$$C^{1,\alpha }$$ -regularity for p-harmonic functions on SU(3) and semi-simple Lie groups

Abstract

In this paper, when \(1<p<2\), we establish the \(C^{1,\alpha }_{\,\textrm{loc}\,}\)-regularity of weak solutions to the degenerate subelliptic p-Laplacian equation

$$\begin{aligned} \triangle _{{{\mathcal {H}}},p}u(x)=\sum \limits _{i=1}^6X^*_i(|{\nabla _{{{\mathcal {H}}}}u}|^{p-2}X_iu)=0 \end{aligned}$$

on SU(3) endowed with the horizontal vector fields \(X_1,\dots ,X_6\). The result can be extended to a class of compact connected semi-simple Lie group.