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Using the Fourier analysis techniques on hyperbolic spaces and Green'sfunction estimates, we confirm in this paper the conjecture given by the sameauthors 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 halfspace of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolevconstant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establisha lower bound of the coefficient of the Hardy term for the $k-$th orderHardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases ofdimension $n$ and $k$-th order derivatives (see Theorem 1.7). Preciseexpressions 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 preciseexpression and optimal pointwise bound of Green's function of the Paneitz andGJMS operators on hyperbolic space, which are of their independent interest(see Theorem 1.10).

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