What is it about?
This article studies how certain function spaces (Besov spaces) can be “embedded” into other spaces when the underlying space is a general metric space with a doubling measure, not just the usual Euclidean space. The authors develop a Sobolev-type embedding theorem in this very general setting, introduce suitable notions of smoothness, and obtain several consequences such as uncertainty inequalities, embeddings into BMO, and criteria for essential continuity of functions.
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Why is it important?
The results extend classical Sobolev embedding theory beyond Euclidean and Q-regular spaces to a wide class of metric spaces with doubling measures, which appear naturally in analysis, geometry and applications. This provides new tools to understand regularity, continuity and integrability properties of functions in highly irregular environments, with potential impact on areas such as PDEs, geometric analysis and analysis on fractals or Lie groups
Perspectives
From my perspective as a coauthor, this work shows that many of the fine properties of Sobolev and Besov spaces remain valid even when the underlying space is very general and only satisfies doubling and k,m-type structural conditions, far from the classical Euclidean case. I am particularly interested in the fact that these results open the door to studying regularity, essential continuity, and uncertainty-type inequalities in geometrically complex contexts, such as Carnot–Carathéodory spaces or weighted Euclidean spaces, providing new tools for the analysis of partial differential equations and geometric analysis in non-standard media.
Walter Andrés Ortíz Vargas
Universidad Internacional de La Rioja
Read the Original
This page is a summary of: A Sobolev type embedding theorem for Besov spaces defined on doubling metric spaces, Journal of Mathematical Analysis and Applications, November 2019, Elsevier,
DOI: 10.1016/j.jmaa.2019.07.032.
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