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 between the bilinear Hilbert transform and a paraproduct satisfies the same estimates as the itself, solving completely a problem introduced by Muscalu et al. (Acta Math. 193:2 (2004), 269–296). Then, we prove that for “locally exponents” the corresponding vector-valued satisfies (again) the same estimates as the 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 spaces, answering a question of Kenig in nonlinear dispersive PDE.
"Multiple vector-valued inequalities via the helicoidal method." Anal. PDE 9 (8) 1931 - 1988, 2016. https://doi.org/10.2140/apde.2016.9.1931