Global fractional Calderón-Zygmund type regularity

Abstract

We obtain a global fractional Calderón-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, a bounded domain with boundary $\partial \Omega$ of class $C^2$, $s\in (0,1)$ and $f\in L^m(\Omega )$ for some $m\ge 1$, we consider the problem $$ \left. \begin{aligned} (-Δ)^s u = f \quad \mbox{in } Ω, \qquad\ u = 0 \quad \mbox{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. $$ and, according to $m$, we find the values of $s\le t<\min \{1,2s\}$ and of $1<p<+\infty$ such that $u\in L^{t,p}({\mathbb{R}}^N)$ and such that $u\in W^{t,p}({\mathbb{R}}^N)$.

20 pages, section 2 removed and presentation improved