*Submitted Paper*

**Inserted:** 8 oct 2021

**Last Updated:** 5 may 2022

**Pages:** 44

**Year:** 2021

**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 $C^{1,\alpha}$.

**Keywords:**
Sub-elliptic equations, quasilinear, p-Laplacian, regularity.

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