We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for$n$>1 which contain a given compact set$K$and at the same time avoid a totally real affine subspace$L$of dimension <$n$, provided that$K$∪$L$is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤$k$<$n$enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension <$n$.

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

On montre que la fonction maximale centrée de Hardy–Littlewood,$M$, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$ , satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute$f$∈$L$^{1}(ℍ^{$n$}), où$A$>0 est une constante indépendante de la dimension$n$.

No abstract uploaded!

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.