Multiple vector-valued inequalities via the helicoidal method

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.