Hypoellipticity for infinitely degenerate elliptic operators
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.
scientific article; zbMATH DE number 4075638