Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane

TL;DR

This paper establishes advanced functional inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane, extending the theory of Sobolev and Adams inequalities to these complex geometric settings.

Contribution

It introduces quaternionic and octonionic Geller's operators and proves higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on these spaces.

Findings

Established factorization theorems on quaternionic and Cayley hyperbolic spaces.

Proved higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities.

Derived Adams and Hardy-Adams inequalities on these spaces.

Abstract

Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincar\'e-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincar\'e-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on these spaces which are of their independent interests. To this end, we need to identify and introduce the ``Quaternionic Geller's operators" and ``Octonionic Geller's operators" which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernel estimates and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincar\'e-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient for us to establish, as a byproduct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works, completes our study of the factorization theorems, higher order Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.