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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 higherorder Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adamsinequalities on real hyperbolic spaces using the method of Helgason-Fourieranalysis on the hyperbolic spaces. The aim of this paper is to establish suchinequalities on the Siegel domains and complex hyperbolic spaces. Firstly, weprove a factorization theorem for the operators on the complex hyperbolic spacewhich is closely related to Geller' operator, as well as the CR invariantdifferential operators on the Heisenberg group and CR sphere. Secondly, byusing, among other things, the Kunze-Stein phenomenon on a closed linear group$SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolicspaces, we establish the Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya inequality onthe Siegel domain $\mathcal{U}^{n}$ and the unit ball$\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adamsinequalities and sharp Adams type inequalities on Sobolev spaces of anypositive fractional order on the complex hyperbolic spaces. The factorizationtheorem we proved is of its independent interest in the Heisenberg group and CRsphere and CR invariant differential operators therein.

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