Harnack inequality for fractional sub-Laplacians in Carnot groups

TL;DR

This paper establishes an invariant Harnack inequality for fractional sub-Laplacians in Carnot groups, extending analysis techniques and explicitly deriving the Poisson kernel for certain degenerate subelliptic equations.

Contribution

It introduces a new invariant Harnack inequality for fractional sub-Laplacians in Carnot groups and provides explicit formulas for the Poisson kernel in specific degenerate cases.

Findings

Proved an invariant Harnack inequality on Carnot-Carathéodory balls.

Derived explicit Poisson kernel formulas for degenerate subelliptic equations.

Extended the technique of Caffarelli and Silvestre to the subelliptic setting.

Abstract

In this paper we prove an invariant Harnack inequality on Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups.