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

Abstract

We provide a boundedness criterion for the integral operator $S_{\phi }$ on the fractional Fock-Sobolev space $F^{s,2}({\mathbb{C}}^n)$, $s\ge 0$, where $S_{\phi }$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_φF(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w}) dλ(w) \end{eqnarray*} with $\phi$ 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 $W_H^{s,2}({\mathbb{R}}^n)$.