Regularity of quasi-linear equations with H\"ormander vector fields of step two

TL;DR

This paper proves that weak solutions to certain quasi-linear equations involving Hörmander vector fields of step two are locally $C^{1,\alpha}$, extending regularity theory to these subelliptic PDEs with $p$-Laplacian growth.

Contribution

It establishes the full interior regularity for quasi-linear equations with Hörmander vector fields of step two, a significant extension of regularity results to subelliptic PDEs.

Findings

Weak solutions are locally $C^{1,\alpha}$.

Regularity holds under $p$-Laplacian growth conditions.

Results apply to equations with vector fields spanning tangent space.

Abstract

If the smooth vector fields $X_1,\ldots,X_m$ and their commutators span the tangent space at every point in $\Omega\subseteq \mathbb{R}^N$ for any fixed $m\leq N$, then we establish the full interior regularity theory of quasi-linear equations $\sum_{i=1}^m X_i^*A_i(X_1u, \ldots,X_mu)= 0$ with $p$-Laplacian type growth condition. In other words, we show that a weak solution of the equation is locally $C^{1,\alpha}$.