Regularity for $p$-Harmonic Functions in the Grušin Plane

Abstract

Let $X=\{X_1,X_2\}$ be the orthogonal complement of a Cartan subalgebra in the Grušin plane, whose orthonormal basis is formed by the vector fields $X_1$ and $X_2$. When $1have the $C^{0,1}_{loc}$, $C^{1,\alpha}_{loc}$ and $W^{2,2}_{X,loc}$-regularities.