Characterisations of Sobolev spaces and constant functions over metric spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908816993419264 |
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| author | Hytönen, Tuomas Korte, Riikka |
| author_facet | Hytönen, Tuomas Korte, Riikka |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterisations of Sobolev spaces and constant functions over metric spaces Hytönen, Tuomas Korte, Riikka Functional Analysis 46E36, 42B35 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. |
| title | Characterisations of Sobolev spaces and constant functions over metric spaces |
| topic | Functional Analysis 46E36, 42B35 |
| url | https://arxiv.org/abs/2508.07801 |