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arXiv · 2508.16247

Nonlocal parabolic De Giorgi classes

Simone Ciani·Kenta Nakamura
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Abstract

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 \`a la John-Nirenberg and is valid regardless of any comparison principle. The regularity program is completed by addressing local H\"older estimates, eventually leading to a Liouville-type theorem.

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Simone Ciani, Kenta Nakamura. 2025-08-22. Nonlocal parabolic De Giorgi classes. https://arxiv.org/abs/2508.16247

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