Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-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 ${\tilde{P}}_{\gamma }\ne P_{\gamma }$ unless $\gamma$ is an integer. We establish the fractional Poincaré-Sobolev inequalities associated with both $P_{\gamma }$ and ${\tilde{P}}_{\gamma }$ on ${\mathbb{H}}^n$. In particular, when $n\ge 3$ and $\frac{n-1}{2}\le \gamma <\frac{n}{2}$, we prove that the sharp constants in the $\gamma$-th order of Poincaré-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 $n\ge 3$ and $\frac{n-1}{2}\le \gamma <\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourier analysis on hyperbolic spaces and delicate analysis of special functions.

Theorem 4.11 is improved to include more general $\nu$