  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.
