De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

Abstract

We extend the De Giorgi--Nash--Moser theory to nonlocal hypoelliptic equations arising in kinetic theory as linearized models for the non-cutoff Boltzmann equation. Assuming that the nonlocal tail in velocity of weak solutions belongs locally to $L_{t,x}^p$ for some $p>N_d/(2s)$, where $N_d$ is the drift homogeneous dimension and $2s\in (0,2)$ is the order of diffusion, we prove a local $L^2$-$L^{\infty }$ estimate, from which we also deduce a nonlocal strong Harnack inequality. The tail summability threshold for the local estimate is optimal: for every $1\le p\le N_d/(2s)$, we construct a family of nonnegative, time-dependent solutions that rules out this estimate even for the pure fractional Kolmogorov equation. The local boundedness estimate also allows for possibly unbounded source terms.

In this revision, we establish the sharp tail integrability threshold for the local boundedness estimate. We also construct a family of solutions showing that the estimate fails at and below the critical exponent