Hardy's inequality and the isotropic Landau equation

Abstract

In this manuscript we establish an L∞ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent γ 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 Poincaré inequality, which in turn implies the propagation of some Lp norms of solutions. From here, the L∞ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.