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

Abstract: We extend the celebrated De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations naturally arising in kinetic theory, which combine a first-order operator with an elliptic one involving fractional derivatives along only part of the coordinates. Provided that the nonlocal tail in velocity of weak solutions is just $p$-summable along the drift variables, we prove the first local $L^2$-$L^\infty$ estimate for kinetic integral equations. Then, we establish the first strong Harnack inequality under the aforementioned tail summability assumption. The latter is in fact naturally implied in literature, e. g., from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and it reveals to be in clear accordance with the very recent counterexample by Kassmann and Weidner \cite{KW24c}. Armed with the aforementioned results, we are able to provide a geometric characterization of the Harnack inequality in the same spirit of the seminal paper by Aronson and Serrin \cite{AS67} for the (local) parabolic counterpart.