Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Abstract

Abstract In [1], for 1 < p < ∞ {1<p<\infty} , we proved the W loc 2 ⁢ s , p {W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} on an arbitrary bounded open set of ℝ N {\mathbb{R}^{N}} . Here we make a more precise and rigorous statement. In fact, for 1 < p < 2 {1<p<2} and s ≠ 1 2 {s\neq\frac{1}{2}} , local regularity does not hold in the Sobolev space W loc 2 ⁢ s , p {W^{2s,p}_{\mathrm{loc}}} , but rather in the larger Besov space ( B p , 2 2 ⁢ s ) loc {(B^{2s}_{p,2})_{\mathrm{loc}}} .