Hardy's inequality and (almost) the Landau equation

Abstract

In this manuscript we establish an $L^\infty$ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent $\gamma$ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincar\'e inequality, which in turn implies the propagation of some $L^p$ norms of solutions. From here, the $L^\infty$ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.