Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincar\'e-Sobolev and Hardy-Sobolev-Maz'ya inequalities

Abstract

Using the scattering theory on the hyperbolic space $\mathbb{H}^n$, we give the explicit formulas of the fractional GJMS operators $P_{\gamma}$ for all $\gamma\in(0,\frac{n}{2})\setminus\mathbb{N}$ on $\mathbb{H}^n$.These $P_{\gamma}$ for $\gamma\in(0,\frac{n}{2})\setminus\mathbb{N}$ are neither conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ nor on $\mathbb{B}^n$ in $\mathbb{R}^{n}$ though $P_{\gamma}$ are conformal to $(-\Delta)^{\gamma}$ via half space model and ball model of hyperbolic spaces when $\gamma\in\mathbb{N}$. To circumvent this, we introduce another family of fractional operators $\tilde{P}_{\gamma}$ on $\mathbb{H}^n$ which are conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ and $\mathbb{B}^n$. It is worthwhile to note that $\tilde{P}_{\gamma}\not =P_{\gamma}$ unless $\gamma$ is an integer. We establish the fractional Poincar\'e-Sobolev inequalities associated with both $P_{\gamma}$ and $\tilde{P}_{\gamma}$ on $\mathbb{H}^n$. In particular, when $n\geq 3$ and $\frac{n-1}{2}\leq \gamma<\frac{n}{2}$, we prove that the sharp constants in the $\gamma$-th order of Poincar\'e-Sobolev inequalities on the hyperbolic space associated with $P_{\gamma}$ and $\tilde{P}_{\gamma}$ coincide with the best $\gamma$-th order Sobolev constant in the $n$-dimensional Euclidean space $\mathbb{R}^n$. We also establish fractional Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_+$ and $\mathbb{B}^n$ and prove that the sharp constants in the $\gamma$-th order Hardy-Sobolev-Maz'ya inequalities on half space $\mathbb{R}^{n}_+$ and unit ball $\mathbb{B}^n$ are the same as the best $\gamma$-th order Sobolev constants in $\mathbb{R}^n$ when $n\geq 3$ and $\frac{n-1}{2}\leq \gamma<\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourier analysis on hyperbolic spaces and delicate analysis of special functions.