Carleson type measures for Fock--Sobolev spaces

TL;DR

This paper characterizes Carleson measures for weighted Fock--Sobolev spaces using Berezin transforms and averaging functions, and applies these results to analyze boundedness and compactness of weighted composition operators.

Contribution

It provides a comprehensive characterization of Carleson measures and operator boundedness in Fock--Sobolev spaces, extending previous results with new integral transform techniques.

Findings

Characterization of $(p,q)$ Fock--Carleson measures via Berezin transforms.

Criteria for boundedness and compactness of weighted composition operators.

Estimates for operator norms and essential norms in terms of integral transforms.

Abstract

We describe the $(p,q)$ Fock--Carleson measures for weighted Fock--Sobolev spaces in terms of the objects $(s,t)$-Berezin transforms, averaging functions, and averaging sequences on the complex space $\mathbb{C}^n$. The main results show that while these objects may have growth not faster than polynomials to induce the $(p,q)$ measures for $q\geq p$, they should be of $L^{p/(p-q)}$ integrable against a weight of polynomial growth for $q<p$. As an application, we characterize the bounded and compact weighted composition operators on the Fock--Sobolev spaces in terms of certain Berezin type integral transforms on $\mathbb{C}^n$. We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.