Vector-valued inequalities with application to bi-parameter problems in linear and bi-linear settings

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 Lp bounds and the tensor product of two bilinear Hilbert transforms is unbounded in any range. In this work we answer a question they raised by proving L p-bounds of the bilinear Hilbert transform tensored with a paraproduct. Our method relies on new vector-valued estimates for a family of bilinear Hilbert transforms. Using the same method we also give new proofs for a few known classical vector-valued inequalities without relying on the full strength of weighted theory. We also give new proofs of estimates for operators with bi-parameter structure without using product theory in the linear setting.