke product. However, B 1/2(Z) :=
B(z) = exp ( 
L1/2 0
~ ~:)
is a singular inner function. Define the exceptional set £ (
(2.16)
£(
{a
:=
E IDl : Ta
0
This exceptional set £ (
2.17. The exceptional set £(
Fa· PROOF.
We follow the proof from [22, p. 53]. For each a
r
I>
r
JaIl)
E
1Dl, we the function
log l<Pa(r()ldm(()
is increasing on [0, 1) [8, p. 9] and so from Theorem 2.14, we see that a E £(
r~rr1aIl) log l<Pa(r()ldm(() < 0. For r E [0,1) and a E 1Dl, let
r
I(r, a):=
JaIl)
log l<Pa(r()ldm(()
and observe how, for fixed r, I(r, a) is a continuous function of a. For fixed r E (0,1) and kEN, let
F(r,k):= {a E 1Dl: I(r, a)
~ ~}.
Notice that F(r, k) is relatively closed in 1Dl. Finally, we observe that
hJ1 n02 F (1) 1  :;;' k 0000
£(
o
which proves the result.
The exceptional set £ (
Ga(z):=
J
log 1 1(_(z z 1 da(()
INDESTRUCTIBLE BLASCHKE PRODUCTS
125
and note that
o(
Ga(Z)
(00,
ZE
J])).
Since K is a compact subset of J])), G a is continuous near
aID and in fact
We will say K has positive logarithmic capacity if there is a positive (nonzero) measure a supported on K such that G a is bounded on J])). Otherwise, we say that K has zero logarithmic capacity. We say a Borel set E c J])) has positive logarithmic capacity if it contains a compact subset of positive logarithmic capacity. For example, if A denotes twodimensional Lebesgue area measure in the plane and A(E) > 0, a computation shows that G A is bounded on J])). Thus any set of positive area has positive logarithmic capacity. However sets of zero logarithmic capacity are much 'thinner'. For example, Borel subsets of logarithmic capacity zero must have zero area and compact subsets of zero logarithmic capacity must be totally disconnected. There are various other ways to define logarithmic capacity, depending on the particular application. However, they all have the same sets of logarithmic capacity zero. Two excellent sources which sort all this out are [11, 29]. This next result of Frostman [9] says that £(1;) is a small set. THEOREM 2.18 (Frostman). For an inner function 1;, £(1;) has logarithmic capacity zero. PROOF. Suppose £(¢) has positive logarithmic capacity. By Theorem 2.14, there is a compact subset K of positive logarithmic capacity such that (2.19)
lim_ ,.~1
r
J alTh
log 11  w~(r(~) I dm(() > 0, w r
\/w E K.
Moreover, by the definition of logarithmic capacity, there is a positive nonzero measure a supported on K such that G a is bounded on J])). We than have
r
0= lim G a (1;(r())dm(() 1'~lJalTh = lim_ ,.>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 nontrivial 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)
~ lIan 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 innerouter 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 nonnegative 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 malmost 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 nonnegative 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 nonnegative 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 nonzero 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 nonnegative superharmonic function log IBal. This function is the residual function covered in the previous section (see eq.(3.4)). Since U a is a nonnegative 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 malmost 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+1z 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 subclasses) of e, the Carles onNewman Blaschke products. These are Blaschke products B whose zeros (an)n~l satisfy the socalled 'conformal invariant' version of the Blaschke condition 00
L(1lan l) <
00,
n=l
i.e., sup
{~(1 
IlP(an)l) : lP E Aut(j[))) } <
00.
