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.Comment: 58 page
