Hankel Bilinear forms on generalized Fock-Sobolev spaces on ${\mathbb C}^n$

TL;DR

This paper characterizes the boundedness, compactness, and Schatten class membership of Hankel bilinear forms and small Hankel operators on generalized Fock-Sobolev spaces on complex n-dimensional space, using kernel decompositions and Littlewood-Paley formulas.

Contribution

It provides a comprehensive characterization of Hankel bilinear forms and small Hankel operators on generalized Fock-Sobolev spaces with weighted norms, including kernel estimates and decomposition techniques.

Findings

Boundedness criteria for Hankel bilinear forms established

Compactness conditions for small Hankel operators derived

Membership in Schatten classes characterized

Abstract

We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock-Sobolev spaces on ${\mathbb C}^n$ with respect to the weight $(1+|z|)^\rho e^{-\frac{\alpha}2|z|^{2\ell}}$, for $\ell\ge 1$, $\alpha>0$ and $\rho\in{\mathbb R}$. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood-Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.