Homogeneous algebras via heat kernel estimates

TL;DR

This paper investigates the structure of homogeneous function spaces on metric measure spaces with heat kernel estimates, establishing algebra properties in smooth manifold and Lie group contexts.

Contribution

It introduces new algebraic properties of homogeneous Besov and Triebel--Lizorkin spaces under heat kernel conditions, extending to Lie groups and Grushin spaces.

Findings

Spaces form algebras under pointwise multiplication on smooth manifolds.

Results apply to nilpotent Lie groups and Grushin-type spaces.

Provides heat kernel-based criteria for algebraic structure.

Abstract

We study homogeneous Besov and Triebel--Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with $L^\infty$ is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.