Fourier Multipliers and Pseudo-differential Operators on Fock-Sobolev Spaces

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