On the Local solvability of a class of degenerate second order operators with complex coefficients

TL;DR

This paper investigates the local solvability of certain degenerate second order operators with complex coefficients, extending previous work to include new cases, and provides solvability results with various regularity levels.

Contribution

It extends existing results on local solvability to include operators with complex coefficients and offers new theorems demonstrating different regularity conditions.

Findings

Establishes local solvability for a broader class of operators.

Demonstrates cases with improved Sobolev regularity.

Provides solvability theorems for degenerate parabolic and Schrödinger type operators.

Abstract

We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex coefficients that were not present in [9]. The class of operators considered here ideally encompasses classes of degenerate parabolic and Schr\"odinger type operators. We will give local solvability theorems. In general, one has $L^2$ local solvability, but also cases of local solvability with better Sobolev regularity will be presented.