Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Helgason-Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we prove in a unified way that sharp constant n−12-th order Hardy-Sobolev-Maz'ya inequality upper half space of dimension n coincides with best Sobolev when is odd n≥5. We will also establish lower bound coefficient Hardy term for k-th remaining cases derivatives. As consequence, thus show strictly less than all n≥2k+2. Precise expressions optimal bounds functions operator −ΔH−(n−1)24 Bn operators product form are given, where (n−1)24 spectral gap Laplacian −ΔH Bn. Finally, give precise expression pointwise Paneitz GJMS terms hypergeometric functions. In fact, formulas operator∏j=0k−1((ν+j)2−(n−1)2/4−ΔH),ν≥0, F(a,b;c,z). Our approach to main theorems using substantially different from dealing first inequalities spaces.