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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.05425 |
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| _version_ | 1866916726979952640 |
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| author | Fraccaroli, Marco Kosz, Dariusz Roncal, Luz |
| author_facet | Fraccaroli, Marco Kosz, Dariusz Roncal, Luz |
| contents | We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^ω$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^ω)$ if and only if $p \geq p_0$, thus reproving classical theorems of Hayes in $\mathbb{R}$. The main novelty is that our $\mathcal{B}$ is a Busemann--Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let $\mathcal{B}$ be a basis in a metric measure space $\mathcal{X}$. If $\mathcal{X}$ is complete, then the set $\{ p \in [1,\infty] : \mathcal{B} \text{ differentiates } L^p(\mathcal{X}) \}$ takes one of the six forms\[ \emptyset, \, \{\infty\}, \, [p_0,\infty], \, (p_0,\infty], \, [p_0,\infty), \, (p_0,\infty) \quad \text{for some} \quad p_0 \in [1,\infty). \] Conversely, for every $p_0 \in [1,\infty)$ and each of the six cases above, we construct a complete space $\mathcal{X}$ and a basis $\mathcal{B}$ illustrating the corresponding range of differentiation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On differentiation of integrals in Lebesgue spaces Fraccaroli, Marco Kosz, Dariusz Roncal, Luz Classical Analysis and ODEs Primary 43A75, 42B25 We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^ω$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^ω)$ if and only if $p \geq p_0$, thus reproving classical theorems of Hayes in $\mathbb{R}$. The main novelty is that our $\mathcal{B}$ is a Busemann--Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let $\mathcal{B}$ be a basis in a metric measure space $\mathcal{X}$. If $\mathcal{X}$ is complete, then the set $\{ p \in [1,\infty] : \mathcal{B} \text{ differentiates } L^p(\mathcal{X}) \}$ takes one of the six forms\[ \emptyset, \, \{\infty\}, \, [p_0,\infty], \, (p_0,\infty], \, [p_0,\infty), \, (p_0,\infty) \quad \text{for some} \quad p_0 \in [1,\infty). \] Conversely, for every $p_0 \in [1,\infty)$ and each of the six cases above, we construct a complete space $\mathcal{X}$ and a basis $\mathcal{B}$ illustrating the corresponding range of differentiation. |
| title | On differentiation of integrals in Lebesgue spaces |
| topic | Classical Analysis and ODEs Primary 43A75, 42B25 |
| url | https://arxiv.org/abs/2505.05425 |