Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces

TL;DR

This paper develops a comprehensive theory of Besov and Triebel-Lizorkin spaces within Dirichlet spaces, utilizing heat kernel methods to establish characterizations and extend to various geometric contexts.

Contribution

It introduces a unified framework for Besov and Triebel-Lizorkin spaces in Dirichlet spaces using heat kernel techniques and frame theory, applicable to Lie groups and Riemannian manifolds.

Findings

Established heat kernel characterizations of function spaces

Developed frames with sub-exponential localization

Extended the theory to geometric settings like Lie groups

Abstract

Classical and non classical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincar\'e inequality. This leads to Heat kernel with small time Gaussian bounds and H\"older continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows to develop Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifold, and other settings.