Translation Invariant Operators on Polyanalytic Sobolev-Fock Spaces

TL;DR

This paper characterizes translation invariant operators on Polyanalytic Sobolev-Fock spaces, extending previous results and providing conditions for their boundedness using time-frequency analysis and pseudo-differential operator theory.

Contribution

It extends the characterization of translation invariant operators to Polyanalytic Sobolev-Fock spaces and establishes boundedness criteria via symbol class analysis.

Findings

Operators have a specific integral form involving va functions.

Extended Cao et al. (2020) results to new function spaces.

Provided sufficient conditions for operator boundedness.

Abstract

We examine translation invariant operators on the Polyanalytic Sobolev-Fock spaces and show that they take the form \begin{align*} S_{\phi} F(z) = \int_{\mathbb{C}^n} F(w)e^{\pi z\cdot \overline{w}}\phi(w-z,\overline{w}-z) e^{\pi |w|^2}\, dw \end{align*} for certain $\phi$, using tools from time-frequency analysis. This extends the results of Cao et al. (2020) to both the Sobolev-Fock spaces and the Polyanalytic Sobolev-Fock spaces. We use results on symbol classes of pseudo-differential operators to give sufficient conditions for boundedness of $S_{\phi}$ on all polyanalytic Sobolev-Fock spaces.