Title: Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces,
Abstract: By using, among other things, the Fourier analysis techniques on hyperbolic
and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for
higher order derivatives on half spaces. The proof relies on a
Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its
independent interest. We also give an alternative proof of Benguria, Frank and
Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality
in the three dimensional upper half space. Finally, we show the sharp constant
in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half space
of dimension five coincides with the Sobolev constant.