Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918283174739968 |
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| author | Flores, Guillermo Garrigós, Gustavo Viviani, Beatriz |
| author_facet | Flores, Guillermo Garrigós, Gustavo Viviani, Beatriz |
| contents | We study differentiability conditions on a complex measure $ν$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_tν=e^{-t\sqrt L}ν$, where $L=-Δ+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $ν$ iff a slightly stronger notion than non-tangential convergence holds for $P_tν$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $σ$-point of $ν$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals Flores, Guillermo Garrigós, Gustavo Viviani, Beatriz Analysis of PDEs Classical Analysis and ODEs 42C10, 35C15, 33C45, 40A10, 31B25, 28A15 We study differentiability conditions on a complex measure $ν$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_tν=e^{-t\sqrt L}ν$, where $L=-Δ+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $ν$ iff a slightly stronger notion than non-tangential convergence holds for $P_tν$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $σ$-point of $ν$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition. |
| title | Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals |
| topic | Analysis of PDEs Classical Analysis and ODEs 42C10, 35C15, 33C45, 40A10, 31B25, 28A15 |
| url | https://arxiv.org/abs/2601.07063 |