Unexpected combinatorial property of all positive measures on the unit square and unit cube

Paraproducts are building blocks of many singular integral operators and the main instrument in proving “Leibniz rule” for fractional derivatives (Kato–Ponce). Also 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. The latter problem (without loss of information) can be often reduced to boundedness of weighted dyadic multi-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—it is a certain combinatorial property of all measures in dimension 2 and 3—and seemingly goes against the well known difference between box and Chang–Fefferman condition that was given by Carleson quilts example of 1974.