Hardy inequality on the tree and bi-tree and some (unexpected) combinatorial properties of planar measures

We consider embeddings of various spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc. These problems can be often moved to a discrete setting, by considering weighted Dirichlet spaces on (poly-) trees and weighted dyadic multi-parameter Hardy operator. We find the necessary and sufficient condition for this operator to be bounded 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.