On the microlocal regularity of the analytic vectors for "sums of squares" of vector fields

TL;DR

This paper establishes the microlocal Gevrey regularity of analytic vectors for sums of squares of vector fields with real analytic coefficients, using FBI-transform techniques, extending previous results to the analytic category.

Contribution

It provides the first microlocal Gevrey regularity result for analytic vectors of sums of squares operators with real analytic coefficients, generalizing Derridj's earlier work.

Findings

Proves optimal microlocal Gevrey regularity of analytic vectors

Uses FBI-transform to analyze regularity in the analytic category

Extends previous Gevrey results to the real analytic setting

Abstract

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of H\"ormander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj in "Local estimates for H\"ormander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients. Nous d\'emontrons , en utilisant la transformation de Fourier-Bros-Iagolnitzer, un r\'esultat de r\'egularit\'e Gevrey microlocale , optimale, des vecteurs analytiques d'op\'erateurs de H\"ormander de type "Sommes de carr\'es de champs de vecteurs" \`a coefficients analytiques sur un ouvert. Ce r\'esultat est, dans le cadre analytique, la version microlocale du r\'esultat de M.Derridj "Local estimates for H\"ormander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", obtenu pour les vecteurs de Gevrey de tels op\'erateurs \`a coefficients Gevrey.