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

Abstract

Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Green's functions of the operator $ -\Delta_{\mathbb{H}}-\frac{(n-1)^{2}}{4}$ on the hyperbolic space $\mathbb{B}^n$ and operators of the product form are given, where $\frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-\Delta_{\mathbb{H}}$ on $\mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).