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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 $R^n$.

Let $D$ be a Hermitian symmetric space of tube type, $S$ its Silov boundary and $G$ the neutral component of the group of bi-holomorphic diffeomorphisms of $D$. Our main interest is in studying the action of $G$ on $S^3=S\times S\times S$. Sections 1 and 2 are part of a joint work with B. Ørsted (see [4]). In Section 1, as a pedagogical introduction, we study the case where $D$ is the unit disc and $S$ is the circle. This is a fairly elementary and explicit case, where one can easily get a flavour of the more general results. In Section...

To each complex number $\lambda$ is associated a representation ${\pi }_{\lambda }$ of the conformal group $S{O}_0(1,n)$ on ${𝒞}^{\infty }\left(S^{n-1}\right)$ (spherical principal series). For three values ${\lambda }_1,{\lambda }_2,{\lambda }_3$, we construct a trilinear form on ${𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)$, which is invariant by ${\pi }_{{\lambda }_1}\otimes {\pi }_{{\lambda }_2}\otimes {\pi }_{{\lambda }_3}$. The trilinear form, first defined for $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ in an open set of ${ℂ}^3$ is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.