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-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 $p\ge 2$, we prove $u\in {\dot{B}}_{p,\infty }^{s+1/p}(\Omega )$ for $s\in [\frac{1}{p^{\prime }},1)$, and $u\in {\dot{B}}_{p,\infty }^{s+\frac{s}{p-1}}(\Omega )$ for $s\in (0,\frac{1}{p^{\prime }})$. 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 $s\in (0,\frac{1}{2})$, with quantitative bounds depending on the source data.