Dans cet article, nous étudions les ensembles d’unicité pour le groupe $\mathrm{Aut}\left(D\right)$ des automorphismes analytiques d’un domaine borné $D$ de ${ℂ}^n$ (resp. pour l’ensemble $H(D,D)$ des fonctions holomorphes de $D$ dans lui-même). Dans les deux cas, nous montrons qu’il existe des ensembles d’unicité contenus dans $D^{n+1}$ ; pour $\mathrm{Aut}\left(D\right)$, nous montrons que ces ensembles d’unicité forment un ensemble dense de $D^{n+1}$, et pour $H(D,D)$, que ce n’est pas le cas en général.

This is a short description of some results obtained by Ewa Damek, Andrzej Hulanicki, Richard Penney and Jacek Zienkiewicz. They belong to harmonic analysis on a class of solvable Lie groups called NA. We apply our results to analysis on classical Siegel domains.

Resnikoff [12] proved that weights of a non trivial singular modular form should be integral multiples of 1/2, 1, 2, 4 for the Siegel, Hermitian, quaternion and exceptional cases, respectively. The θ-functions in the Siegel, Hermitian and quaternion cases provide examples of singular modular forms (Krieg [10]). Shimura [15] obtained a modular form of half-integral weight by analytically continuing an Eisenstein series. Bump and Bailey suggested the possibility of applying an analogue of Shimura's...

We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that the germ...

Holomorphic bundles, with fiber ${ℂ}^n$, defined on open sets in $ℂ$ by locally constant transition automorphisms, are shown to extend to holomorphic bundles on the Riemann sphere. In particular, it allows us to give an example of a non-Stein holomorphic bundle on the unit disc, with polynomial transition automorphisms.