Multiple Vector Valued Inequalities via the Helicoidal Method

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.

arXiv comment56 pages, 7 figures

References in corpus (6)

• $L^p$ estimates for the Hilbert transforms along a one-variable vector field
• Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems
• Pointwise convergence of vector-valued Fourier series
• The Vector Valued Quartile Operator
• Banach-valued multilinear singular integrals
• Mixed Estimates for Degenerate Multilinear Operators Associated to Simplexes

Cited by in corpus (18)

• Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform
• Domination of multilinear singular integrals by positive sparse forms
• Quantitative estimates and extrapolation for multilinear weight classes
• Weighted mixed weak-type inequalities for multilinear operators
• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• Limited range multilinear extrapolation with applications to the bilinear Hilbert transform
• Bilinear Calderón-Zygmund theory on product spaces
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Banach-valued multilinear singular integrals
• Vector-valued extensions of operators through multilinear limited range extrapolation
• Commutators of bilinear bi-parameter singular integrals
• Genuinely multilinear weighted estimates for singular integrals in product spaces
• Smoothing Properties of Bilinear Operators and Leibniz-Type Rules in Lebesgue and Mixed Lebesgue Spaces
• The non-resonant bilinear Hilbert--Carleson operator
• Multi-parameter flag Leibniz rules of arbitrary complexity in mixed-norm spaces
• The Kato-Ponce Inequality with Polynomial Weights
• A bilinear Rubio de Francia inequality for arbitrary squares
• Real-Variable Characterizations of New Anisotropic Mixed-Norm Hardy Spaces