Carleson type measures for Fock--Sobolev spaces

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.