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On the - boundedness of operators
Authors:Stefano Meda   Peter Sjö  gren   Maria Vallarino
Affiliation:Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano--Bicocca, Via Cozzi, 53, 20125 Milano, Italy ; Department of Mathematical Sciences, University of Gothenburg, SE-412 96 Göteborg, Sweden; and Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden ; Laboratoire MAPMO UMR 6628, Fédération Denis Poisson, Université d'Orléans, UFR Sciences, Bâtiment de mathématiques -- Route de Chartres, B.P. 6759 -- 45067 Orléans cedex 2, France
Abstract:We prove that if $ q$ is in $ (1,infty)$, $ Y$ is a Banach space, and $ T$ is a linear operator defined on the space of finite linear combinations of $ (1,q)$-atoms in $ mathbb{R}^n$ with the property that

$displaystyle sup{Vert{Ta}Vert {Y}: hbox{$a$ is a $(1,q)$-atom} } < infty, $

then $ T$ admits a (unique) continuous extension to a bounded linear operator from $ H^1({mathbb{R}^n})$ to $ Y$. We show that the same is true if we replace $ (1,q)$-atoms by continuous $ (1,infty)$-atoms. This is known to be false for $ (1,infty)$-atoms.

Keywords:BMO   atomic Hardy space   extension of operators.
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