Vol. 9, No. 8, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Multiple vector-valued inequalities via the helicoidal method

Cristina Benea and Camil Muscalu

Vol. 9 (2016), No. 8, 1931–1988
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.

Keywords
vector-valued estimates for singular and multilinear operators, tensor products in mixed norms, Leibniz rule, AKNS systems
Mathematical Subject Classification 2010
Primary: 42A45, 42B15, 42B25, 42B37
Milestones
Received: 20 January 2016
Revised: 12 June 2016
Accepted: 12 July 2016
Published: 11 December 2016
Authors
Cristina Benea
Université de Nantes
Laboratoire de Mathématiques
Nantes 44322
France
Camil Muscalu
Department of Mathematics
Cornell University
Ithaca, NY 14853
United States