On hypoelliptic operators with multiple characteristics of odd order
type: Item
instance of: scholarly article
title: On hypoelliptic operators with multiple characteristics of odd order
zbMATH Open document ID: 0544.35024
author: Yoshinori Morimoto
published in: Osaka Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Denote by \(\bar S^ m\) the set of functions p(x,\(\xi)\in {\mathcal B}^{\infty}({\mathbb{R}}^ n_ x\times {\mathbb{R}}^ n_{\xi})\) such that p(x,\(\xi)\) is positively homogeneous of degree m with respect to \(\xi\) if \(| \xi | \geq 1/2\). The author studies the hypoellipticity of pseudodifferential operators \(L(x,D)=P(x,D)+A(x,D)\) where P and A have principal symbols \(p_ 1^{\ell}\) (with \(p_ 1\in \bar S^ 1)\) and \(a\in \bar S^{\ell -1}\) respectively. The author shows that if certain conditions (too detailed to state here) hold, L satisfies suitable a priori estimates which imply that it is hypoelliptic. These estimates are obtained by adapting the techniques developed in the proof of the Egorov- Hörmander subellipticity theorem. A counterexample is given which shows that if one of the conditions does not hold, L may fail to be hypoelliptic.
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 3865798
zbMATH Keywords: multiple characteristics of odd order
zbMATH Keywords: hypoellipticity
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: a priori estimates
zbMATH Keywords: Egorov-Hörmander subellipticity theorem
reviewed by: Paul Godin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3865798