On boundedness of Bergman projection operators in Banach spaces of holomorphic functions in half-plane and harmonic functions in half-space
type: Item
instance of: scholarly article
title: On boundedness of Bergman projection operators in Banach spaces of holomorphic functions in half-plane and harmonic functions in half-space
zbMATH Open document ID: 1386.30051
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-01-08T00:00:00Z
review text: Let \(A^p(\mathbb{D})\) denote the Bergman space on the unit disk \(\mathbb{D}\); the space of holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^p dA(z) \] is finite; here \(dA(z)=dxdy\) is the usual area measure. It is well known that for a finite \(p>1\), the Bergman projection from \(L^p(\mathbb{D})\) to \(A^p(\mathbb{D})\) is bounded. There has been a lot of interest to extend this result to regions other than the unit disk, as well as to the variable exponent Bergman space. This latter space consists of holomorphic functions in a given domain \(\Omega\) for which the integral \[ \int_{\Omega}|f(z)|^{p(z)} dxdy \] is finite. In the paper under review, the authors are mainly concerned with the upper half-space \(\mathbb{R}^n_+\) for \(n\geq 2\), and the upper half-plane when \(n=2\). For \(f\in L^1(\mathbb{R}^2_+)\), the Bergman projection is given by \[ Bf(z)=-\frac{1}{\pi}\int_{\mathbb{R}^2_+}\frac{f(w)}{(z-\overline w)^2}dA(w),\quad z\in \mathbb{R}^2_+, \] where \[ K(z,w)=\frac{-1}{\pi(z-\overline w)^2} \] is the Bergman kernel function for the upper half-plane. It is also known that the Bergman projection operator \(B:L^p(\mathbb{R}^2_+) \to A^p(\mathbb{R}^2_+)\) is bounded if \(p\in (1, \infty)\). In a similar fashion, one can define the harmonic Bergman space on the upper half-space. For \(n\geq 2\), this is the space \(h^p(\mathbb{R}^n_+)\) consisting of all harmonic functions in \(\mathbb{R}^n_+\) that belong to \(L^p(\mathbb{R}^n_+, dA(x))\) where \(dA(x)\) is the usual volume measure in \(\mathbb{R}^n_+\). It is known that the Bergman projection \[ bf(x)=\int_{\mathbb{R}^n_+}R(x,y)f(y)dA(y),\quad x\in \mathbb{R}^n_+, \] from \(L^p(\mathbb{R}^n_+)\) to \(h^p(\mathbb{R}^n_+)\) is bounded; here \(R(x,y)\) stands for the corresponding kernel function. The main purpose of the current paper is to extend the aforementioned results both to the variable exponent Bergman space on the upper half-plane, and to the variable exp…
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 6822963
zbMATH Keywords: Bergman projection
zbMATH Keywords: Bergman space, harmonic Bergman space
author: Alexey N. Karapetyants
author: S. G. Samko
MaRDI profile type: Publication
full work available at URL: s10958-017-3538-6
OpenAlex ID: W2756369031
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DOI: 10.1007/S10958-017-3538-6
description: scientific article; zbMATH DE number 6822963