Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities
Guozhen Lu, Qiaohua Yang
Abstract
Using the scattering theory on the hyperbolic space $\mathbb{H}^n$, we give the explicit formulas of the fractional GJMS operators $P_γ$ for all $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ on $\mathbb{H}^n$.These $P_γ$ for $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ are neither conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ nor on $\mathbb{B}^n$ in $\mathbb{R}^{n}$ though $P_γ$ are conformal to $(-Δ)^γ$ via half space model and ball model of hyperbolic spaces when $γ\in\mathbb{N}$. To circumvent this, we introduce another family of fractional operators $\tilde{P}_γ$ 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 $\tilde{P}_γ\not =P_γ$ unless $γ$ is an integer. We establish the fractional Poincaré-Sobolev inequalities associated with both $P_γ$ and $\tilde{P}_γ$ on $\mathbb{H}^n$. In particular, when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$, we prove that the sharp constants in the $γ$-th order of Poincaré-Sobolev inequalities on the hyperbolic space associated with $P_γ$ and $\tilde{P}_γ$ coincide with the best $γ$-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 $γ$-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 $γ$-th order Sobolev constants in $\mathbb{R}^n$ when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourier analysis on hyperbolic spaces and delicate analysis of special functions.