Bellman function and bilinear embedding theorem for Schrödinger-type operators

Dragičević, Oliver; Volberg, Alexander

doi:10.1016/j.crma.2009.02.004

Harmonic Analysis/Functional Analysis

Bellman function and bilinear embedding theorem for Schrödinger-type operators

Presented by: Gilles Pisier

Dragičević, Oliver ¹; Volberg, Alexander ^2,³

¹ Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
³ School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK

Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 537-540.

Abstract
Résumé

We discuss bilinear embedding theorems for a certain class of Schrödinger operators on $L^{p}$ . The obtained estimates are dimension-free and linear in p. We outline a uniform proof of the theorem which relies on establishing three crucial properties of the concrete Bellman function we consider.

On considère un théorème de plongement bilinéaire pour une classe des opérateurs de Schrödinger sur $L^{p}$ . Le résultat ne depend pas de dimension et il est p-linéaire. On fait une esquisse de la démonstration basée sur trois observations concernant la fonction de Bellman spécifique.

Received: 2009-01-26
Accepted: 2009-01-27
Published online: 2009-03-13

DOI: 10.1016/j.crma.2009.02.004

Author's affiliations:

Dragičević, Oliver ¹; Volberg, Alexander ^{2, 3}

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     title = {Bellman function and bilinear embedding theorem for {Schr\"odinger-type} operators},
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Dragičević, Oliver; Volberg, Alexander. Bellman function and bilinear embedding theorem for Schrödinger-type operators. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 537-540. doi : 10.1016/j.crma.2009.02.004. http://www.numdam.org/articles/10.1016/j.crma.2009.02.004/

References
Cited by

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[3] Dragičević, O.; Volberg, A. Bellman functions and dimensionless estimates of Littlewood–Paley type, J. Oper. Theory, Volume 56 (2006) no. 1, pp. 167-198

[4] Dragičević, O.; Volberg, A. Linear dimension-free estimates for the Hermite–Riesz transforms | arXiv

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[6] Nazarov, F.; Treil, S. The Hunt for a Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis, St. Petersburg Math. J., Volume 8 (1997) no. 5, pp. 721-824

[7] Simon, B. Kato's inequality and the comparison of semigroups, J. Funct. Anal., Volume 32 (1979) no. 1, pp. 97-101

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