Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914391017914368 |
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| author | Hashash, Paz Poliakovsky, Arkady |
| author_facet | Hashash, Paz Poliakovsky, Arkady |
| contents | Let $X$ be a metric space and $μ$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u \in B^r_{q,\infty}(X,μ;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a $σ$-finite set with respect to the Hausdorff measure $\mathcal{H}^{s - rq}$. The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions $u$ in the fractional Sobolev space $W^{r,q}(X,μ;Y)$, almost every point with respect to $\mathcal{H}^{s - rq}$ is an {\it average Lebesgue point} of $u$. Finally, if $Y$ is also complete, we prove that for $u \in B^r_{q,\infty}(X,μ;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s - rq$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12954 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces Hashash, Paz Poliakovsky, Arkady Functional Analysis Analysis of PDEs Let $X$ be a metric space and $μ$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u \in B^r_{q,\infty}(X,μ;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a $σ$-finite set with respect to the Hausdorff measure $\mathcal{H}^{s - rq}$. The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions $u$ in the fractional Sobolev space $W^{r,q}(X,μ;Y)$, almost every point with respect to $\mathcal{H}^{s - rq}$ is an {\it average Lebesgue point} of $u$. Finally, if $Y$ is also complete, we prove that for $u \in B^r_{q,\infty}(X,μ;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s - rq$. |
| title | Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces |
| topic | Functional Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2603.12954 |