There are several equivalent definitions of e. For example, BEe {::} B is the finite product of interpolating Blaschke products {::} the measure L:n(1la n I2 )6an is a Carleson measure. The standard reference for this is [12] but another nice
132
WILLIAM T. ROSS
exposition with further references is [24, Theorem 2.2]. Two important examples of Blaschke products which belong to e are 'J, the thin Blaschke products B which satisfy the condition and
~,
the Frostman Blaschke products which satisfy the condition
~
sup 6 n=l
(E8l!}
lIan I < 00. Ie  an I
An example of a thin Blaschke product is one whose zeros lim n>oo
llan +11 1  lanl
=
(an)n~l
satisfy
°
[13, Prop. 1.1], while an example of a Frostman Blaschke product is one with zeros an = rneiOn, where (rn)n~l C (0, 1), (On)n~l C (0,1),
e::: n
sup { On+l
~
1} < 1
and
~ l  rn 6   <00 On n=l
[2, p. 130]. The thin Blaschke products relate to Douglas algebras and the structure of the bounded analytic functions [31] as well as composition operators on the Bloch space [4]. The Frostman Blaschke products turn out to be the only inner multipliers of the space of Cauchy transforms of measures on 8]]J) [19] (see also [2]). Following the definition of £(B), the exceptional set of a Blaschke product in eq.(2.16), define
£e(B) £'J(B)
:=
{a
E]]J) :
:=
{a {a
E]]J) :
£~(B) :=
E]]J) :
Ba Ba Ba
~
e}; ~ 'J}; ~ ~}.
Gorkin and Mortini [14, Lemma 3.2] use a result of Tolokonnikov [31, p. 884] to show the following. THEOREM 5.1. If BE 'J then £'J(B) = 0. The current author and Matheson [20] use the theory of inner multipliers for the space of Cauchy transforms and the ideas of Tolokonnikov [31] and Pekarskii [28] to prove the following. THEOREM 5.2. If B E
~,
then
£~(B)
= 0.
Nicolau [25] states necessary and sufficient conditions, in terms of the zeros of
B, so that £e(B) = 0  which are a bit technical to get into here. We do point out the following. (1) If B is a Blaschke product, then £e(B) is closed in]]J). (2) Given any 0 < s < 1, there is aBE e such that £e(B) = {s ~ Izl < I}.
THEOREM 5.3.
REMARK 5.4. (1) Compare this to £(B) which is an F(7 set of logarithmic capacity zero (Proposition 2.17 and Theorem 2.18). (2) The first result of the above theorem is contained in [25, Lemma 1]. A version of the second result is found in [25, §3]. See [14, Theorem 4.2] for the version we state here.
INDESTRUCTIBLE BLASCHKE PRODUCTS
133
There is a sizable literature of deep results which relate the class of Blaschke products P:= {B : £e(B)
= 0}
to many ideas in function algebras. We refer the reader to [24, p. 287] for a discussion of this and for the exact references. So far we have discussed when a Blaschke product has the property that all its Frostman shifts belong to a certain class of Blaschke products. We point out two papers [14, 24] which discuss when an inner function
134
WILLIAM T. ROSS
20. A. Matheson and W. T. Ross, An observation about Frostman shifts, Compo Math. Punet. Thry. 7 (2007), 111  126. 21. R. McLaughlin, Exceptional sets for inner functions, J. London Math. Soc. (2) 4 (1972), 696700. MR 0296309 (45 #5370) 22. R. McLaughlin and G. Piranian, The exceptional set of an inner function, Osterreich. Akad. Wiss. Math.Naturwiss. Kl. S.B. II 185 (1976), no. 13,5154. MR 0447585 (56 #5895) 23. H. S. Morse, Destructible and indestructible Blaschke products, Trans. Amer. Math. Soc. 257 (1980), no. 1, 247253. MR 549165 (80k:30034) 24. R. Mortini and A. Nicolau, Frostman shifts of inner functions, J. Anal. Math. 92 (2004), 285326. MR 2072750 (2005e:30088) 25. A. Nicolau, Finite products of interpolating Blaschke products, J. London Math. Soc. (2) 50 (1994), no. 3, 520531. MR 1299455 (95k:30072) 26. K. Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 28, SpringerVerlag, Berlin, 1960. MR 0133464 (24 #A3295) 27. M. Parreau, Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Ann. Inst. Fourier Grenoble 3 (1951), 103197 (1952). MR 0050023 (14,263c) 28. A. A. Pekarskil, Estimates of the derivative of a Cauchytype integral with meromorphic density and their applications, Mat. Zametki 31 (1982), no. 3, 389402, 474. MR 652843 (83e:30047) 29. T. Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001) 30. W. Rudin, Real and complex analysis, McGrawHill Book Co., New York, 1966. MR 0210528 (35 #1420) 31. V. A. Tolokonnikov, Carleson's Blaschke products and Douglas algebras, Algebra i Analiz 3 (1991), no. 4, 186197. MR 1152609 (93c:46098) DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OF RICHMOND, RICHMOND, VIRGINIA 23173 Email address: wrosslDrichmond. edu
Contemporary Mathematics Volume 454, 2008
On Taylor Coefficients and Multipliers in Fock Spaces James Tung ABSTRACT. Several necessary or sufficient conditions for a function to be a member of the Fock space Fl: are proved, which improve previously known results. We also give conditions for a sequence of complex numbers to be a coefficient multipliers from the Fock space F~ to el .
