On the $H^1$–$L^1$ boundedness of operators
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- by Stefano Meda, Peter Sjögren and Maria Vallarino PDF
- Proc. Amer. Math. Soc. 136 (2008), 2921-2931 Request permission
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 \[ \sup \left \{ \Vert Ta \Vert Y: \text {$a$ is a $(1,q)$-atom} \right \} < \infty , \] 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.References
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Additional Information
- Stefano Meda
- Affiliation: Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano–Bicocca, Via Cozzi, 53, 20125 Milano, Italy
- Email: stefano.meda@unimib.it
- Peter Sjögren
- Affiliation: 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
- Email: peters@math.chalmers.se
- Maria Vallarino
- Affiliation: 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
- Email: maria.vallarino@unimib.it
- Received by editor(s): June 18, 2007
- Published electronically: April 3, 2008
- Additional Notes: This work was partially supported by the Progetto Cofinanziat “Analisi Armonica”.
- Communicated by: Andreas Seeger
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2921-2931
- MSC (2000): Primary 42B30, 46A22
- DOI: https://doi.org/10.1090/S0002-9939-08-09365-9
- MathSciNet review: 2399059