Hypoelliptic functional inequalities

Abstract: In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain several versions of local and global weighted Trudinger-Moser inequalities with remainder terms, critical Hardy and weighted Gagliardo-Nirenberg inequalities, which appear to be new also in the case of the sub-Laplacian. Curiously, we also show the equivalence of many of these critical inequalities as well as asymptotic relations between their best constants. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.