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2016 Multiple vector-valued inequalities via the helicoidal method
Cristina Benea, Camil Muscalu
Anal. PDE 9(8): 1931-1988 (2016). DOI: 10.2140/apde.2016.9.1931


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 BHTΠ between the bilinear Hilbert transform BHT and a paraproduct Π satisfies the same Lp estimates as the BHT itself, solving completely a problem introduced by Muscalu et al. (Acta Math. 193:2 (2004), 269–296). Then, we prove that for “locally L2 exponents” the corresponding vector-valued BHT satisfies (again) the same Lp estimates as the BHT 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 Lp spaces, answering a question of Kenig in nonlinear dispersive PDE.


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Cristina Benea. Camil Muscalu. "Multiple vector-valued inequalities via the helicoidal method." Anal. PDE 9 (8) 1931 - 1988, 2016.


Received: 20 January 2016; Revised: 12 June 2016; Accepted: 12 July 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1361.42005
MathSciNet: MR3599522
Digital Object Identifier: 10.2140/apde.2016.9.1931

Primary: 42A45 , 42B15 , 42B25 , 42B37

Keywords: AKNS systems , Leibniz rule , tensor products in mixed norms , vector-valued estimates for singular and multilinear operators

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.9 • No. 8 • 2016
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