# The pseudo-differential calculus in a Bargmann setting

## TL;DR

This paper develops a foundational framework for Berezin's analytic pseudo-differential operators within the Bargmann setting, establishing continuity results and linking them to modulation space operators.

## Contribution

It introduces a new approach to analyze Berezin's analytic pseudo-differential operators using Bargmann images of Pilipović spaces, extending continuity results to weighted Lebesgue spaces.

## Key findings

- Established continuity of Berezin's pseudo-differential operators in weighted Lebesgue spaces.
- Connected analytic pseudo-differential operators to real pseudo-differential operators with modulation space symbols.
- Provided a theoretical foundation for further analysis of operators in Bargmann and modulation space frameworks.

## Abstract

We give a fundament for Berezin's analytic $\Psi$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{\'c} spaces. We deduce basic continuity results for such $\Psi$do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply well-known continuity results for real $\Psi$do with symbols in modulation spaces, when acting on other modulation spaces.