1. Introduction
For 0 < p < 00 and a function J analytic in either the complex plane C or the unit disk ][]) in C, the integral means are Mp(r,f)
For a
> 0, the
(2~ 10
=
1
27r
IJ(reiOW de)
P
Fock space F'{; consists of all entire functions J such that 1
IIJllFg = (~: lIJ(z)e'i,z,2IP dA(Z)) P < 00. The following theorems are known to hold for functions in F'{; [10]. THEOREM A (HausdorffYoung Theorem for F'{;). Let 1 ~ p ~ conjugate index, and let J(z) = 2:::::"=0 anz n be an entire Junction.
(i)
00,
let pi be its
Forl
f
E
FE =
t;; lanl
P'
(:!) .; n~l <
00.
For the case p = 1, we have
(ii)
For2~p~00,
LI an I 00
n=l
pI
(n') ~ 
.
an
n
£..._1 I 4
2
<
00 ===}
J E F'{;.
2000 Mathematics Subject Classification. Primary 30BlO, 30D15. Key words and phrases. Fock space, entire function, Taylor coefficients. ©2008 American Mathematical Society
135
JAMES TUNG
136
THEOREM B (HardyLittlewood Theorem for Fg). Let J(z) an entire junction, and fix a > O. (i) ForO
~ lanl (:~) ~ n~+~ < * P
00
(ii) For 2'5. P < n! )
~ lanl P an 00
(
=
2:~=o anz n be
J E
F~ * ~ lanl (:~) ~ n¥~ <
J E
F~
P
00.
00,
~ n :!E 2 < 00 * 3
4
*
~
n! )
~ lanl P an (
~ n
E
4
1
+2 <
00.
Note that the conditions involving the sums in Theorems A and B can be restated in terms of membership in various p spaces. In this paper we show that the theorems above can be improved in several ways. First we restrict our attention to functions with lacunary Taylor series; that is, those functions of the form J(z) = 2:::'=1 akznk, where there is a number>' > 1 such that for every positive integer k, nk+1/nk ~ >.. 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 vectorvalued 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 HausdorffYoung 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 It xae x is decreasing on [a, 00); together with the estimate X 4
00
:':.1<.!'.

and Stirling's formula, we have
> 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<.!'. 2
In;p
Jn kP 
2
l'
> C (nk!) 2 ni+~. 
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)ecrlzI2 dA(z)1
=C
11 t 00
akCnk r2nk ecrr2 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 <
00.
k=1 Using the idea of the proof for Theorem 2.2, we give a new proof below of Theorem E for the case 1 ::; p ::; 2. As in the case of F[!" the HardyLittlewood theorem for AP (see [6], Chapter 3) already gives the sufficiency part of Theorem E for 1 ::; p ::; 2 and the necessity part for 2 ::; p < 00.
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES THEOREM 2.5. Let 1
:=:;
p
141
2, {nd be a lacunary sequence, and f(z) =
:=:;
2:~=o anz n be a function in AP. Then
for some constant C independent of f. PROOF. Let gration, we have
~
f
E AP. By Cauchy's formula and breaking the domain of inte
00
c
L
00
ianki P n;l (e 1 />.  e 1 )
= cL
k=l
ianki P n;l.
k=l
o We omit the proof for the following sufficiency condition for membership in AP when p ~ 2, which is based on a duality argument similar to that found in Theorem 2.4. THEOREM 2.6. Let 2 :=:; p < sequence of numbers such that
00,
and {nd be a lacunary sequence. If {ad is a
00
L
iaki P n;l <
00,
k=l then the function f(z)
= 2:akznk
is in AP.
