Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities
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2023
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| author | Lu, Guozhen Yang, Qiaohua |
| author_facet | Lu, Guozhen Yang, Qiaohua |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_15973 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities Lu, Guozhen Yang, Qiaohua Analysis of PDEs Differential Geometry 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. |
| title | Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2310.15973 |