  We prove regularity results for solutions of the equation \[div(< AXu,X
u>^{(p-2)/2} AX u) = 0,\] $1<p<\infty$, where $X=(X_1,...,X_m)$ is a family of
vector fields satisfying H\"ormander's ellipticity condition, $A$ is an
$m\times m$ symmetric matrix that satisfies degenerate ellipticity conditions.
If the degeneracy is of the form \[\lambda w(x)^{2/p}|\xi|^2\leq <
A(x)\xi,\xi>\leq \Lambda w(x)^{2/p}|\xi|^2,\] $w \in A_p$, then we show that
solutions are locally H\"older continuous. If the degeneracy is of the form \[
k(x)^{-2/p'}|\xi|^2\leq < A(x)\xi,\xi>\leq k(x)^{2/p}|\xi|^2, \] $k\in
A_{p'}\cap RH_\tau$,where $\tau$ depends on the homogeneous dimension, then the
solutions are continuous almost everywhere, and we give examples to show that
this is the best result possible. We give an application to maps of finite
distortion.

Analysis of PDEs;Classical Analysis and ODEs