Bilinear Calderón--Zygmund theory on product spaces
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We develop a wide general theory of bilinear bi-parameter singular integrals $T$. This includes general Calder\'on--Zygmund type principles in the bilinear bi-parameter setting: easier bounds, like estimates in the Banach range, imply boundedness in the full bilinear range $L^p \times L^q \to L^r$, $1/p + 1/q = 1/r$, $1 < p,q \le \infty$, $1/2 < r < \infty$, weighted estimates, mixed-norm estimates, and so on. We build this Calder\'on--Zygmund theory using the very useful perspective of dyadic representation theorems that hold under testing conditions. New weighted estimates are developed and used effectively throughout. We also develop commutator decompositions and use them to 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 commutator method contains several novelties -- one is that the new method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also quickly show commutator lower bounds by using and developing the recent median method.