Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

TL;DR

This paper extends higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams, and Hardy-Adams inequalities to complex hyperbolic spaces and Siegel domains using Helgason-Fourier analysis, factorization theorems, and CR invariant operators.

Contribution

It establishes new inequalities on complex hyperbolic spaces and Siegel domains, including sharp Hardy-Adams inequalities, with novel factorization theorems and analysis techniques.

Findings

Proved a factorization theorem for operators on complex hyperbolic space.

Established Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequalities on Siegel domain and complex hyperbolic space.

Derived sharp Hardy-Adams and Adams inequalities for fractional Sobolev spaces.

Abstract

This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain $\mathcal{U}^{n}$ and the unit ball $\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein.