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arXiv · 2508.07801

Characterisations of Sobolev spaces and constant functions over metric spaces

Tuomas Hytönen·Riikka Korte
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Abstract

In a doubling metric measure space $(X,ρ,μ)$ supporting a Poincaré inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of the finiteness of certain integrals through a new approach. As a key tool of independent potential, we introduce a novel ``macroscopic'' Poincaré inequality, whose right-hand side has oscillations of the same form as the left-hand side, but at a smaller macroscopic scale $r\in(0,R)$. Besides intrinsic interest, these results are motivated by applications to quantitative compactness properties of commutators $[f,T]$ of pointwise multipliers and singular integrals. With pivotal use of the present results, a characterisation of commutator mapping properties, over the same class of general domains $(X,ρ,μ)$, is obtained in a companion paper.

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Tuomas Hytönen, Riikka Korte. 2026-02-06. Characterisations of Sobolev spaces and constant functions over metric spaces. https://arxiv.org/abs/2508.07801

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