Local solvability of a class of degenerate second order operators

TL;DR

This paper investigates the local solvability of a class of degenerate second order linear PDEs, extending known results to cases with non-smooth coefficients and analyzing the effects of degeneracy and singularities.

Contribution

It introduces new local solvability results for a generalized class of degenerate second order operators, including cases with non-smooth coefficients, expanding previous understanding.

Findings

Established local solvability for a broad class of degenerate operators

Extended solvability results to operators with non-smooth coefficients

Analyzed the impact of degeneracy and singularities on solvability

Abstract

In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smooth coefficients which are a variation of the main class presented above.