Embedding theorems and integration operators on Bergman spaces with exponential weights
type: Item
instance of: scholarly article
title: Embedding theorems and integration operators on Bergman spaces with exponential weights
zbMATH Open document ID: 1421.30072
DOI: 10.1215/20088752-2018-0013
author: Xiaofen Lv
published in: Annals of Functional Analysis
publication date: 2019-03-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.afa/1547629228
review text: Let $\mathbb{D}$ be the open unit disk in the complex plane, and let $\rho$ be a continuous real-valued function on $\mathbb{D}$ satisfying $\lim_{|z|\to 1}\rho(z)=0$, and \[ \|\rho\|_L=\sup\left\{\frac{|\rho(z)-\rho(w)|}{|z-w|}:z,w\in\mathbb{D},z\neq w\right \}<\infty. \] If $\rho$ has the extra property that for each $\epsilon>0$, there exists a compact subset $E\subset\mathbb{D}$ such that \[ \rho(z)-\rho(w)|\leq \epsilon |z-w|,\quad z,w\in\mathbb{D}\setminus E, \] then $\rho$ is said to belong to the class $\mathcal{L}_0$. Using this notation, the author then defines the class $\mathcal{W}_0$ consisting of all real-valued functions $\varphi\in C^2(\mathbb{D})$ such that the Laplacian $\Delta \varphi$ is positive, and there is a $\rho\in\mathcal{L}_0$ and a constant $C>0$ such that \[ \frac{1}{C\sqrt{\Delta \varphi(z)}}\le \rho(z)\le \frac{C}{\sqrt{\Delta \varphi(z)}},\quad z\in\mathbb{D}. \] It turns out that the class $\mathcal{W}_0$ of weight functions is bigger than the class of weights already introduced by \textit{A. Borichev} et al. in [J. Funct. Anal. 242, No. 2, 563--606 (2007; Zbl 1115.46019)]. \par The main objective of the paper is to study boundedness of compactness of some integral operators on Bergman spaces with exponential weights $\varphi\in\mathcal{W}_0$. Let $\varphi\in\mathcal{W}_0$, and let $p$ be a positive number. Consider the class $L^p_\varphi$ of functions on $\mathbb{D}$ with \[ \| f\|_{p,\varphi}=\left (\int_{\mathbb{C}}|f(z)|^p e^{-p\varphi(z)}dA(z)\right )^{1/p}<\infty, \] where $dA(z)$ represents the usual area measure in the complex plane. The weighted Bergman space is then defined as $A^p_\varphi:=L^p_\varphi \cap Hol(\mathbb{C})$. Let $K(z,w)$ be the reproducing kernel for the Hilbert space $A^2_\varphi$. The orthogonal projection $P:L^2_\varphi\to A^2_\varphi$ is given by \[ Pf(z)=\int_\mathbb{D} K(z,w)f(w)e^{-2\varphi(w)}dA(w),\quad z\in\mathbb{D}. \] For a positive Borel measure $\mu$ on the unit disk satisfying \[ \int_\math…
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 47B34
zbMATH DE Number: 7045490
zbMATH Keywords: Bergman spaces with exponential weights
zbMATH Keywords: Carleson measures
zbMATH Keywords: boundedness and compactness of integral operators
MaRDI profile type: Publication
OpenAlex ID: W2900939108
Wikidata QID: Q115517608
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description: scientific article; zbMATH DE number 7045490