Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911525612027904 |
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| author | Hyde, Matthew |
| author_facet | Hyde, Matthew |
| contents | Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when $X = \mathbb{R}^n$; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17590 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces Hyde, Matthew Classical Analysis and ODEs Metric Geometry Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when $X = \mathbb{R}^n$; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space. |
| title | Quantitative harmonic approximations and Dorronsoro's Theorem in metric measure spaces |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2603.17590 |