Some remarks on analytic pseudodifferential operators

TL;DR

This paper discusses recent advances in the theory of analytic pseudodifferential operators, utilizing the Bargmann transform to connect classical and analytic calculus, with results expressed via Bargmann images of Pilipović spaces.

Contribution

It introduces new results on analytic pseudodifferential operators using the Bargmann transform and Pilipović spaces, bridging real and complex analysis.

Findings

Bargmann transform links classical and analytic pseudodifferential calculus.

Results are formulated in terms of Bargmann images of Pilipović spaces.

Hermite functions' Bargmann transform relates to formal power series in the complex domain.

Abstract

We report some recent results on analytic pseudodifferential operators, also known as Wick operators. An important tool in our study is the Bargmann transform which provides a coupling between the classical (real) and analytic pseudodifferential calculus. Since the Bargmann transform of Hermite functions gives rise to formal power series in the complex domain, the results are formulated in terms of the Bargmann images of Pilipovi\'c spaces.