Semiclassical Gevrey operators in the complex domain

TL;DR

This paper investigates semiclassical Gevrey pseudodifferential operators acting on weighted spaces of entire functions, extending symbols almost holomorphically and establishing bounded realizations with optimal remainders for Gevrey indices up to 2.

Contribution

It introduces a framework for analyzing Gevrey pseudodifferential operators in the complex domain using almost holomorphic extensions, providing new boundedness results.

Findings

Operators are bounded on weighted spaces for Gevrey index ≤ 2

Optimal small remainders are achieved in the operator realizations

Extension of symbols to complex neighborhoods is successfully implemented

Abstract

We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is $\leq 2$.