Equivalence of weak and viscosity solutions for nonlocal p-Laplace type equations on the Heisenberg group

Abstract

We establish the equivalence between weak and viscosity solutions for a broad class of nonlocal $p$-Laplace type equations on the Heisenberg group whose kernels satisfy standard symmetry, ellipticity, and left-translation invariance assumptions. As a particular case, our results apply to the fractional Heisenberg $p$-Laplacian. The proof combines intrinsic approximation by modified infimal convolutions, Heisenberg mollification, and a nonlocal integration-by-parts argument adapted to the sub-Riemannian geometry. The main analytical difficulty stems from the noncommutative structure of the Heisenberg group, which prevents a direct extension of the Euclidean theory and requires new localization and approximation techniques.