Second order Sobolev regularity for p-harmonic functions in SU(3)

Abstract

Let u be a weak solution to the degenerate subelliptic p-Laplacian equation $$ \Delta_{\mathcal{H},p}u(x)=\sum_{i=1}^6 X_i(|\nabla_{\mathcal{H}}u |^{p-2}X_iu)=0, $$ where \(\mathcal{H}\) is the orthogonal complement of a Cartan subalgebra in SU(3) and its orthonormal basis is composed of the vector fields \(X_1,\ldots, X_6\). We prove that when \(1<p<7/2\), the solution u has the second order horizontal Sobolev\(W^{2,2}_{\mathcal{H},\rm loc}\)-regularity.