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

Sparse bilinear forms for Bochner Riesz multipliers and applications

We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced

A bilinear Rubio de Francia inequality for arbitrary squares

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane$$\left(f, g \right)\mapsto \left( \sum_{\omega \in \Omega}\left| \int_{\mathbb{R}^2} \hat{f}(\xi) \hat{g}(\eta) \Phi_{\omega}(\xi, \eta) e^{2 \pi i x\left(\xi+\eta \right)} d \xi d \eta\right|^r \right)^{1/r},$$ provided $r>2$. More exactly, we show that the above operator maps $L^p \times L^q \to L^s$ whenever $p, q, s'$ are in the ``local $L^{r'}$" range, i.e. $\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{s'}=1$, $\displaystyle0 \leq \frac{1}{p}, \frac{1}{q} <\frac{1}{r'}$, and $\displaystyle\frac{1}{s'}<\frac{1}{r'}$. Note that we allow for negative values of $s'$, which correspond to quasi-Banach spaces $L^s$