Abstract. We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form ut − Iu = f , where the operator I is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain C 1,α interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.

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In this article we further advance in the theory of singular fully nonlinear operators modeled on the q-laplacian proving a Harnack inequality. We provide also several applications of this inequality and the ideas used for proving it. In doing so we have left various open questions, all of them related to the fact that the operator is not sub-linear. Mathematics Subject Classification (2000)Primary 35D40 · Secondary 35B45

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Abstract. In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide an Oscillation Lemma and a Harnack Inequality which can be used to prove higher regularity estimates.