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Multiple vector-valued
inequalities via the helicoidal method
Cristina Benea and Camil Muscalu |
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Vol. 9 (2016), No. 8, 1931–1988
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Abstract |
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We develop a new method of proving vector-valued estimates in harmonic analysis, which we call “the helicoidal method”. As a consequence of it, we are able to give affirmative answers to several questions that have been circulating for some time. In particular, we show that the tensor product between the bilinear Hilbert transform and a paraproduct satisfies the same estimates as the itself, solving completely a problem introduced by Muscalu et al. (Acta Math. 193:2 (2004), 269–296). Then, we prove that for “locally exponents” the corresponding vector-valued satisfies (again) the same estimates as the itself. Before the present work there was not even a single example of such exponents. Finally, we prove a biparameter Leibniz rule in mixed norm spaces, answering a question of Kenig in nonlinear dispersive PDE. |
Keywords
vector-valued estimates for singular and multilinear
operators, tensor products in mixed norms, Leibniz rule,
AKNS systems
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Mathematical Subject Classification 2010
Primary: 42A45, 42B15, 42B25, 42B37
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Milestones
Received: 20 January 2016
Revised: 12 June 2016
Accepted: 12 July 2016
Published: 11 December 2016
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Authors |
| Cristina Benea | |
| Université de Nantes Laboratoire de Mathématiques Nantes 44322 France |
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| Camil Muscalu | |
| Department of Mathematics Cornell University Ithaca, NY 14853 United States |
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