Global hypoellipticity property of a differential operator
type: Item
instance of: scholarly article
title: Global hypoellipticity property of a differential operator
zbMATH Open document ID: 0729.35033
DOI: 10.1007/BF01137558
author: Nguyen Minh Thi
published in: Mathematical Notes
publication date: 1991-01-01T00:00:00Z
review text: A differential operator P in a domain \(Q\subset {\mathbb{R}}^ n\) is called globally hypoelliptic if for any distribution \(u\in D'(Q)\) with \(Pu\in C^{\infty}(Q)\) we have \(u\in C^{\infty}(Q)\). It is known that there are non-hypoelliptic operators with the property of global hypoellipticity (Fedij, 1972). In this paper the following theorem is proved. Theorem. Let \(X_ j(j=1,...,m)\) be real vector fields in a neighbourhood of the closure of Q. If the differential operator \(P=\sum^{n}_{j=1}X_ j^ m\) is elliptic in some neighbourhood of the boundary of Q and the Lie algebra, generated by the fields \(X_ 1,...,X_ m\), coincides with the whole space \({\mathbb{R}}^ n\) (that is, it has rank n), then the operator P is globally hypoelliptic.
Mathematics Subject Classification ID: 35H10
zbMATH DE Number: 4204907
zbMATH Keywords: hypoelliptic operator
reviewed by: N. N. Tarkhanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4204907