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

Abstract

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains and complex hyperbolic spaces using the method of Helgason-Fourier analysis on the complex hyperbolic spaces. Firstly, we give a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller's operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. (See Theorem 1.3.) 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 Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain Un and the unit ball BCn. (See Theorems 1.4, 1.5.) 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. (See Theorems 1.9, 1.10 and 1.12.) The factorization theorem we prove on the complex hyperbolic spaces is considerably more difficult than in the case of real hyperbolic spaces and should be of its independent interest in the analysis on Heisenberg group and CR sphere and CR invariant differential operators therein.