Optimal elliptic regularity: a comparison between local and nonlocal equations

Given L >= 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-loc(alpha). This problem can be formulated both in the classical and non-local framework. In the classical case it is known that alpha(L) greater than or similar to exp(-CL beta), for some C, beta >= 1 depending on the dimension N >= 3. We show that in the non-local case, alpha(L) greater than or similar to L-1-delta for all delta > 0.