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We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

This is a note based on the paper [32] written in collaboration with M. Ritoré. The purpose of this note is to present and discuss the Bernstein type problems in the sub-Finsler Heisenberg group H1. We give a general idea of the state of the art and we prove that a complete, stable, (X,Y)-Lipschitz surface is a vertical plane.

We present some recent results obtained on the isoperimetric problem in a class of Carnot-Carathéodory spaces, related to the Heisenberg group. This is the framework of Pansu’s conjecture about the shape of isoperimetric sets. Two different approaches are considered. On one hand we describe the isoperimetric problem in Grushin spaces, under a symmetry assumption that depends on the dimension and we provide a classification of isoperimetric sets for special dimensions. On the other hand, we present some results about the isoperimetric problem in a family of Riemannian manifolds approximating the Heisenberg group. In this context we study constant mean curvature surfaces. Inspired by Abresch and Rosenberg techniques on holomorphic quadratic differentials, we classify isoperimetric sets under a topological assumption.

We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N.

The classical Steiner formula expresses the volume of the ∈-neighborhood Ω∈ of a bounded and regular domain Ω⊂Rn as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltàn Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildrick

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre [8], as established in [14]. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space ℍn × ℝ+.