First, we study the interior regularity of solutions to Lu=f in B1. We prove that if f is Cα then u belong to 4e158e73166921e858d3ad94" title="Click to view the MathML source">Cα+2s whenever α+2s is not an integer. In case 14e8cde178d1b" title="Click to view the MathML source">f∈L∞, we show that the solution u is C2s when s≠1/2, and C2s−ϵ for all ϵ>0 when s=1/2.
Then, we study the boundary regularity of solutions to 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.