The parabolic Harnack inequality for nonlocal equations

Abstract

We complete the local regularity program for weak solutions to linear parabolic nonlocal equations with bounded measurable coefficients. Within the variational framework we prove the parabolic Harnack inequality and H\"older regularity estimates. We discuss in detail the shortcomings of previous results in this direction. The key element of our approach is a fine study of the nonlocal tail term.