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

Abstract

Let the vector fields $X_1, \ldots , X_{6}$ form an orthonormal basis of $\mathcal {H}$, the orthogonal complement of a Cartan subalgebra (of dimension 2) in ${{\,\mathrm{SU}\,}}(3)$. We prove that weak solutions u to the degenerate subelliptic p-Laplacian

$$\begin{aligned} \Delta _{\mathcal {H},{p}} u(x)=\sum _{i=1}^{6} X_i^{*}\left( |\nabla _{\!{\mathcal {H}}}u|^{p-2}X_{i}u \right) =0, \end{aligned}$$

have Hölder continuous horizontal derivatives $\nabla _{\!{\mathcal {H}}}u=(X_1u, \ldots , X_{6}u)$ for $p\ge 2$. We also prove that a similar result holds for all compact connected semisimple Lie groups.