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. (ActaMath.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.
Keywords
vector-valued estimates for singular and multilinear
operators, tensor products in mixed norms, Leibniz rule,
AKNS systems
