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

We provide a boundedness criterion for the integral operator $$S_{\varphi }$$ on fractional Fock–Sobolev space $$F^{s,2}({{\mathbb {C}}}^n)$$ , $$s\ge 0$$ where (introduced by Zhu [18]) is given $$\begin{aligned} S_{\varphi }F(z):= \int _{{\mathbb {C}}^n} F(w) e^{z \cdot \bar{w}} \varphi (z- \bar{w}) d\lambda (w) \end{aligned}$$ with $$\varphi $$ in Fock $$F^2({{\mathbb {C}}^n})$$ and $$d\lambda (w): = \pi ^{-n} e^{-|w|^2} dw$$ Gaussian measure complex $${\mathbb {C}}^{n}$$ . This extends recent result Cao et al. (Adv Math 363: 107001, 33 pp, 2020). The main approach to develop multipliers Hermite–Sobolev $$W_H^{s,2}({{\mathbb {R}}}^n)$$