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We provide a boundedness criterion for the integral operator $S_{\varphi}$ onthe 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 Gaussianmeasure on the complex space $\mathbb{C}^{n}$. This extends the recent resultin Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on thefractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.

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