$C^{1,\alpha }$-subelliptic regularity on SU(3) and compact, semi-simple Lie groups

Abstract

Let the vector fields $X_1,...,X_6$ form an orthonormal basis of $\mathcal{H}$, the orthogonal complement of a Cartan subalgebra (of dimension $2$) in SU(3). We prove that weak solutions $u$ to the degenerate subelliptic $p$-Laplacian $${\Delta }_{\mathcal{H},p}u(x)=\sum _{i=1}^6{X}_i^{\ast }\left(|{\nabla }_{\mathcal{H}}u{|}^{p-2}{X}_{i}u\right)=0,$$ have Hölder continuous horizontal derivatives ${\nabla }_{\mathcal{H}}u=(X_{1}u,\dots ,X_{6}u)$ for $p\ge 2$. We also prove that a similar result holds for all compact connected semisimple Lie groups.

32 pages