Hypoellipticity for infinitely degenerate elliptic operators
type: Item
instance of: scholarly article
title: Hypoellipticity for infinitely degenerate elliptic operators
zbMATH Open document ID: 0658.35039
author: Yoshinori Morimoto
published in: Osaka Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: In Section 1 we state our main theorem, which is formulated for a differential operator \(P=a(x,y,D_ x)+g(x')b(x,y,D_ y)\) in \(R^ n_{x,y}\), \(x=(x',x'')\). In Section 2 we give a new criterion of the hypoellipticity, which is composed of five apriori estimates. In Section 3 we show that P satisfies each estimate. The discussion of Section 4 is similar to the one of Section 5 of a previous work by the author [J. Math. Soc. Japan 30, 327-358 (1978; Zbl 0369.35016)] and is employed to estimate the commutator between P and the cut off function of y variables. In Section 5 we estimate the commutator between P and the cut off function of x'' variables. As studied by the author (loc. cit.), the method of this paper seems to be extendible to infinitely degenerate elliptic operators of higher order, which will be investigated in the future. Finally we remark that the method of the present paper does not apply to all results by \textit{S. Kusuoka} and \textit{D. Stroock} [J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 1-76 (1985; Zbl 0568.60059)], for example, the hypoellipticity of an operator \(x^ 2_ 2\partial^ 2_{x_ 1}+\partial^ 2_{x_ 2}+\sigma (x_ 1)^ 2(\partial^ 2_{y_ 1}+y^ 2_ 1\partial^ 2_{y_ 2}).\) In the future we will show the hypoellipticity for this operator by extending our method.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 4075638
zbMATH Keywords: hypoellipticity
zbMATH Keywords: apriori estimates
zbMATH Keywords: infinitely degenerate
zbMATH Keywords: higher order
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4075638