MULTIPLE VECTOR-VALUED INEQUALITIES VIA THE HELICOIDAL METHOD

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 circle times Pi between the bilinear Hilbert transform BHT and a paraproduct. 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 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 L-p spaces, answering a question of Kenig in nonlinear dispersive PDE.