Harnack's inequality for fractional operators

Details Harnack's inequality for fractional operators Harnack's inequality for fractional operators

2012-03-07

arxiv.org/abs/1203.1518v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

We prove interior Harnack's inequalities for fractional powers of second order partial differential operators. Our examples include 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 operators are obtained.

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