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

TL;DR

This paper characterizes Fourier multipliers and pseudo-differential operators on $L^2( ^n)$ as bounded operators on Fock spaces via the Bargmann transform, and studies their boundedness on Fock-Sobolev spaces.

Contribution

It identifies operators on Fock spaces that correspond to classical Fourier multipliers and pseudo-differential operators, extending their analysis to Fock-Sobolev spaces.

Findings

Characterization of operators on Fock space corresponding to Fourier multipliers.

Identification of pseudo-differential operators in the Fock space setting.

Boundedness results of these 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).$