First, we study the interior regularity of solutions to a99131bf" title="Click to view the MathML source">Lu=f in B1. We prove that if f is Cα then u belong to Cα+2s whenever α+2s is not an integer. In case f∈L∞, we show that the solution u is C2s when s≠1/2, and 05a5be64c8fcbdca43b88347" title="Click to view the MathML source">C2s−ϵ for all ϵ>0 when 05af30b484c704fea2db743c98521" title="Click to view the MathML source">s=1/2.
Then, we study the boundary regularity of solutions to a99131bf" title="Click to view the MathML source">Lu=f in Ω, u=0 in Rn∖Ω, in C1,1 domains Ω. We show that solutions u satisfy for all ϵ>0, where d is the distance to ∂Ω.
Finally, we show that our results are sharp by constructing two counterexamples.