$$C^{1,\alpha }$$C1,α -subelliptic regularity on $${{\,\mathrm{SU}\,}}(3)$$SU(3) and compact, semi-simple Lie groups

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 Holder 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.