Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems

TL;DR

This paper establishes new vector-valued inequalities for bilinear Hilbert transforms and their tensor products, addressing open questions about boundedness in bi-parameter settings and expanding the understanding of these operators.

Contribution

It introduces novel vector-valued estimates for bilinear Hilbert transforms, enabling analysis of their tensor products with paraproducts and resolving previous boundedness questions.

Findings

Proved boundedness of tensor products involving bilinear Hilbert transforms and paraproducts.

Developed new vector-valued estimates for families of bilinear Hilbert transforms.

Extended the range of $L^p$ bounds for hybrid bilinear operators.

Abstract

Muscalu, Pipher, Tao and Thiele \cite{MPTT} showed that the tensor product between two one dimensional paraproducts (also known as bi-parameter paraproduct) satisfies all the expected $L^p$ bounds. In the same paper they showed that the tensor product between two bilinear Hilbert transforms is unbounded in any range. They also raised the question about $L^p$ boundedness of the bilinear Hilbert transform tensor product with a paraproduct. We answer their question by obtaining a wide range of estimates for this hybrid bilinear operator. Our method relies on new vector valued estimates for a family of bilinear Hilbert transforms.