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

Abstract

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional derivatives only in part of the variables. Under the sole assumption that the nonlocal tail in velocity of weak solutions is $p$-summable along the drift variables, we prove a local $L^2$-$L^\infty$ estimate for kinetic integral equations and a corresponding strong Harnack inequality. The tail condition is satisfied in standard kinetic regimes considered in the literature, for instance under the usual boundedness of the mass density in the Boltzmann equation without cut-off, and it is consistent with the recent counterexample by Kassmann and Weidner (Adv. Math. 2024). These estimates further lead to a geometric characterization of the Harnack inequality, in the spirit of the seminal work of Aronson and Serrin (Arch. Ration. Mech. Anal. 1967) for the local parabolic counterpart.