Carleson embedding theorem is a building block for many singular integral operators and the main instrument in proving “Leibniz rule” for fractional derivatives (Kato–Ponce, Kenig). It is also an essential step in all known “corona theorems’’. Multi-parameter embedding is a tool to prove more complicated Leibniz rules that are also widely used in well-posedness questions for various PDEs. Alternatively, multi-parameter embedding appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc.

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 (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. I will present results obtained jointly by Arcozzi, Holmes, Mozolyako, Psaromiligkos, Zorin-Kranich and myself.