Global fractional Calder\'on-Zygmund type regularity

Abstract

We obtain a global fractional Calder\'on-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $\Omega \subset \mathbb{R}^N$, $N \geq 2$, a bounded domain with boundary $\partial \Omega$ of class $C^2$, $s \in (0,1)$ and $f \in L^m(\Omega)$ for some $m \geq 1$, we consider the problem $$ \left. \begin{aligned} (-\Delta)^s u = f \quad \mbox{in } \Omega, \qquad\ u = 0 \quad \mbox{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. $$ and, according to $m$, we find the values of $s \leq t < \min\{1,2s\}$ and of $1 < p < +\infty$ such that $u \in L^{t,p}(\mathbb{R}^N)$ and such that $u \in W^{t,p}(\mathbb{R}^N)$.