Bounded and compact multipliers between Bergman and Hardy spaces
type: Item
instance of: scholarly article
title: Bounded and compact multipliers between Bergman and Hardy spaces
zbMATH Open document ID: 0939.47025
DOI: 10.1007/BF01225524
author: S. M. Buckley
author: M. S. Ramanujan
author: Dragan Vukotić
published in: Integral Equations and Operator Theory
publication date: 2000-07-05T00:00:00Z
review text: Let \(D\) denote the open unit disc \(\{z\in\mathbb{C}: |z|<1\}\) of the complex plane \(\mathbb{C}\). If \(0< p\leq\infty\), \(0<q,s<\infty\), then \(H(p,q,s)\) is associated with the weighted Hardy space \(H^p_s= H(p,\infty,s)\), and contains elements of the form \(f\) in the space of holomorphic functions \(H(D)\), with norm \[ \|f\|_{p,q,s}= \Biggl(\int^1_0 M_p(r, f)^q(1- r)^{sq-1} dr\Biggr)^{1/q};\quad M_p(r, f)= \Biggl((2\pi)^{-1} \int^1_0|f(re^{i\theta})|^p d\theta\Biggr)^{1/p}. \] The weighted Bergman space \(A^p_\alpha\) is identifiable as \(H(p,p,(1+\alpha)/p)\), \(\alpha>-1\). Let \(X\), \(Y\) be function spaces consisting of functions of the form \(f\) with \(f(z)= \sum^\infty_{j=0} a_nz^n\). If \(\gamma= \{\gamma_n, n= 0,1,2,\dots\}\) is a preassigned sequence, and the operator \(T_\gamma\) is defined by \(T_\gamma(f)(z)= \sum^\infty_{n=0} \gamma_na_nz^n\), then \(\gamma\) is said to be a multiplier in \((X,Y)\) if \(T_\gamma: X\to Y\), and \(T_\gamma\) is a bounded operator. The fractional operator \(D^t\), \(t>0\), is defined as a multiplier operator with \(\gamma_j= j^t\), \(j= 0,1,2,\dots\)\ . Let \(\ell_{p,q}\) denote the space of sequences of the form \(a= (a_j)\) for which \(\{\left(\sum^{2^k- 1}_{j= 2^{k-1}}|a_j|^p\right)^{1/p}\), \(k= 1,2,\dots\}\) has finite \(\ell_q\)-norm, and let \(K(X,Y)\) denote the class of compact operators from \(X\) into \(Y\). The main results of this paper identify and indicate inclusions involving the spaces \((H^p,H^q)\) and \((A^p,A^q)\). In particular, it is shown in Theorem 5 that if \(1<p<2\), \(1/r= (1/p)-{1\over 2}= (2-p)/2p\), then \(D^{-1/r}\ell^{r,\infty,\infty}\subseteq (A^p,A^2)\subseteq D^{-1/r}\ell^\infty\). It is also shown in Theorem 14 that if \(p>1\), \(q\geq 1\), then \({\mathcal L}(A^p,A^q)= K(A^p,A^q)\) if and only if \(q<p\), where \({\mathcal L}(X,Y)\) is the space of bounded linear operators from \(X\) into \(Y\). The paper contains references to papers which include related results.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 1346928
zbMATH Keywords: weighted Hardy space
zbMATH Keywords: space of holomorphic functions
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: fractional operator
zbMATH Keywords: multiplier operator
zbMATH Keywords: class of compact operators
zbMATH Keywords: inclusions
reviewed by: George O. Okikiolu
MaRDI profile type: Publication
cites work: Duality and multipliers for mixed norm spaces
cites work: Q4043323
cites work: Coefficient multipliers and solid spaces
cites work: Coefficient Multipliers of Bloch Functions
cites work: Operators on weighted Bergman spaces \((0 < p \leqslant 1)\) and applications
cites work: Multipliers on Spaces of Analytic Functions
cites work: Relative Solidity for Spaces of Holomorphic Functions
cites work: A survey of Hp multipliers as related to classical function theory
cites work: Q5618011
cites work: On the Multipliers of H p Spaces
cites work: Properties of H p (0 < p < 1) and Its Containing Banach Space
cites work: Coefficient multipliers of \(H^ p\) and \(B^ p\) spaces
cites work: Mean growth and coefficients of \(H^ p\) functions
cites work: Mean values of power series
cites work: On the Rate of Growth of Mean Values of Holomorphic and Harmonic Functions
cites work: The dual of an inequality of Hardy and Littlewood and some related inequalities
cites work: Some properties of fractional integrals. II
cites work: Q5580593
cites work: Zeros of functions in the Bergman spaces
cites work: Coefficient Multipliers of Mixed Norm Spaces
cites work: Contributions to the theory of the classical Banach spaces
cites work: Theorems on Fourier Series and Power Series (II)†
cites work: Coefficient Multipliers of Bergman Spaces Ap, II
cites work: $L^{p}$-behaviour of the integral means of analytic functions
cites work: Q3884381
cites work: Q3779231
cites work: Remarks on a Theorem of Paley
cites work: Q5634458
cites work: Some Results About Spaces of Analytic Functions Introduced by Hardy and Littlewood
cites work: Some Inequalities Related to the Hausdorff-Young Theorem
cites work: Q4178022
cites work: On the Coefficient Multipliers of Bergman Spaces
cites work: Q3995204
cites work: On Multipliers into Bergman Spaces and Nevanlinna Class
cites work: Q4020721
cites work: Q3999007
cites work: Q3254471
description: scientific article; zbMATH DE number 1346928