Asian Journal of Mathematics

Volume 27 (2023)

Number 4

BV functions and fractional Laplacians on Dirichlet spaces

Pages: 441 – 466

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n4.a1

Authors

Patricia Alonso Ruiz (Texas A&M University)

Fabrice Baudoin (Aarhus University)

Li Chen (Aarhus University)

Luke Rogers (University of Connecticut)

Nageswari Shanmugalingam (University of Cincinnati)

Alexander Teplyaev (University of Connecticut)

Abstract

We study bounded variation (BV) and fractional Sobolev functional spaces, $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet Metric Measure Spaces that was introduced by the authors in previous works.

Keywords

Besov critical exponents, isoperimetric and Sobolev inequalities, fractional Laplacians, metric measure spaces, heat semigroup, Dirichlet spaces

2010 Mathematics Subject Classification

26A45, 28A80, 31C25, 46E35, 60J45

Full Text (PDF format)

The authors dedicate this paper to the memory of Ka-Sing Lau, whose important contributionsto the general theory of Markov processes and Dirichlet forms on fractals and metricmeasure spaces, especially in the papers [43, 44, 48, 49], influenced the development of thiswork.

Received 31 December 2022

Accepted 25 July 2023

Published 10 July 2024