Characterizations of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces

TL;DR

This paper provides optimal characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces with doubling properties, including pointwise descriptions and conditions for nontriviality under Poincaré inequalities.

Contribution

It introduces new characterizations of these function spaces on metric measure spaces, expanding understanding of their structure and properties.

Findings

Established optimal characterizations of Besov and Triebel-Lizorkin spaces

Provided a pointwise characterization of these spaces

Discussed conditions for nontriviality under Poincaré inequalities

Abstract

On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel-Lizorkin spaces, including a pointwise characterization. Moreover, we discuss their (non)triviality under a Poincar\'e inequality.