Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions

This article is the last in a series of three papers, whose scope is to give new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and Meyer. Here we extend the results of the previous two papers to the polydisc setting. In particular, we solve completely an open question of Coifman from the early eighties.Comment: 25 page

Sparse domination via the helicoidal method

Using exclusively the localized estimates upon which the helicoidal method was built, we show how sparse estimates can also be obtained. This approach yields a sparse domination for multiple vector-valued extensions of operators as well. We illustrate these ideas for an $n$-linear Fourier multiplier whose symbol is singular along a $k$-dimensional subspace of $\Gamma=\lbrace \xi_1+\ldots+\xi_{n+1}=0 \rbrace$, where $k<\dfrac{n+1}{2}$, and for the variational Carleson operator.Comment: 60 page

Multiple Vector Valued Inequalities via the Helicoidal Method

We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to some 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 in a paper of Muscalu, Pipher, Tao and Thiele. Then, we prove that for "locally $L^2$ exponents" the corresponding vector valued $\overrightarrow{BHT}$ 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 bi-parameter Leibniz rule in mixed norm $L^p$ spaces, answering a question of Kenig in nonlinear dispersive PDE.Comment: 56 pages, 7 figure

Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited I. First commutator and generalizations

This article is the first in a series of three papers, whose scope is to give new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and Meyer. Here we treat the case of the first commutator and some of its generalizations.Comment: 23 pages, 1 figur