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Bi-parameter Carleson embeddings with product weights

Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.