We study finite perimeter sets in step 2 Carnot groups. In this way we extend the classical De Giorgi’s theory, developed in Euclidean spaces by De Giorgi, as well as its generalization, considered by the authors, in Heisenberg groups. A structure theorem for sets of finite perimeter and consequently a divergence theorem are obtained. Full proofs of these results, comments and an exhaustive bibliography can be found in our preprint (2001).

Soit $T$ une distribution dissipative sur un groupe de Lie $G$ et soit $\pi$ une représentation fortement continue de $G$ dans un espace de Banach. Supposons $T$ à support compact. Il y a deux façons évidentes de définir un opérateur fermé $\pi \left(T\right)$: une faible et une forte. Le résultat principal de cet article est que l’on obtient le même résultat et que $\pi \left(T\right)$ engendre un semi-groupe fortement continu d’opérateurs.

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Let $H_n$ be the Heisenberg group of dimension $2n+1$. Let ${\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n$ be the partial sub-Laplacians on $H_n$ and $T$ the central element of the Lie algebra of $H_n$. We prove that the kernel of the operator $m\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n,-iT\right)$ is in the Schwartz space $S\left(H_n\right)$ if $m\in S\left({\mathbb{R}}^{n+1}\right)$. We prove also that the kernel of the operator $h\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n\right)$ is in $S\left(H_n\right)$ if $h\in S\left({\mathbb{R}}^n\right)$ and that the kernel of the operator $g\left(\mathcal{L},-iT\right)$ is in $S\left(H_n\right)$ if $g\in S\left({\mathbb{R}}^2\right)$. Here $\mathcal{L}={\mathcal{L}}_1+\mathrm{\dots }+{\mathcal{L}}_n$ is the Kohn-Laplacian on $H_n$.

Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. $\left|{\nabla }_{{\mathbb{H}}^n}d\left(z,t\right)\right|=1$ in the classical sense for all $\left(z,t\right)\in {\mathbb{H}}^n$ with $z\ne 0$, where $d$ is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.

We obtain some matrix elements of basis transformations in a representation space of the unimodular pseudo-orthogonal group. Using these elements, we derive some formulas for special functions.

On étudie diverses convergences des sommes de Riesz des fonctions de puissance pième sommable sur un groupe de Lie compact. On montre que $\frac{n-1}{2}$, où $n$ est la dimension du groupe, est un indice critique pour la classe $L^1$. On donne également un théorème de multiplicateurs qui redonne le résultat classique de Marcinkiewicz pour le tore. On établit enfin un lien entre les multiplicateurs des groupes de Lie compacts et certains multiplicateurs de $R^n$.