Harnack's inequality for fractional nonlocal equations

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schr\"odinger operators proved by C. E. Guti\'errez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.