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 the sharp constant in the n-1/2-th order HardySobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n-1/2-th order Sobolev constant when n is odd and n >= 5. 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. As a consequence, we thus show that the sharp constant in the k-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n is strictly less than the best k-th order Sobolev constant for all n >= 2k + 2. Precise expressions and optimal bounds for Green's functions of the operator -triangle(H) - (n-1)(2)/4 on the hyperbolic space B-n and operators of the product form are given, where (n-1)(2)/4 is the spectral gap for the Laplacian -triangle(H) on 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 in terms of hypergeometric functions. In fact, we will establish the precise formulas of the Green's functions of the operator Pi(k-1)(j=0) ((nu + j)(2) - (n - 1)(2)/4 - triangle(H)), nu >= 0, in terms of hypergeometric function F(a, b; c, z). Our approach to establish the main theorems is using the Helgason-Fourier analysis on hyperbolic spaces and are substantially different from that in dealing with the first order Hardy-Sobolev-Maz'ya inequalities on upper half spaces. (c) 2021 Elsevier Inc. All rights reserved.