Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

TL;DR

This paper explores the relationship between pseudo-differential operators and Wick operators using the Bargmann transform, providing new formulas, characterizations, and results on composition, hypoellipticity, and continuity.

Contribution

It introduces a formula linking Wick and pseudo-differential symbols, characterizes Wick symbols of Shubin type, and establishes hypoellipticity transfer and continuity results.

Findings

Derived a formula for Wick symbols via short-time Fourier transform

Provided a series expansion of Wick operators in anti-Wick operators

Established a sharp Gårding inequality and hypoellipticity transition

Abstract

We study the link between pseudo-differential operators and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of pseudo-differential operators of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in anti-Wick operators which leads to a sharp G{\aa}rding inequality and transition of hypoellipticity between Wick and and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.