Carleson Type Measures for Fock–Sobolev Spaces

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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 $q\ge p$ , 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.

Communicated by Harald Woracek.