  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.
