On the Necessity of Logarithmic Estimates for Hypoellipticity

TL;DR

This paper establishes a logarithmic necessary condition for hypoellipticity of certain degenerate operators, clarifying restrictions on their degeneracy and closing longstanding gaps in the theory.

Contribution

It introduces a logarithmic criterion for hypoellipticity of operators with degeneracies, advancing understanding of necessary conditions in higher dimensions.

Findings

Logarithmic criterion for hypoellipticity established

Necessary conditions restrict degeneracy of operator $L_2$

Results close gaps in the theory since the 1980s

Abstract

This paper is focused on necessary conditions for hypoellipticity of an operator $L$ of the form $L=L_1(x)+g(x)L_2(y)$, where the operator $L_1$ is either elliptic or parabolic, $L_2$ is degenerately elliptic and $g(x)$ may itself vanish adding further degeneracy. First, we establish a logarithmic criterion: if the operator $L$ above is hypoelliptic and $L_1$ has a family of spectral solutions we define in the paper, then the remaining part $L_2$ must gain a power of a logarithm of a derivative. Such a property can be thought of as a restriction on degeneracy of the operator $L_2$. We then use this criterion to examine degenerate elliptic and parabolic operators closing gaps between sufficiency and necessity that have been open since 1980s in three and higher dimensions.