Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

On Gevrey vectors of L. Hörmander's operators


Author: Makhlouf Derridj
Journal: Trans. Amer. Math. Soc. 372 (2019), 3845-3865
MSC (2010): Primary 35B45, 35B65
DOI: https://doi.org/10.1090/tran/7387
Published electronically: June 17, 2019
MathSciNet review: 4009421
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the regularity of Gevrey vectors of L. Hörmander's operators:

$\displaystyle P=\sum _{j=1}^{m} X_j^2+X_0+c,$    

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$:
$\displaystyle \vert\vert v \vert\vert^2_{p/q}\leq C(\vert(Pv,v)\vert+\vert\vert v \vert\vert^2), \; \forall v \in \mathcal D(\Omega _0),$     (1)

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35B45, 35B65

Retrieve articles in all journals with MSC (2010): 35B45, 35B65


Additional Information

Makhlouf Derridj
Affiliation: 5, rue de la Juvinière, 78350 Les Loges en Josas, France
Email: maklouf.derridj@numericable.fr

DOI: https://doi.org/10.1090/tran/7387
Keywords: Gevrey vectors, H\"ormander's operators, subelliptic estimates
Received by editor(s): January 22, 2017
Received by editor(s) in revised form: August 17, 2017
Published electronically: June 17, 2019
Article copyright: © Copyright 2019 American Mathematical Society