The overall goal of the thesis is to investigate hypoelliptic and subelliptic functional inequalities of different types and the corresponding function spaces on homogeneous nilpotent Lie groups, which include the cases of Rn, Heisenberg, stratified and more general graded Lie groups. In such settings we derive a variety of functional inequalities including Hardy, Rellich, Hardy-Littllewood-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, uncertainty and other related inequalities, and their extensions.

The approach for obtaining hypoelliptic functional inequalities relies on establishing integral versions of Hardy inequalities on homogeneous Lie 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.

Moreover, we describe Euler–Hilbert–Sobolev, Sobolev–Lorentz–Zygmund, Besov type, Morrey and generalised Morrey spaces on homogeneous Lie groups. We also study the boundedness of the Bessel–Riesz operators, generalised Bessel–Riesz operators, generalised fractional integral operators and Olsen type inequalities.

The obtained results on homogeneous Lie groups in this thesis give new statements in the Euclidean setting of Rn when we are working with anisotropic differential structures. Furthermore, even in the isotropic situation in Rn, one novelty of all the obtained results is in the arbitrariness of the choice of any homogeneous quasi-norm, and some estimates are also new in the usual isotropic structure of Rn with the Euclidean norm.