A regularity theorem for fully nonlinear maximally subelliptic PDE

TL;DR

This paper establishes a precise interior regularity theorem for fully nonlinear maximally subelliptic PDEs in both isotropic and non-isotropic Sobolev spaces, extending classical results and previous work by Street.

Contribution

It introduces a sharp regularity theorem for these PDEs in non-isotropic Sobolev spaces, complementing existing results and recovering classical theorems in elliptic PDEs.

Findings

Proves a sharp interior regularity theorem in non-isotropic Sobolev spaces.

Recovers classical regularity results for fully nonlinear elliptic PDEs.

Extends regularity theory to maximally subelliptic PDEs in isotropic Sobolev spaces.

Abstract

We prove a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in non-isotropic Sobolev spaces adapted to maximally subelliptic operators. This result complements the regularity theorem proven by Street for adapted Zygmund-H\"older spaces. This also recovers the classical regularity theorem for fully nonlinear elliptic differential operators for classical Sobolev spaces. We also obtain a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in isotropic Sobolev spaces.