A universal heat semigroup characterisation of Sobolev and BV spaces in Carnot groups

TL;DR

This paper introduces a new way to characterize Sobolev and BV spaces in Carnot groups using heat semigroup properties, overcoming the lack of explicit heat kernel formulas in sub-Riemannian geometry.

Contribution

It provides a novel heat semigroup characterization of Sobolev and BV spaces in Carnot groups based on an integral decoupling property of the heat kernel.

Findings

Established a heat semigroup characterization in Carnot groups.

Overcame challenges due to non-explicit, non-symmetric heat kernels.

Provided tools for analysis in sub-Riemannian geometry.

Abstract

In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and $BV$ spaces in a Carnot group by means of an integral decoupling property of the heat kernel.