The effect of the Hardy potential in some Calderon-Zygmund properties for the fractional Laplacian

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems {(-Delta)(s)u - lambda u/vertical bar x vertical bar(2s) = f(x, u) in Omega, u = 0 in R-N\Omega, u > 0 in Omega, where (-Delta)(s), s is an element of (0,1), is the fractional Laplacian operator, Omega subset of R-N is a bounded domain with Lipschitz boundary such that 0 is an element of Omega and N > 2s. We will mainly consider the solvability in two cases: (1) The linear problem, that is, f(x, t) = f(x), where according to the summability of the datum f and the parameter lambda we give the summability of the solution u. (2) The problem with a nonlinear term f(x, t) = h(x)/t(sigma) for t > 0. In this case, existence and regularity will depend on the value of sigma and on the summability of h. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new. (C) 2016 Elsevier Inc. All rights reserved.