[Paper Review] Improved higher order Poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

Elvise Berchio, Debdip Ganguly|arXiv (Cornell University)|Nov 2, 2015

Advanced Harmonic Analysis Research21 references3 citations

TL;DR

This paper establishes improved higher-order Poincaré inequalities on hyperbolic space by introducing $k$ Hardy-type remainder terms for the ratio of $L^2$ norms of iterated gradients of order $k$ and $l$, with $k > l$. The key contribution is the explicit construction of sharp remainder terms involving inverse powers of the geodesic distance $r$, extending previous results for $k=1,2$ and proving optimality of constants via variational and spherical harmonic techniques.

ABSTRACT

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} ight)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} | abla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} | abla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.

Motivation & Objective

• To improve the classical higher-order Poincaré inequality on hyperbolic space by adding non-trivial remainder terms.
• To address the strictness of the inequality (1.1) by identifying explicit Hardy-type remainder terms that quantify the gap.
• To extend previous results for $k=1, l=0$ and $k=2, l=0$ to general $k > l \geq 0$, particularly focusing on $k=2, l=1$.
• To prove sharpness of the constants in the remainder terms using variational and asymptotic analysis.
• To derive improved Rellich-type inequalities on the upper half-space of Euclidean space via the hyperbolic results.

Proposed method

• Utilizes spherical harmonics decomposition to analyze the spectral structure of higher-order differential operators on $\mathbb{H}^N$.
• Applies a novel construction of supersolutions to derive lower bounds for the difference between the $k$-th and $l$-th order gradient norms.
• Employs the upper half-space model of $\mathbb{H}^N$ with the metric $ds^2 = y^{-2}(dx^2 + dy^2)$ to transform the problem into Euclidean coordinates.
• Derives explicit expressions for $\int_{\mathbb{H}^N} |\nabla_{\mathbb{H}^N}^k u|^2 \, dv_{\mathbb{H}^N}$ and $\int_{\mathbb{H}^N} |\nabla_{\mathbb{H}^N}^l u|^2 \, dv_{\mathbb{H}^N}$ in terms of $v(x,y)$ via change of variables $u(x,y) = y^\alpha v(x,y)$.
• Uses the Laplacian expression $\Delta_{\mathbb{H}^N} = y^2 \Delta - (N-2)y \partial_y$ to relate hyperbolic and Euclidean Laplacians.
• Employs variational arguments and comparison with known Hardy inequalities to establish sharpness of constants in the remainder terms.

Experimental results

Research questions

• RQ1 Can the classical higher-order Poincaré inequality on $\mathbb{H}^N$ be improved by adding $k$ Hardy-type remainder terms for $k > l \geq 0$?
• RQ2 For $k=2, l=1$, does a Poincaré-Hardy inequality with a remainder term of the form $C \int \frac{u^2}{r^\gamma} \, dv_{\mathbb{H}^N}$ hold, and what are the optimal constants?
• RQ3 Is the constant $\frac{(N-1)^2}{16}$ in the remainder term for $k=2, l=0$ sharp, and can it be improved further?
• RQ4 Can the improved inequalities on $\mathbb{H}^N$ be used to derive new Rellich-type inequalities with non-standard remainder terms on $\mathbb{R}^N_+$?
• RQ5 How do the remainder terms depend on the dimension $N$ and the choice of $k,l$?

Key findings

• For $k=1, l=0$, the paper proves the inequality $\int_{\mathbb{H}^N} |\nabla_{\mathbb{H}^N} u|^2 \, dv_{\mathbb{H}^N} - \left(\frac{N-1}{2}\right)^2 \int_{\mathbb{H}^N} u^2 \, dv_{\mathbb{H}^N} \geq \frac{1}{4} \int_{\mathbb{H}^N} \frac{u^2}{r^2} \, dv_{\mathbb{H}^N}$ with sharp constants for $N > 2$.
• For $k=2, l=0$, the inequality $\int_{\mathbb{H}^N} (\Delta_{\mathbb{H}^N} u)^2 \, dv_{\mathbb{H}^N} - \left(\frac{N-1}{2}\right)^4 \int_{\mathbb{H}^N} u^2 \, dv_{\mathbb{H}^N} \geq \frac{(N-1)^2}{8} \int_{\mathbb{H}^N} \frac{u^2}{r^2} \, dv_{\mathbb{H}^N} + \frac{9}{16} \int_{\mathbb{H}^N} \frac{u^2}{r^4} \, dv_{\mathbb{H}^N}$ holds with sharp constants for $N > 4$.
• The constant $\frac{(N-1)^2}{16}$ in the remainder term for $k=2, l=1$ is proven to be optimal, as any smaller value leads to contradiction with the known sharp Poincaré inequality.
• The sharpness of the constant $\frac{(N-1)^2}{8}$ in the $r^{-2}$ term is confirmed via comparison with the $L^2$-norm of the gradient and the Hardy-Maz’ya inequality.
• The paper derives improved Rellich-type inequalities on the upper half-space $\mathbb{R}^N_+$ by transforming the hyperbolic results into Euclidean coordinates using the $y^\alpha$-scaling.
• The method yields non-standard remainder terms involving $y^{-2}$ and $y^{-4}$ weights, which are not present in classical Rellich inequalities.