Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 8 (2016), 1931-1988.

Multiple vector-valued inequalities via the helicoidal method

Cristina Benea and Camil Muscalu

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Abstract

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.

Article information

Source
Anal. PDE, Volume 9, Number 8 (2016), 1931-1988.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843379

Digital Object Identifier
doi:10.2140/apde.2016.9.1931

Mathematical Reviews number (MathSciNet)
MR3599522

Zentralblatt MATH identifier
1361.42005

Subjects
Primary: 42A45: Multipliers 42B15: Multipliers 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]

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

Citation

Benea, Cristina; Muscalu, Camil. Multiple vector-valued inequalities via the helicoidal method. Anal. PDE 9 (2016), no. 8, 1931--1988. doi:10.2140/apde.2016.9.1931. https://projecteuclid.org/euclid.apde/1510843379


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