Multi-parameter Poincaré inequality, multi-parameter Carleson embedding: Box condition versus Chang--Fefferman condition

Abstract: 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.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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