Removable sets for Lipschitz harmonic functions on Carnot groups

Abstract

Let $$\mathbb {G}$$ be a Carnot group with homogeneous dimension $$Q \ge 3$$ and let $${\mathcal L}$$ be a sub-Laplacian on $$\mathbb {G}$$ . We prove that the critical dimension for removable sets of Lipschitz $${\mathcal L}$$ -harmonic functions is $$(Q-1)$$ . Moreover we construct self-similar sets with positive and finite $$\mathcal {H}^{Q-1}$$ measure which are removable.