On the microlocal regularity of the Gevrey vectors for second order partial differential operators with non negative characteristic form of first kind

Abstract

Abstract We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations $$\begin{aligned} P(x,D) = \sum _{\ell ,j=1}^{n} a_{\ell ,j}(x) D_{\ell } D_{j} + \sum _{\ell =1}^{n} i b_{\ell }(x) D_{\ell } +c(x), \end{aligned}$$ where $$a_{\ell ,j}(x) = a_{j,\ell }(x)$$ , $$b_{\ell }(x)$$ , $$\ell ,j \in \lbrace 1,\dots ,\, n\rbrace $$ , 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 Derridj in (Complex Anal. Synerg. 6:10, 2020). The class of operators considered in this work generalizes the classes studied by Braun Rodrigues-Chinni-Cordaro-Jahnke (Proc. Am. Math. Soc. 144:5159–5170, 2016) and Chinni-Derridj (Math. Z. 302:1983–2003, 2022).