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

We provide a boundedness criterion for the integral operator $S_φ$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_φ$ (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 $φ$ in the Fock space $F^2(\mathbb C^n)$ and $dλ(w): = π^{-n} e^{-|w|^2} dw$ 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 $W_H^{s,2}(\mathbb R^n)$.