3. Mixed norm sequence spaces We prove the following proposition, which can be viewed as an analogue of the HausdorffYoung theorem with the domains restricted to lacunary blocks.
142
JAMES TUNG PROPOSITION
Ak = {z: ~ ::;
h = [nk, nk+d n Nand
3.1. Let {nd be lacunary, and let
Izl < Vn;l}.
sup lanl nEh
Then for any entire function f,
r:;;;. n! ::; C r If(reiB)1 e'}lzI 2 dA(z) V~ JA k
for some constant C independent of k. PROOF. Let f(z) = L~=o anz n . We mimic the proof of the HausdorffYoung Theorem for Fock spaces (see [10]) and look at the integral
r zn f(z)ealzI2 dA(z).
JAk
For all sufficiently large k and nk ::; n < nk+1,
r
IJA k zn f(z)ealzI2 dA(z)1
r
::; sup Izl n e'}l zI2 If(z)1 zEAk JAk
= (~) ~ ae
r
JAk
e~lzI2 dA(z)
If(z) I e'}l zI2 dA(z).
For the other direction, we calculate
IJAr
k
zn f(z)ealzI2 dA(z)1 = ciani Jv'nk+1/0l r2nealzl2 rdr v'nk/a nk + 1 1 = c lanlune u duo an nk
I
Now fix an integer k, and let Jk = [nk, nk+1), and for each n E h, let following interval:
Jk =
[max {nk, n 
Jk be the
Vn}, min {nk+1, n + Vn}) .
Then Jk C J k , and since {nd is lacunary, the width of the intervals Jk is comparable to a constant multiple of Vn when k is sufficiently large. The realvalued function cp(u) = une U has maximum at u = n and points of inflection at u = n ± Vn, so that for every n E h, the function cp is concave down on Jk . Thus the area under the graph of cp on Jk can be estimated from below by the area of a triangle with base Jk and height cp( n); that is,
l~k+l une u du 2: c (~) n Vn,
nE
h,
for some constant c independent of k and n. Thus
Ilk zn f(z)e0lIzI2 dA(z)l2: ciani
(:ef Vn·
Combining the two inequalities we obtain, for all large k and n E h,
lanl
(~) ~ n~
::; C
r If(z)1
ae JA k and the result follows from the fact that as n ;
( aen)~ n'i 1
rvC
e ~ Izl2
dA(z),
00,
IN'
1 n4. an
o
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES COROLLARY 3.2. Let 0: > 0, {nd be a lacunary sequence, and h N. For every f(z) = L:~=oanzn E Fl, we have
143
= [nk' nk+l)n
for some constant C; that is, a necessary condition for an entire function f to belong to Fl is that its coefficients {an} satisfy
PROOF. This follows immediately from the Proposition by summing over k.
0
We remark that since £00,1 C £00, this is an improvement of the know HausdorffYoung inequality (see Theorem A, part (i)). We also remark that the following implication holds
f
E
F~
===}
an
=0
(
~ni)
,
which improves the bigO condition in Corollary 5 of [10]. We use interpolation theory to fill in the result between Fl and F~. In the following proof we will use the weighted spaces £P,q (J.L), where where J.L is the discrete measure
J.L ({O}) = 1,
J.L({n})
1
= ,jii' n=1,2, ....
The spaces £p,q(J.L) can be thought of as an Lq space consisting of £P(J.L)valued functions, and a sequence {an} is in £p,q (J.L) if
The space £00,1 (J.L) coincides with £00,1 because J.L is a discrete measure, and the essential sup norm defined by J.L is the same as that defined by the counting measure. THEOREM 3.3. Let 1 < p :::; 2, and p' be its conjugate index. Let {nd be a lacunary sequence, and h = [nk, nk+1) n N. For every f(z) = L:~=o anz n E Fg, we have
that is,
We remark that this is an improvement of the HausdorffYoung theorem (Theorem A, part (i)), since £p',p (J.L) c £P' (J.L).
