Generalized Hardy spaces on tube domains over cones

Gustavo Garrigos¹

¹ Université d'Orléans MAPMO–BP 6759 45067 Orléans Cedex 2, France and Facultad de Ciencias C-XV Universidad Autónoma de Madrid 28049 Madrid, Spain

Colloquium Mathematicum, Tome 90 (2001) no. 2, pp. 213-251

Résumé

We define a class of spaces $H^p_\mu$, $0 p \infty$, of holomorphic functions on the tube, with a norm of Hardy type: $$\|F\|^p_{H^p_\mu}= \sup_{y\in{\mit\Omega}}\int_{{\overline{{\mit\Omega}}}}\int_{{\mathbb R}^n} |F(x+i(y+t))|^p\,dx\,d\mu(t). $$ We allow $\mu$ to be any quasi-invariant measure with respect to a group acting simply transitively on the cone. We show the existence of boundary limits for functions in $H^p_\mu$, and when $p\geq1$, characterize the boundary values as the functions in $L^p_\mu$ satisfying the tangential CR equations. A careful description of the measures $\mu$ when their supports lie on the boundary of the cone is also provided.

Détail

DOI : 10.4064/cm90-2-4

Keywords: define class spaces infty holomorphic functions tube norm hardy type sup mit omega int overline mit omega int mathbb y allow quasi invariant measure respect group acting simply transitively cone existence boundary limits functions geq characterize boundary values functions satisfying tangential equations careful description measures their supports lie boundary cone provided