Nonlocal parabolic De Giorgi classes

We propose a new paradigm for the point-wise regularity theory of parabolic nonlocal problems, by addressing directly the elements of a wide parabolic energy class.

First we carry on a refined analysis of their local boundedness under optimal tail conditions, and then prove several weak Harnack estimates through a purely measure-theoretical framework.

Then we give a novel proof of the nonlocal parabolic Harnack inequality, that avoids any covering argument or lemma à la John-Nirenberg and is valid regardless of any comparison principle.

The regularity program is completed by addressing local Hölder estimates, eventually leading to a Liouville-type theorem.