# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## On Gevrey vectors of L. Hörmander’s operatorsHTML articles powered by AMS MathViewer

by Makhlouf Derridj
Trans. Amer. Math. Soc. 372 (2019), 3845-3865 Request permission

## Abstract:

We study the regularity of Gevrey vectors of L. Hörmander’s operators: \begin{equation*} P=\sum _{j=1}^{m} X_j^2+X_0+c, \end{equation*} where $X_0$, $X_1$, …, $X_m$ are real vector fields in an open set $\Omega \subset \mathbb {R}^n$ and $c$ is a smooth function. More precisely, we prove the following: If the coefficients of $P$ are in the Gevrey class $G^k(\Omega )$, $k\in \mathbb N$, $k\geq 1$, and $P$ satisfies the following estimate with $p/q$ rational, $0<p\leq q$: \begin{eqnarray} ||v ||^2_{p/q}\leq C(|(Pv,v)|+||v ||^2), \; \forall v \in \mathcal D(\Omega _0), \end{eqnarray} for some open subset $\Omega _0\subset \overline {\Omega _0}\subset \Omega$, then $G^k(P, \Omega _0)\subset G^{k\frac {q}{p}}(\Omega _0)$. This provides in particular a local version of a recent result of N. Braun Rodrigues, G Chinni, P. D. Cordaro, and M. R. Jahnke, giving a global such result, with $k\geq 1$ not necessarily integer, for Hörmander’s operators on a torus.
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