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 ∈ (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 [21], 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 ∈ Ḃ^{s+1/p}{p,∞}(Ω) for s ∈ [1/p', 1), and u ∈ Ḃ^{s + s/(p-1)}{p,∞}(Ω) for s ∈ (0, 1/p'). In the subquadratic case 1 < p < 2, we show u ∈ Ḃ^{s+1/2}{p,∞}(Ω) for s ∈ [1/2, 1), and u ∈ Ḃ^{2s}{p,∞}(Ω) for s ∈ (0, 1/2), with quantitative bounds depending on the source data.