What is it about?
This article studies when Sobolev-type inequalities hold for functions defined on very general metric spaces equipped with a doubling measure. The key point is to characterize these embeddings in terms of a simple geometric condition on the measure, the so-called non-collapsing condition, which requires that all balls of radius one have a uniformly positive measure. The paper also develops a unified framework for Hajłasz–Besov spaces built on rearrangement-invariant and quasi-Banach function spaces, and derives new embedding results in this abstract setting.
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Why is it important?
Sobolev and Besov embeddings are a cornerstone in the analysis of partial differential equations, but many classical results rely heavily on Euclidean structure and regularity assumptions that fail in irregular or fractal-type spaces. By linking Sobolev embeddings for Hajłasz–Besov spaces to a transparent non-collapsing condition, this work provides sharp and verifiable criteria that apply to a wide class of doubling metric measure spaces, without assuming Poincaré inequalities or strong regularity. The general formulation in terms of rearrangement-invariant and quasi-Banach spaces also unifies and extends several known embedding theorems, and yields new results even in the Euclidean case, which can be useful for future developments in geometric analysis and PDEs on non-smooth spaces.
Perspectives
I hope this article helps make what many would see as a highly technical and abstract corner of functional analysis feel more intuitive, relevant, and even a bit exciting. At its heart, the work is about a very simple question with wide-reaching consequences: when does the underlying measure of a space give us enough “room” for Sobolev and Besov embeddings to hold, even if the space itself is very rough or irregular. More than anything, if nothing else, the wish is that readers find the paper thought-provoking: that it sparks new questions about which measure-theoretic conditions are truly essential, how far these ideas can be pushed in probabilistic or fractal contexts, and how abstract function space machinery can translate into concrete advances in analysis on rough spaces.
Walter Andrés Ortíz Vargas
Universidad Internacional de La Rioja
Read the Original
This page is a summary of: Non-collapsing condition and Sobolev embeddings for Hajłasz–Besov spaces, Positivity, March 2025, Springer Science + Business Media,
DOI: 10.1007/s11117-025-01115-1.
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