A new class of Carleson measures and integral operators on Bergman spaces

TL;DR

This paper introduces a new class of Carleson measures and studies the boundedness, compactness, and Hilbert-Schmidt properties of generalized integral and weighted composition operators on Bergman spaces, with applications to differential equations.

Contribution

It develops a novel class of Sobolev-Carleson measures and characterizes the boundedness and compactness of generalized Volterra and weighted composition operators on Bergman spaces.

Findings

Characterization of boundedness and compactness of $I_{ extbf{g}}^{(n)}$ operators.

Introduction of Sobolev-Carleson measures for Bergman spaces.

Conditions for solutions of differential equations in Bergman spaces.

Abstract

Let $n$ be a positive integer and $\mathbf{g}=(g_0,g_1,\cdots,g_{n-1})$, with $g_k\in H(\mathbb{D})$ for $k=0,1,\cdots,n-1$. Let $I_{\mathbf{g}}^{(n)}$ be the generalized Volterra-type operators on $H(\mathbb{C})$, which is represented as $$ I_{\mathbf{g}}^{(n)}f=I^n\left(fg_0+f'g_1+\cdots+f^{(n-1)}g_{n-1}\right), $$ where $I$ denotes the integration operator $$(If)(z)=\int_0^zf(w)dw,$$ and $I^n$ is the $n$th iteration of $I$. This operator is a generalization of the operator that was introduced by Chalmoukis in \cite{Cn}. In this paper, we study the boundedness and compactness of the operator $I_{\mathbf{g}}^{(n)}$ acting on Bergman spaces to another. As a consequence of these characterizations, we obtain conditions for certain linear differential equations to have solutions in Bergman spaces. Moreover, we study the boundedness, compactness and Hilbert-Schmidtness of the following sums of generalized weighted composition operators: Let $\mathbf{u}=(u_0,u_1,\cdots,u_n)$ with $u_k\in H(\mathbb{D})$ for $0\leq k\leq n$ and $\varphi$ be an analytic self-map of $\mathbb{D}.$ The sums of generalized weighted composition operators is defined by $$L_{\mathbf{u},\varphi}^{(n)}=\sum_{k=0}^nW_{u_k,\varphi}^{(k)},$$ where $$W_{u_k,\varphi}^{(k)}f=u_k\cdot f^{(k)}\circ\varphi.$$ Our approach involves the study of new class of Sobolev-Carleson measures for classical Bergman spaces on unit disk which appears in the first main Theorems \ref{Theorem1.1} and \ref{Theorem1.2}.