Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces

Though there has been extensive study on Hardy-Sobolev-Maz'ya inequalities on upper half spaces for first order derivatives, whether an analogous inequality for higher order derivatives holds has still remained open. By using, among other things, the Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. Moreover, we derive sharp Poincar\'e-Sobolev inequalities (namely, Sobolev inequalities with a substraction of a Hardy term) for the Paneitz operators on hyperbolic spaces which are of their independent interests and useful in establishing the sharp Hardy-Sobolev-Maz'ya inequalities. Our sharp Poincar\'e-Sobolev inequalities for the Paneitz operators on hyperbolic spaces improve substantially those Sobolev inequalities in the literature. The proof of such Poincar\'e-Sobolev inequalities relies on hard analysis of Green's functions estimates, Fourier analysis on hyperbolic spaces together with the Hardy-Littlewood-Sobolev inequality on the hyperbolic spaces. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for the bi-Laplacian in the upper half space of dimension five coincides with the best Sobolev constant. This is an analogous result to that of the sharp constant in the first order Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half spaces.