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

We study the Dirichlet problem for a class of fractional p-Laplacian operators of order s∈(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≥2, we prove u∈B˙s+1/pp,∞(Ω) for s∈[1p′,1), and u∈B˙s+sp−1p,∞(Ω) for s∈(0,1p′). In the subquadratic case 1<p<2, we show u∈B˙s+1/2p,∞(Ω) for s∈[12,1), and u∈B˙2sp,∞(Ω) for s∈(0,12), with quantitative bounds depending on the source data.