Sums of squares III: hypoellipticity in the infinitely degenerate regime

TL;DR

This paper extends the theory of sums of squares and hypoellipticity to more general, infinitely degenerate operators, providing new regularity results and broadening the scope of previous theorems.

Contribution

It introduces a C^{2,delta} generalization of Christ's sum of squares theorem and proves a hypoellipticity theorem for nondiagonal and more degenerate operators.

Findings

Established a C^{2,delta} sum of squares theorem

Proved hypoellipticity for a broader class of operators

Extended previous results to include nondiagonal degeneracies

Abstract

This is the third in a series of papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^2,delta generalization of M. Christ's sum of squares theorem, and use a bootstrap argument with the sum of squares theorem for matrix functions in the second paper of this series, in order to prove a hypoellipticity theorem generalizing work in the infinitely degenerate regime to include nondiagonal operators and more general degeneracies.