Nonlocal parabolic De Giorgi classes

We study the local behavior of the elements of a specific energy class of functions, called the nonlocal parabolic ($p$-homogenous) De Giorgi class. First we carry on an analysis of their local boundedness under optimal tail conditions, and then prove several weak Harnack inequalities, measure theoretical propagation lemmas, and a parabolic Harnack inequality. We show a full proof of the local Hölder continuity, eventually establishing a Liouville-type rigidity property. Finally, as an application of our method, we prove a state-of-the-art nonlocal Harnack inequality for nonnegative solutions of the nonlocal Trudinger equation.

💡 Research Summary

The paper develops a comprehensive regularity theory for the nonlocal parabolic (p‑homogeneous) De Giorgi class, denoted PDG_{s,p}^{±}(Ω_T,γ_{DG},ε). After introducing the precise definition—functions satisfying a Caccioppoli‑type energy inequality on every space‑time cylinder together with a nonlocal “tail” term that quantifies the influence of values outside the cylinder—the authors establish a series of sharp local estimates.

The first main result (Theorem 1.1) is a refined L∞–Lν bound. For any σ∈(0,1) and ν∈(0,p] the supremum of u^{±} on a reduced cylinder Q_{σR,σθ} is controlled by a power of (1−σ)^{-1}, an integral of the tail in L^{p‑1+ε}, and the ν‑average of u. This improves earlier bounds by allowing the tail to appear in an L^{p‑1+ε} norm rather than an L∞‑supremum and by making ν arbitrarily small.

Next, the authors prove two nonlocal weak Harnack inequalities. The first (Theorem 1.3) involves the negative part u⁻ and yields a bound of the form (∫{B_R} u^{β})^{1/β} ≤ C inf{B_{2R}} u + C