p-Laplace Equation in the Heisenberg Group

Conformality and Q-harmonicity in sub-Riemannian manifolds

“…Under these assumptions one has that solutions in the range p ≥ 2 have Hölder regular horizontal gradient. This is a formidable achievement in itself, building on contributions by several authors [Cap97,Dom04,DFDFM05,MM07,DM09a,MZGZ09,DM09b,Ric15], with the final result being established eventually by Zhong in [Zho10]. Beyond the Heisenberg group one has some promising results due to Domokos and Manfredi [Dom08,DM10b,DM10a] in the range of p near 2.…”

Mean Value Property and Harmonicity on Carnot-Carathéodory Groups

“…For further studies about this equation and the associated linear potentials we refer, for instance, to the book of Ricciotti [21].…”

“…The following result holds in the Carnot group setting similarly to the Euclidean setting, cf. Gilbarg-Trudinger [12, Chapter 7.11], Hörmander [15], Manfredi-Mingione [20], Capogna [5] and Ricciotti [21]. [20]).…”

“…Moreover, if the first assertion holds, then D Z h u converges strongly toZu in L s (Ω ′ ), as h → 0. For another proof of the lemma, we refer to [21,Theorem 2.11], where it is stated and proved for the Heisenberg group. However, one easily observes that the proof holds for any Carnot group as well.…”

Random walks and random tug of war in the Heisenberg group

“…Above, we denoted by C p a constant depending only on p, whereas C is a constant depending only on the quantities |∇ H v (q)| and ∇ 2 v C(B (q)) , that remain uniformly bounded in q ∈D for small . Since v is the sum of the smooth on U function q → |q| s K , and a p-harmonic function u that is also smooth in virtue of its non vanishing horizontal gradient (see [33]), we obtain that:…”