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

TL;DR

This paper uses Fourier analysis on hyperbolic spaces to confirm conjectures about sharp Hardy-Sobolev-Maz'ya inequalities and provides explicit Green's functions for Paneitz and GJMS operators, revealing their precise bounds and constants.

Contribution

It establishes the exact Green's functions and sharp constants for high-order operators on hyperbolic spaces, confirming conjectures and deriving bounds for Hardy-Sobolev-Maz'ya inequalities.

Findings

Confirmed the conjecture relating sharp constants in Hardy-Sobolev-Maz'ya inequalities.

Derived explicit Green's functions for Paneitz and GJMS operators on hyperbolic space.

Established optimal bounds and expressions for Green's functions of relevant operators.

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).