Optimal elliptic regularity: a comparison between local and nonlocal equations

Given $Lgeq 1$, we discuss the problem of determining the highest $alpha=alpha(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^alpha_{rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $alpha(L)gtrsim {rm exp}(-CL^beta)$, for some $C, betageq 1$ depending on the dimension $Ngeq 3$. We show that in the non-local case, $alpha(L)gtrsim L^{-1-delta}$ for all $delta>0$.