For regular non-local Dirichlet forms without killing, the strong parabolic Harnack inequality is equivalent to the weak parabolic Harnack inequality plus upper jumping smoothness, proved analytically.
The parabolic Harnack inequality for nonlocal equations
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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.
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The parabolic Harnack inequality on non-local Dirichlet spaces in the view of pure analysis
For regular non-local Dirichlet forms without killing, the strong parabolic Harnack inequality is equivalent to the weak parabolic Harnack inequality plus upper jumping smoothness, proved analytically.