Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces

Abstract

Any bounded linear operator $ T $ on $ L^2(\mathbb{R}^n) $ gives rise to the operator $ S= B \circ T \circ B^\ast $ on the Fock space $ \mathcal{F}(\C^n) $ where $ B $ is the Bargmann transform. In this article we identify those $ S $ which correspond to Fourier multipliers and pseudo-differential operators on $ L^2(\mathbb{R}^n)$ and study their boundedness on the Fock-Sobolev spaces $ \mathcal{F}^{s,2}(\C^n).$