Multiple Vector Valued Inequalities via the Helicoidal Method

TL;DR

This paper introduces the helicoidal method for vector-valued harmonic analysis estimates, proving new results on tensor products, vector-valued inequalities, and a bi-parameter Leibniz rule, advancing understanding in harmonic analysis and PDEs.

Contribution

The paper develops the helicoidal method, enabling new vector-valued inequalities and solving longstanding open problems in harmonic analysis and PDEs.

Findings

Tensor product $BHT \otimes \Pi$ satisfies same $L^p$ estimates as $BHT$

Vector-valued $\overrightarrow{BHT}$ satisfies $L^p$ estimates for new exponents

Established a bi-parameter Leibniz rule in mixed norm $L^p$ spaces

Abstract

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.