Bilinear Calder\'on-Zygmund theory on product spaces

Abstract

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$ estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like $[b_1,T]_1$ and $[b_2, [b_1, T]_1]_2$, where $b_1$ and $b_2$ are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.