Strange property of positive measures and bi-linear estimates on multi-trees

Carleson embedding theorems often serve as a first building block for interpolation in complex domain, for the theory of Hankel operators and in PDE. The embedding of certain spaces of holomorphic functions on n-polydisc can be reduced (without loss of information) to the boundedness of weighted multi-parameter dyadic Carleson embedding. We find the necessary and sufficient condition for this Carleson embedding in n-parameter case, when n is 1, 2, or 3. The main tool is the harmonic analysis on graphs with cycles. 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. The main tool is an unexpected combinatorial properties of positive measures on cube in dimensions 1, 2, 3.
I will present results obtained jointly by Arcozzi, Holmes, Mozolyako, Psaromiligkos, Zorin-Kranich and myself.