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

Muscalu, Pipher, Tao and Thiele showed that the tensor product between two one-dimensional paraproducts (also known as a bi-parameter paraproduct) satisfies all the expected L-p bounds. In the same paper, they showed that the tensor product of two bilinear Hilbert transforms is unbounded in any range. They also raised a question concerning the L-p boundedness of the bilinear Hilbert transform tensored 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.