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

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.