144
JAMES TUNG
PROOF. For f(z)
= 2:::=0 anz n Tf
=
in FJ;" let T be the operator
;;;; n { an VIn!
1}
4.
•
The wellknown equation 2 ~ IlflIF~ = L
lanl
2
n! an'
n=O
shows T is a bounded operator from F; to £2(Jl) = £2,2(Jl). Corollary 3.2 shows T is a bounded operator from F~ to £00,1 (Jl). We now use the complex interpolation method (see Chapter 4 of [4]) to get the desired result. More specifically, Janson, Peetre and Rochberg ([7], Theorem 7.3) give the interpolation spaces between the various FJ;, spaces: for 0 < () < 1, [F~, F~]o = Fg,
where ~ = (1 ())/2 + (). For the interpolation spaces between £2,2(Jl) and £00 ,1 (Jl), we apply the theorem concerning interpolation between vectorvalued LP spaces (see 5.1.2 in Bergh and Lofstrom [4]) to conclude that
[P,2(Jl),£00,1(Jl)]O = £q,P(Jl), where 1P = (1  ())/2 + () and 1q = (1  ())/2 = 1" so that q is simply the conjugate P index of p. We apply complex interpolation method to conclude that T is a bounded operator from FJ;, to £p',p (Jl) for 1 s:; p s:; 2. 0 The following sufficient condition for membership in FJ;, when p ?: 2 can be proved by a duality argument, and the proof is omitted. THEOREM 3.4. Let 2 s:; p < 00, and p' be its conjugate index. Let {nk} be a lacunary sequence, and Ik = [nk,nk+l) nN. For every sequence {an} such that
4. Coefficient Multipliers THEOREM 4.1. A sufficient condition for a sequence of complex numbers {An} to be a multiplier from
F~ to
(2) The exponent 
i is sharp.
£1
is {>"n/ff n i
}
E £1,00; that is,
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES PROOF. Let J(z)
f
=
145
~:=o anz n be in F~. Then
f (L IAnl nEh ~ f nEh ~ cf (1
t ; n  i lanl {ff;n i )
IAnanl =
k=1
n=~
k=1
(L IAnl V~;r z (LnEh IAnl t;
(sup lanl {ff;n i )
n i )
nE1k
Q
n i ) , k=1 Ak by Proposition 3.1. Continuing with the estimate and applying the hypothesis, we have IJ(reilJ)1 e'il I2 dA(Z))
n~" IAnanl OS C (,~p ~, IAn I t; n ~
c IIJIIF~
<
I) (~l,
If(re") I,,1,1' dA(Z))
00.
We now show that the exponent  ~ is sharp; that is, given any c > 0, there is a sequence {An} that satisfies
'" IAnl Vran ;r, nn.
sUP.L... k
nEh
41
e: <
00
but is not a multiplier. Let c > 0, and let {An} be the lacunary sequence where \
{{if o ! an
_
An 
Look at the function J(z) an
=
= ~:=o
ni+e: , otherwise.
anz n , where
nie:/2, { Vr;;;; rtf
o
otherwise.
Then J(z) is in F~ by Theorem 2.2, but ~n IAnanl = ~k (nk)e:/2 = REMARK 4.2. The case p theorem by letting \
_
An 
=
00.
o
1 of Theorem 2.2 follows as a corollary of the above
{{if o !
on
ni
n
=
nk for some k;
otherwise.
In the converse direction, we are able to give a necessary condition that is off by a factor of ..;n. We first prove the following lemma that gives a condition on a sequence to belong to the mixed norm sequence space £1,00. This lemma shows that the results in [5] and [6] can be described in terms of mixed norm spaces. For the following lemma and theorem, we assume that the mixednorm spaces are formed by a lacunary sequence {nd satisfying the addition property that {nk+ Iink} is bounded above. LEMMA 4.3. Let Cn be a sequence oj nonnegative numbers, and 8 > O. Suppose is a mixednorm space Jormedfrom a lacunary sequence {nd where {nk+l/nk} is bounded above. IJ~:=1 n"cn = O(N"), then {cn} E £1,00.
