Embedding theorems and integration operators on Bergman spaces with exponential weights
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 \[ \int_\mathbb{D} |K(z,w)|^2e^{-2\varphi(w)}d\mu(w)<\infty,\quad z\in\mathbb{D}, \] the author considers the following integral operators: \[ I_\mu (f)(z)=\int_\mathbb{D} f(w)K(z,w)e^{-2\varphi(w)}d\mu(w),\quad z\in\mathbb{D}, \] \[ J_\mu (f)(z)=\int_\mathbb{D} |f(w)K(z,w)|e^{-2\varphi(w)}d\mu(w),\quad z\in\mathbb{D}. \] The main result of the paper under review says that for $p,q\in (1, \infty)$, the following statements are equivalent: \begin{itemize} \item $I_\mu:A^p_\varphi\to A^q_\varphi$ is bounded (resp. compact), \item $J_\mu:A^p_\varphi\to L^q_\varphi$ is bounded (resp. compact), \item $\mu$ is an $s$-Carleson measure (resp. vanishing $s$-Carleson measure), where $s$ satisfies $\frac{1}{s}=1-\frac{1}{q}+\frac{1}{p}$. \end{itemize}
boundedness and compactness of integral operators
scientific article; zbMATH DE number 7045490
scientific article