Harnack's inequality for fractional nonlocal equations

Abstract

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operatorsarising in classical orthogonal expansions and the radial Laplacian.To get the results we use an analytic method based on ageneralization of the Caffarelli--Silvestre extension problem, theHarnack's inequality for degenerate Schrödinger operators provedby C. E. Gutiérrez, and a transference method. In this manner weapply local PDE techniques to nonlocal operators. Onthe way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.