# Bilinear pseudo-differential operators with Gevrey-H\"ormander symbols

## TL;DR

This paper studies bilinear pseudo-differential operators with Gevrey-Hörmander symbols, establishing their invariance and continuity properties on modulation and Gelfand-Shilov spaces, using short-time Fourier transform techniques.

## Contribution

It introduces a new symbol class description via short-time Fourier transform and proves invariance and continuity results for these bilinear operators.

## Key findings

- Symbols characterized by sub-exponential growth and Gevrey regularity.
- Operators are invariant under certain transformations.
- Continuity established on modulation and Gelfand-Shilov spaces.

## Abstract

We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurling and Roumieu type symbol classes and Gelfand-Shilov spaces.