Sobolev and BV functions on $\mathrm{RCD}$ spaces via the short-time behaviour of the heat kernel

TL;DR

This paper characterizes Sobolev and BV functions on finite-dimensional RCD spaces using the short-time behavior of the heat kernel, linking local and nonlocal functionals.

Contribution

It provides a novel characterization of Sobolev and BV spaces on RCD spaces through heat kernel analysis, extending previous understanding.

Findings

Sobolev spaces characterized via heat flow behavior

BV functions characterized through heat kernel limits

Cheeger energies and total variations linked to nonlocal functionals

Abstract

In the setting of finite-dimensional $\mathrm{RCD}(K,N)$ spaces, we characterize the $p$-Sobolev spaces for $p\in(1,\infty)$ and the space of functions of bounded variation in terms of the short-time behaviour of the heat flow. Moreover, we prove that Cheeger $p$-energies and total variations can be computed as limits of nonlocal functionals involving the heat kernel.