Reproducing kernels and radial differential operators for holomorphic and harmonic Besov spaces on unit balls: a unified view
type: Item
instance of: scholarly article
title: Reproducing kernels and radial differential operators for holomorphic and harmonic Besov spaces on unit balls: a unified view
zbMATH Open document ID: 1213.46026
DOI: 10.1007/BF03321777
published in: Computational Methods and Function Theory
publication date: 2011-01-25T00:00:00Z
review text: The author provides a short survey of the basic characteristics of the two--parameter Besov spaces \(B^p_q,\, 1\leq q <\infty, q\in\mathbb R\), (respectively \(b^p_q\)) of holomorphic (respectively harmonic) functions on the unit ball \(\mathbb B\) in \(\mathbb C^n\) (respectively \(\mathbb R^n\)). For \(q>-1\), these become the weighted Bergman spaces of holomorphic (respectively harmonic) functions in \(L^p_q(\mathbb B) = L^p(d\nu_q)\), where \(d\nu_q\) is the measure \((1-|z|^2)^qd\nu\) (respectively \((1-|x|^2)^qd\nu) \) on \(\mathbb B\). For \(q\leq -1\), the spaces \(B^p_q\) as well as \(b^p_q\) are defined in terms of the radial differential operators \(D^t_s\) of fractional order \(t\in\mathbb R\). For both holomorphic functions \(f\) and harmonic functions \(u\), the operator \(D^t_s\) is defined in terms of the expansion of \(f\) in terms of homogeneous holomorphic polynomials \(f_k(z)\), and the expansion of \(u\) in terms of homogeneous harmonic polynomials \(u_m(x)\). Thus for \(q\in\mathbb R\), \(1\leq p<\infty\), \(B^p_q\) is the space of holomorphic functions \(f\) on \(\mathbb B\) for which \((1-|z|^2)^tD^t_sf(z) \in L^p_q(\mathbb B)\), where \(t\) is chosen such that \(q+pt >-1\), with a similar definition for \(b^p_q\). The paper stresses similarities as well as contrasts between the two spaces. It also investigates some relationships between the reproducing kernels of the Hilbert spaces \(B^2_q\) and \(b^2_q\).
reviewed by: Manfred Stoll
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 47B34
zbMATH DE Number: 5840921
zbMATH Keywords: Besov space
zbMATH Keywords: Dirichlet space
zbMATH Keywords: Drury-Arveson space
zbMATH Keywords: Hardy space
zbMATH Keywords: Bergman space
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: radial differential operator
zbMATH Keywords: spherical harmonic
author: H. Turgay Kaptanoğlu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5840921