£1,00
JAMES TUNG
146
PROOF. Let n be in the kth lacunary block, that is, nk nk+l
Cn%+l ~
L nDc
n
Lc
~ (nk)D
n=l
~
n
< nk+1. Then
n·
nEh
Since nk+l/nk ~ C, we have L:nEh Cn ~ CD, so {c n } E £1,=.
D
THEOREM 4.4. Let a > O. Suppose £1,= is a mixednorm space formed from a lacunary sequence {nd where {nk+l/nd is bounded above. A necessary condition for a sequence of complex numbers {An} to be a multiplier
f rom
F1a to £1 is
{Anyr;;;; nr n~} E £1,=." that is
(3)
L
sup
IAnl
nElk
k
~ n~ V;;:r
<
00.
PROOF. Fix r < 1, and let fr(z)
e~(rz)2,
=
9r(Z)
=
ze~(rz)2.
fr and 9r are in F~, because they are of growth less than (2, a/2) (see p.15, [9]). A calculation shows C Ilfr(z)IIF~ ~ (1 r)l/2'
and
C 119r(z)IIF~ ~ 1  r'
Note that fr(z) = L::=o Q;:r~~ z2m, and 9r(Z) {An} is a multiplier, we have
=
a m r2m IA2ml 2mm!
=
a m r2m IA2m+11 2mm!
fo fo
=
L::=o ~:r~~ z2m+l. Then since
~ C IlfrllF~ ~
C (1  r)l/2'
~ C 119r11F~ ~
C (1 r)'
Now let N be an integer, and choose r = 1  l/N. Then
v'fj> "" IA ~

2m
I a m (1 1/N)2m > c "" IA 2mm!
~

m~N
I am 2m 2mm!
m~N
and similarly
Then by Lemma 4.3,
am ! m { A2m+1 2mm
I}
E
F§
£1,= .
Now consider the expression ni, as in the statement of the theorem. For large n = 2m, the expression is comparable to
J(~~!
(2m)i
rv ( ; ; ) m
m 1
rv
2~:! m~.
ON TAYLOR COEFFICIENTS AND MULTIPLIERS IN FOCK SPACES
For large n
147
= 2m + 1, we have
(2~:+~)! (2m + 1)~
rv
C::J
m
m~
rv
2::! m 1
a 2m +1 (2m (2m + I)!
:c:
3} E e
+ 1)4
and the theorem follows by adding the two sequences.
1 ,00
' D
References [1] O. Blasco, Introduction to vector valued Bergman spaces, in FUnction spaces of operator theory (Joensuu, 2003), Univ. Joensuu Dept. Math. Rep. Ser. 8 (2005), 930. [2] O. Blasco and A. Galbis, On Taylor coefficients of entire functions integrable against exponential weights, Math. Nachr. 223 (2001), 52l. [3] S. Buckley, P. Koskela and D. Vukotic, F'ractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Camb. Phil. Soc. 126 (1999), 369385. [4] J. Bergh and J. Lofstrom, Interpolation Spaces, SpringerVerlag, Berlin, 1976. [5] P. L. Duren, Theory of HP Spaces, Dover Publications, Mineola, New York, 2000. [6] P. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, Providence, RI, 2004. [7] S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Revista Mat. Iberoamericana,3 (1987),61129. [8] C. N. Kellogg, An extension of the HausdorffYoung Theorem, Michigan Math. J., 18 (1971), 121127. [9] J. 'lUng, Fock Spaces, Ph.D. thesis, University of Michigan, 2005. [10] J. 'lUng, Taylor coefficients of functions in Fock spaces, J. Math. Anal. Appl., 318 (2006), no. 2, 397409. DEPARTMENT OF MATHEMATICAL SCIENCES, DEPAUL UNIVERSITY, CHICAGO, ILLINOIS
Email address: ytung0depaul. edu
60614
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ISBN 9780821842683
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