Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

Abstract

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form L1(x)+g(x)L2(y), where L1(x) is one-dimensional and g(x) may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation, and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our methods do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.