Generalized Hardy spaces on tube domains over cones

Tom 90 / 2001

Gustavo Garrigos Colloquium Mathematicum 90 (2001), 213-251 MSC: 42B30, 32M15. DOI: 10.4064/cm90-2-4


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.


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

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