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

TL;DR

This paper demonstrates that for certain degenerately elliptic operators, hypoellipticity necessitates more than a logarithmic derivative gain, extending previous results to more general operators with non-analytic coefficients.

Contribution

It generalizes the necessity of super-logarithmic derivative gain for hypoellipticity to a broader class of operators with non-smooth coefficients, without requiring explicit analytic constructions.

Findings

Super-logarithmic derivative gain is necessary for hypoellipticity.

Results apply to operators with non-analytic, possibly vanishing coefficients.

Analysis uses spectral projections and interpolation methods.

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 $L_1(x)+g(x)L_2(y)$, where $L_1(x)$ 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 results 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.