Hardy inequalities

Hardy inequalities is a very rich area with lots of works in it. We are interested in analysis of this and related inequalities (Hardy, Rellich, Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg, Trudinger, etc.) in the noncommutative settings of nilpotent groups: Heisenberg group, stratified groups, graded groups, and general homogeneous groups.

Some of the history and some “notes of our observations” on this topic can be found in the forthcoming book “Hardy Inequalities on Homogeneous Groups” which is dedicated to 100 years of Hardy inequalities (1918-2018).

Here we give only very few examples.

Theorem. (Hardy inequalities on homogeneous groups) Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q,$ and let $|\cdot|$ be any homogeneous quasi-norm on $\mathbb{G}$ . Let $\mathcal{R}=\frac{d}{d|x|}$ be the radial derivative, with the radial direction taken with respect to the quasi-norm $|\cdot|.$

(i) Let $f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ be a complex-valued function. Then we have the following Hardy inequality on homogeneous group $\mathbb{G}:$

$\left \| \frac{f}{|x|} \right\|_{L^p(\mathbb{G})} \leq \frac{p}{Q-p}\|\mathcal{R}f\|_{L^p(\mathbb{G})}, \quad 1<p<Q,$

when the constant $\frac{p}{Q-p}$ is sharp. Moreover, the equality above is attained if and only if $f =0$

(ii) For a real valued function $f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ and with the notations

$u:=u(x)= - \frac{p}{Q-p}\mathcal{R}f(x),$

$v:= v(x)= \frac{f(x)}{|x|},$

we have the exact form of the relainder:

$\|u\|^p_{L^p(\mathbb{G})} - \|v\|^p_{L^p(\mathbb{G})}= p \int_{\mathbb{G}} I_p(v,u)|v-u|^2dx,$

where

$I_p(h,g)=(p-1)\int_0^1 |\xi h +(1-\xi)g|^{p-2}\xi d\xi.$

(iii) For $Q\geq 3$ , for all complex-valued functions $f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ we have

$\| \mathcal{R} f\|^2_{L^p(\mathbb{G})} = \left( \frac{Q-2}{2} \right)^2 \left\| \frac{f}{|x|}\right\|^2_{L^2(\mathbb{G})} + \left\| \mathcal{R}f + \frac{Q-2}{2} \frac{f}{|x|}\right\|^2_{L^2(\mathbb{G})},$

that is, when $p=2$ , Part (ii) holds for complex-valued functions as well.

Of course, dropping the nonnegative remainder terms in (ii) and (iii), we obtain (i).

More explanations and details can be found in

Ruzhansky M., Suragan D., Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math., 317 (2017), 799-822. download open access, also on arxiv

Durvudkhan Suragan

Michael Ruzhansky

There are other papers on different aspects of such analysis that can be downloaded here.

Ruzhansky M., Suragan D., Yessirkegenov N Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability and superweights for $L^{p}$ -weighted Hardy inequalities, Trans. Amer. Math. Soc. Ser. B, 5 (2018), 32-62. offprint (open access)

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 $p+q\geq r$ , $\delta\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ 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
$\||x|^{c}f\|_{L^{r}(\mathbb{R}^{n})} \leq \left|\frac{p}{n-p(1-a)}\right|^{\delta} \left\||x|^{a}\nabla f\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})}$
for $n\neq p(1-a)$ , where the constant $\left|\frac{p}{n-p(1-a)}\right|^{\delta}$ is sharp for $p=q$ with $a-b=1$ or $p\neq q$ with $p(1-a)+bq\neq0$ .
In the critical case $n=p(1-a)$ we have
$\left\||x|^{c}f\right\|_{L^{r}(\mathbb{R}^{n})} \leq p^{\delta} \left\||x|^{a}\log|x|\nabla f\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})}.$
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. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. 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 $\beta$ .
There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

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