Besov regularity of solutions to the Dirichlet problem for the Bessel $(p,s)$-Laplacian

Abstract

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-Calder\'on spaces, Besov embeddings, and an adaptation of Nirenberg's difference quotient method, originally developed by Savar\'e, 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 $u \in \dot{B}_{p,\infty}^{s+1/p}(\Omega)$ for $s \in [\frac{1}{p'},1)$, and $u \in \dot{B}_{p,\infty}^{s+\frac{s}{p-1}}(\Omega)$ for $s \in (0,\frac{1}{p'})$. In the subquadratic case $1<p<2$, we show $u \in \dot{B}_{p,\infty}^{s+1/2}(\Omega)$ for $s \in [\frac{1}{2},1)$, and $u \in \dot{B}_{p,\infty}^{2s}(\Omega)$ for $s \in (0,\frac12)$, with quantitative bounds depending on the source data.