Functional inequalities

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations.
G. H. Hardy

This research section is devoted to the exposition of the developments at the intersection of two active fields of mathematics: Hardy inequalities and related analysis, and the noncommutative analysis in the setting of nilpotent Lie groups of different types. This topic can be found in the book “Hardy Inequalities on Homogeneous Groups” which is dedicated to 100 years of Hardy inequalities (1918-2018).

Hardy inequality

G. H. Hardy and Harald Bohr (pictures from Wikipedia)

G. H. Hardy reported Harald Bohr as saying ’all analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove’.

The Hardy inequalities has now been a fascinating subject of continuous research by numerous mathematicians for 100 years. The origin of the Hardy inequality comes from the following interesting fact:
In the theory of integral equations, D. Hilbert found a beautiful fact that the series
$\displaystyle \sum^{\infty}_{m,n=1} \frac{a_m a_n}{m+n}$
with positive entries $a_{n}\geq 0$ is convergent whenever $\sum_{m=1}^{\infty} a_{m}^2$ is convergent.

In a couple years, there were published at least four different proofs of this fact. The original proof was given by H. Weyl, who was a doctoral student of D. Hilbert in 1908 and gave it in his doctoral dissertation, a proof by F. Wiener in 1910, and two proofs by I. Schur in 1911.
However, for G. H. Hardy all these proofs were not enough elementary, so he came up in 1918 with yet another proof which seemed to him ”to lack nothing but simplicity”.

Theorem A. If the series $\sum_{m=1}^{\infty} a_{m}^2$ is convergent and we get $A_n = a_1 + \dots + a_n$ , then also series
$\displaystyle \sum^{\infty}_{n=1} \left( \frac{A_n}{n} \right)^2$
is convergent. Thus, this moment could be considered as the birth of what is now known as Hardy’s inequalities.

The communication of G. H. Hardy and M. Riesz leads to the following generalisation of Hardy’s result
Theorem B. If $p>1$ and $\sum_{m=1}^{\infty} a_{m}^p$ is convergent and $A_n = a_1 + \dots + a_n$ , then also the series
$\displaystyle \sum^{\infty}_{n=1} \left( \frac{A_n}{n} \right)^p$
is convergent. This gave the birth of well-known Lp-Hardy’s inequality.

Frigyes Riesz and Marcel Riesz (picture from Wikipedia)

Also, the exact value of the best constant in the inequality was given by G. H. Hardy as follows
$\displaystyle \int_a^{\infty}\left( \frac{\int_a^{x} f(t)dt }{x}\right)^p dx \leq \left(\frac{p}{p-1}\right)^p \int_a^{\infty} f(x)^p dx,$
where $a>0$ and $f(x)$ is any positive function. Interestingly, G. H. Hardy called his own proof for the best constant ”unnecessarily complicated”, so he published another simpler proof that was ”sent to him by Prof. Schur by letter”. The modern version of the one-dimensional Hardy inequality is often given by
$\displaystyle \int_0^{\infty}\left( \frac{F(x) }{x}\right)^p dx \leq \left(\frac{p}{p-1}\right)^p \int_0^{\infty} |F'(x)|^p dx,$
where $F(x) = \int_a^x f(t)dt$ and $\forall F \in C_0^{\infty}(0,\infty)$ .

In 1934, Leray investigated the well-posedness of the weak solutions of the Navier Stokes equation, by applying a Hardy inequality with partial derivatives
$\displaystyle \int_{\mathbb{R}^3} |\nabla u|^2 dx \geq \frac{1}{4} \int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2}dx, \,\,\, \forall u\in C_0^{\infty}(\mathbb{R}^3).$
This inequality was extended to any dimension $n\geq 3$
$\displaystyle \int_{\mathbb{R}^n} |\nabla u|^2 dx \geq \left(\frac{n-2}{2}\right)^2 \int_{\mathbb{R}^n} \frac{|u|^2}{|x|^2}dx, \,\,\, \forall u\in C_0^{\infty}(\mathbb{R}^n),$
where $(n-2)^2/4$ is the largest optimal constant and never attained, i.e. no minimizer exists.

In 1954 at ICM held in Amsterdam, Franz Rellich firstly presented the following inequality for $n \geq 5$
$\displaystyle \int_{\mathbb{R}^n} |\Delta u|^2 dx \geq \frac{n^2(n-4)^2}{16} \int_{\mathbb{R}^n} \frac{|u|^2}{|x|^4}dx, \,\,\, \forall u\in C_0^{\infty}(\mathbb{R}^n\backslash \{0\}).$
The Rellich inequality has undergone extensive further developments beginning with the L^p version by Davies-Hinz, in particular, being an important tool to study spectrum of biharmonic-type operators.

In 2007, Tertikas-Zographopoulos have discovered the following Hardy-Rellich inequality for $n \geq 5$
$\displaystyle \int_{\mathbb{R}^n} |\Delta u|^2 dx \geq \frac{n^2}{4} \int_{\mathbb{R}^n} \frac{|\nabla u|^2}{|x|^2}dx, \,\,\, \forall u\in C_0^{\infty}(\mathbb{R}^n\backslash \{0\}).$
It turns out that it is a special case of Pitt’s inequality from harmonic analysis and it is also equivalent to the Hardy-Leray inequality for curl-free vector fields.

Hardy Inequalities on Homogeneous Groups

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 $\displaystyle \mathcal{R}=\frac{d}{d|x|}$ be the radial derivative, with the radial direction taken with respect to the quasi-norm $|\cdot|.$
(i) Let $\displaystyle f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ be a complex-valued function. Then we have the following Hardy inequality on homogeneous group $\mathbb{G}:$
$\displaystyle \left \| \frac{f}{|x|} \right\|_{L^p(\mathbb{G})} \leq \frac{p}{Q-p}\|\mathcal{R}f\|_{L^p(\mathbb{G})}, \quad 1<p<Q,$
when the constant $\displaystyle \frac{p}{Q-p}$ is sharp. Moreover, the equality above is attained if and only if $\displaystyle f =0$
(ii) For a real valued function $\displaystyle f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ and with the notations
$\displaystyle u:=u(x)= - \frac{p}{Q-p}\mathcal{R}f(x),$
$\displaystyle v:= v(x)= \frac{f(x)}{|x|},$
we have the exact form of the relainder:
$\displaystyle \|u\|^p_{L^p(\mathbb{G})} - \|v\|^p_{L^p(\mathbb{G})}= p \int_{\mathbb{G}} I_p(v,u)|v-u|^2dx,$
where
$\displaystyle I_p(h,g)=(p-1)\int_0^1 |\xi h +(1-\xi)g|^{p-2}\xi d\xi.$
(iii) For $\displaystyle Q\geq 3$ , for all complex-valued functions $\displaystyle f \in C_0^{\infty}(\mathbb{G} \backslash \{0\})$ we have
$\displaystyle \| \mathcal{R} f\|^2_{L^p(\mathbb{G})} = \left( \frac{Q-2}{2} \right)^2 \left\| \frac{f}{|x|}\right\|^2_{L^2(\mathbb{G})} + \left\| \mathcal{R}f + \frac{Q-2}{2} \frac{f}{|x|}\right\|^2_{L^2(\mathbb{G})},$
that is, when $\displaystyle p=2$ , Part (ii) holds for complex-valued functions as well.
Of course, dropping the nonnegative remainder terms in (ii) and (iii), we obtain (i).