On The Weak Harnack Estimate For Nonlocal Equations

Abstract

We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for $p\in (1,\infty)$ and $s \in (0,1)$. Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional $p-$Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.