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

We provide a boundedness criterion for the integral operator Sϕ on the fractional Fock–Sobolev space F^s,2(Cⁿ) , s≥ 0 , where Sϕ (introduced by Zhu [18]) is given by SϕF(z):=∫_CnF(w)e^z·w¯ϕ(z-w¯)dλ(w) with ϕ in the Fock space F²(Cⁿ) and dλ(w):=π^-ne^-|w|2dw the Gaussian measure on the complex space Cⁿ. This extends the recent result in Cao et al. (Adv Math 363: 107001, 33 pp, 2020). The main approach is to develop multipliers on the fractional Hermite–Sobolev space W_H^s,2(Rⁿ).