On the Microlocal Regularity of the Gevrey Vectors for second order partial differential operators with non negative characteristic form of first kind

Abstract

We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i b_{\ell}(x) D_{\ell} +c(x), \end{align*} where $a_{\ell ,j}(x)=a_{j,\ell }(x)$, $b_{\ell }(x)$, $\ell ,j\in \{1,\dots ,n\}$, are real valued real Gevrey functions of order $s$ and $c(x)$ is a Gevrey function of order $s$, $s\ge 1$, on $\Omega$ open neighborhood of the origin in ${\mathbb{R}}^n$. Thus providing a microlocal version of a result due to M. Derridj in "Gevrey regularity of Gevrey vectors of second order partial differential operators with non negative characteristic form", Complex Anal. Synerg. $6$, 10 (2020), https://doi.org/10.1007/s40627-020-00047-8.