A universal heat semigroup characterization of Sobolev and BV spaces in Carnot groups
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. In particular, they are not a function of the control distance, nor they are, in general, spherically symmetric in any of the layers of the Lie algebra. Despite this, the heat kernels possess two unexpected “one-dimensional” symmetries. In this talk I present some notable consequences of these properties in a Carnot group of arbitrary step. I also discuss a dimensionless heat based version of the famous Bourgain-Brezis-Mironescu phenomenon. This is joint work with Giulio Tralli.