#### Vol. 9, No. 8, 2016

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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 $BHT\otimes \Pi$ between the bilinear Hilbert transform $BHT$ and a paraproduct $\Pi$ satisfies the same ${L}^{p}$ 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 ${L}^{2}$ exponents” the corresponding vector-valued $\stackrel{⃗}{}BHT$ satisfies (again) the same ${L}^{p}$ 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 ${L}^{p}$ 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