Besov regularity of solutions to the Dirichlet problem for the Bessel $(p,s)$-Laplacian
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917316213604352 |
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| author | Borthagaray, Juan Pablo Del Pezzo, Leandro M. Niño, José Camilo Rueda |
| author_facet | Borthagaray, Juan Pablo Del Pezzo, Leandro M. Niño, José Camilo Rueda |
| contents | We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis combines the framework of Lions-Calderón spaces, Besov embeddings, and an adaptation of Nirenberg's difference quotient method, originally developed by Savaré, to the fractional Riesz setting. As a main result, we establish global Besov regularity estimates for weak solutions. Concretely, in the superquadratic regime $p \geq 2$, we prove $u \in \dot{B}_{p,\infty}^{s+1/p}(Ω)$ for $s \in [\frac{1}{p'},1)$, and $u \in \dot{B}_{p,\infty}^{s+\frac{s}{p-1}}(Ω)$ for $s \in (0,\frac{1}{p'})$. In the subquadratic case $1<p<2$, we show $u \in \dot{B}_{p,\infty}^{s+1/2}(Ω)$ for $s \in [\frac{1}{2},1)$, and $u \in \dot{B}_{p,\infty}^{2s}(Ω)$ for $s \in (0,\frac12)$, with quantitative bounds depending on the source data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05298 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Besov regularity of solutions to the Dirichlet problem for the Bessel $(p,s)$-Laplacian Borthagaray, Juan Pablo Del Pezzo, Leandro M. Niño, José Camilo Rueda Analysis of PDEs We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis combines the framework of Lions-Calderón spaces, Besov embeddings, and an adaptation of Nirenberg's difference quotient method, originally developed by Savaré, to the fractional Riesz setting. As a main result, we establish global Besov regularity estimates for weak solutions. Concretely, in the superquadratic regime $p \geq 2$, we prove $u \in \dot{B}_{p,\infty}^{s+1/p}(Ω)$ for $s \in [\frac{1}{p'},1)$, and $u \in \dot{B}_{p,\infty}^{s+\frac{s}{p-1}}(Ω)$ for $s \in (0,\frac{1}{p'})$. In the subquadratic case $1<p<2$, we show $u \in \dot{B}_{p,\infty}^{s+1/2}(Ω)$ for $s \in [\frac{1}{2},1)$, and $u \in \dot{B}_{p,\infty}^{2s}(Ω)$ for $s \in (0,\frac12)$, with quantitative bounds depending on the source data. |
| title | Besov regularity of solutions to the Dirichlet problem for the Bessel $(p,s)$-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.05298 |