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 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).

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