Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces

TL;DR

This paper establishes a criterion for the boundedness of a class of integral operators on fractional Fock-Sobolev spaces, extending recent results and utilizing multipliers on fractional Hermite-Sobolev spaces.

Contribution

It introduces a boundedness criterion for integral operators on fractional Fock-Sobolev spaces, extending prior work and developing multipliers on fractional Hermite-Sobolev spaces.

Findings

Provides a boundedness criterion for $S_{\varphi}$ on $F^{s,2}(\mathbb{C}^n)$

Extends recent results by Cao et al.

Develops multipliers on fractional Hermite-Sobolev spaces.

Abstract

We provide a boundedness criterion for the integral operator $S_{\varphi}$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_{\varphi}$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_{\varphi}F(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w}) d\lambda(w) \end{eqnarray*} with $\varphi$ in the Fock space $F^2(\mathbb C^n)$ and $d\lambda(w): = \pi^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $\mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.