[Paper Review] Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^{p}$-weighted Hardy inequalities

Michael Ruzhansky, Durvudkhan Suragan+1 more|arXiv (Cornell University)|Jan 5, 2017

Nonlinear Partial Differential Equations5 citations

TL;DR

This paper extends the classical Caffarelli-Kohn-Nirenberg inequalities by replacing the full gradient with the radial derivative, establishing sharp constants and remainder estimates for $L^p$-weighted Hardy inequalities on homogeneous groups. It introduces novel superweights of the form $\frac{(a+b|x|^\alpha)^{\beta/p}}{|x|^m}$, derives critical logarithmic Hardy inequalities, and proves higher-order inequalities with sharp constants via iterative integration by parts and Hölder's inequality in the setting of homogeneous Lie groups.

ABSTRACT

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\\infty$, $0<r<\\infty$ with $p+q\\geq r$, $\\delta\\in[0,1]\\cap\\left[\\frac{r-q}{r},\\frac{p}{r}\ ight]$ with $\\frac{\\delta r}{p}+\\frac{(1-\\delta)r}{q}=1$ and $a$, $b$, $c\\in\\mathbb{R}$ with $c=\\delta(a-1)+b(1-\\delta)$, and for all functions $f\\in C_{0}^{\\infty}(\\mathbb{R}^{n}\\backslash\\{0\\})$ we have $$ \\||x|^{c}f\\|_{L^{r}(\\mathbb{R}^{n})} \\leq \\left|\\frac{p}{n-p(1-a)}\ ight|^{\\delta} \\left\\||x|^{a}\ abla f\ ight\\|^{\\delta}_{L^{p}(\\mathbb{R}^{n})} \\left\\||x|^{b}f\ ight\\|^{1-\\delta}_{L^{q}(\\mathbb{R}^{n})} $$ for $n\ eq p(1-a)$, where the constant $\\left|\\frac{p}{n-p(1-a)}\ ight|^{\\delta}$ is sharp for $p=q$ with $a-b=1$ or $p\ eq q$ with $p(1-a)+bq\ eq0$. In the critical case $n=p(1-a)$ we have $$ \\left\\||x|^{c}f\ ight\\|_{L^{r}(\\mathbb{R}^{n})} \\leq p^{\\delta} \\left\\||x|^{a}\\log|x|\ abla f\ ight\\|^{\\delta}_{L^{p}(\\mathbb{R}^{n})} \\left\\||x|^{b}f\ ight\\|^{1-\\delta}_{L^{q}(\\mathbb{R}^{n})}. $$ Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for $L^{p}$-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of $\\mathbb{R}^{n}$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and stability estimates. We also establish sharp Hardy type inequalities in $L^{p}$, $1<p<\\infty$, with superweights, i.e. with the weights of the form $\\frac{(a+b|x|^{\\alpha})^{\\frac{\\beta}{p}}}{|x|^{m}}$ allowing for different choices of $\\alpha$ and $\\beta$.

Motivation & Objective

• To extend the classical Caffarelli-Kohn-Nirenberg inequalities by replacing the full gradient with the radial derivative, improving sharpness and applicability.
• To derive remainder estimates and stability bounds for $L^p$-weighted Hardy inequalities in the context of homogeneous groups.
• To establish critical Hardy inequalities of logarithmic type and uncertainty principles on homogeneous groups.
• To introduce and analyze $L^p$-Hardy inequalities with superweights, defined by arbitrary homogeneous quasi-norms and flexible parameters $\alpha$, $\beta$, $m$, $a$, $b$, enabling broader applicability.
• To investigate the relationship between critical and subcritical Hardy inequalities and prove higher-order versions via iterative methods.

Proposed method

• Derives anisotropic $L^p$-weighted Hardy inequalities using the radial derivative $\mathcal{R}f = \frac{x}{|x|} \cdot \nabla f$ in place of $|\nabla f|$, improving sharpness and stability.
• Applies integration by parts and Hölder's inequality in the setting of homogeneous Lie groups $\mathbb{G}$ with homogeneous dimension $Q$, using radial integration in polar coordinates.
• Introduces superweights $\frac{(a+b|x|^\alpha)^{\beta/p}}{|x|^m}$, where $a,b>0$, $\alpha,\beta \in \mathbb{R}$, and $|x|$ is a homogeneous quasi-norm, to generalize classical Hardy weights.
• Uses iterative application of the main inequality to derive higher-order Hardy-type inequalities involving $\mathcal{R}^k f$, with sharp constants depending on $Q$, $p$, $m$, $\alpha$, $\beta$.
• Establishes sharpness of constants by testing equality conditions in Hölder's inequality using homogeneous functions $f(x) = |x|^C$, $C \neq 0$.
• Derives critical Hardy inequalities in the case $n = p(1-a)$ by replacing $|\nabla f|$ with $|\nabla f| \log |x|$ to account for loss of compactness.

Experimental results

Research questions

• RQ1 Can the Caffarelli-Kohn-Nirenberg inequality be extended with the radial derivative instead of the full gradient, and what are the sharp constants in this setting?
• RQ2 What are the remainder estimates and stability bounds for $L^p$-weighted Hardy inequalities on homogeneous groups?
• RQ3 How can critical Hardy inequalities with logarithmic weights be derived in the limiting case $n = p(1-a)$?
• RQ4 What are the conditions under which $L^p$-Hardy inequalities with superweights $\frac{(a+b|x|^\alpha)^{\beta/p}}{|x|^m}$ are sharp and valid?
• RQ5 How do higher-order Hardy inequalities with radial derivatives relate to the first-order case, and what are their sharp constants?

Key findings

• For $n \neq p(1-a)$, the inequality $\||x|^c f\|_{L^r} \leq \left|\frac{p}{n - p(1-a)}\right|^\delta \||x|^a \mathcal{R}f\|_{L^p}^\delta \||x|^b f\|_{L^q}^{1-\delta}$ holds with sharp constant $\left|\frac{p}{n - p(1-a)}\right|^\delta$ under the given parameter constraints.
• In the critical case $n = p(1-a)$, the inequality becomes $\||x|^c f\|_{L^r} \leq p^\delta \||x|^a \log|x| \mathcal{R}f\|_{L^p}^\delta \||x|^b f\|_{L^q}^{1-\delta}$, with a logarithmic correction term.
• The constant $\frac{Q - pm - p + \alpha\beta}{p}$ in the superweight inequality is sharp, achieved when equality holds in Hölder’s inequality for homogeneous functions $f(x) = |x|^C$.
• For homogeneous groups $\mathbb{G}$, the higher-order inequality $\left[\prod_{j=0}^{k-1} \left(\frac{Q-p}{p} - (m+j)\right)\right] \left\| \frac{(a+b|x|^\alpha)^{\beta/p}}{|x|^{m+k}} f \right\|_{L^p} \leq \left\| \frac{(a+b|x|^\alpha)^{\beta/p}}{|x|^m} \mathcal{R}^k f \right\|_{L^p}$ holds under $\alpha\beta > 0$ and $pm \leq Q - p$.
• A dual higher-order inequality holds for $\alpha\beta < 0$ and $pm - \alpha\beta \leq Q - p$, with modified constant $\frac{Q - p + \alpha\beta}{p} - (m+j)$, proving the robustness of the method across parameter regimes.