# Multi-parameter paraproducts: Box condition versus Chang--Fefferman condition for weighted estimates

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Paraproducts are building blocks of many singular integral operators and the main instrument in proving ``Leibniz rule" for fractional derivatives (Kato--Ponce). Multi-parameter paraproducts are tools to prove more complicated Leibniz rules that are also widely used in well posedness questions for various PDEs. Alternatively, multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc.

Those Carleson embedding theorems often serve as a first building block for interpolation in complex space and also for corona type results. The embedding of spaces of holomorphic functions on n-polydisc can be reduced to problem (without loss of information) of boundedness of weighted dyadic n-parameter paraproducts.

We find the necessary and sufficient condition for this boundedness in n-parameter case, when n is 1, 2, or 3. The answer is quite unexpected and seemingly goes against the well known difference between box and Chang--Fefferman condition that was given by Carleson quilts example of 1974.