[Paper Review] The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups

Van Hoang Nguyen|arXiv (Cornell University)|Aug 30, 2017

Advanced Harmonic Analysis Research35 references3 citations

TL;DR

This paper establishes sharp higher-order Hardy–Rellich type inequalities on homogeneous groups using integral identities involving higher-order derivatives. It derives new weighted $L^p$ and critical inequalities, including sharp global scaling-invariant versions, correcting and extending prior results in Euclidean space $ ^n$.

ABSTRACT

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the subcritical and critical inequalities on the homogeneous groups. We also prove several uncertainty principles on the homogeneous groups. Our results seem to be new even in the case of Euclidean space $\mathbb R^n$ and give a simple proof of several classical Hardy--Rellich type inequalities in $\mathbb R^n$.

Motivation & Objective

• To derive sharp higher-order Hardy–Rellich type inequalities on homogeneous groups using integral identities.
• To generalize classical Hardy–Rellich inequalities in $ ^n$ to higher-order derivatives and weighted $L^p$ settings.
• To establish new critical inequalities with sharp constants, correcting previous results in the literature.
• To prove uncertainty principles and scaling-invariant inequalities in the critical case.
• To provide a unified framework that simplifies and extends known results in Euclidean space

Proposed method

• Derives exact integral identities for higher-order derivatives on homogeneous groups, serving as the foundation for the inequalities.
• Applies these identities to prove sharp weighted $L^2$ and $L^p$ Hardy–Rellich inequalities with optimal constants.
• Introduces a scaling-invariant framework using the transformation $f_R(x) = f(Rx/|x|)$ to derive global critical inequalities.
• Uses radial decomposition and radial derivatives ($ abla_r$, $ abla_r^k$) to analyze radial symmetry and sharpness.
• Employs variational techniques and test functions (e.g., $f_ ho(x) = ( ext{ln}(R/|x|))^{1/2 - ho}$) to verify sharpness of constants.
• Establishes equivalence between critical and subcritical inequalities in higher dimensions, extending known results

Experimental results

Research questions

• RQ1 What are the sharp constants in higher-order weighted $L^p$ Hardy–Rellich inequalities on homogeneous groups?
• RQ2 How can critical Hardy–Rellich inequalities be formulated in a scaling-invariant form on $ ^n$?
• RQ3 In what way do the new inequalities correct or improve upon previously claimed sharp constants, such as in Adimurthi and Santra (2019)?
• RQ4 Can uncertainty principles be derived from integral identities involving higher-order derivatives on homogeneous groups?
• RQ5 What is the relationship between subcritical and critical Hardy–Rellich inequalities in higher dimensions?

Key findings

• The paper proves sharp $L^p$ Hardy–Rellich inequalities for higher-order derivatives on homogeneous groups, with explicit optimal constants.
• For $n = 4k$, the inequality $\left(2^{2k-2}(2k-1)! ight)^2 \sup_{R>0} \int_{\r^n} \frac{|f - f_R|^2}{|x|^n |\ln(R/|x|)|^2} dx \leq \int_{\r^n} |\Delta_r^k f|^2 dx$ holds and is sharp.
• For $n = 4k+2$, the inequality $\left(2^{2k-1}(2k)! ight)^2 \sup_{R>0} \int_{\r^n} \frac{|f - f_R|^2}{|x|^n |\ln(R/|x|)|^2} dx \leq \int_{\r^n} |\partial_r \Delta_r^k f|^2 dx$ is sharp and corrects an earlier claim in Adimurthi and Santra.
• The inequality $\left(\frac{n-1}{n}\right)^n \int_{\r^n} \frac{|f - f_R|^n}{|x|^n |\ln(R/|x|)|^n} dx \leq \int_{\r^n} \left|\frac{x}{|x|} \cdot \nabla f\right|^n dx$ is established as sharp and scaling-invariant.
• The results extend to $L^p$-versions with weights, yielding new inequalities such as $\left(\prod_{i=0}^{k-1} \frac{n + p'(2i + \alpha))(n - p(2i + 2 + \alpha))}{pp'}\right)^p \int \frac{|f|^p}{|x|^{p(2k + \alpha)}} dx \leq \int \frac{|\Delta_r^k f|^p}{|x|^{p\alpha}} dx$.
• The paper proves that the critical inequality (6.18) is sharp via test functions, resolving a discrepancy in prior work