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General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences
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Sums of moduli for holomorphic functions E. Doubtsov St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences |
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Abstract: nformally, we give an answer to the following question: How do the sums To formalize the above question, assume that Given a radial weight $$c w(z) < |f| + |g| < C w(z)$$ for all The main result of the talk gives an explicit description of those radial weights for which the problem is solvable. Also, we have an answer when two functions are replaced by a finite set of holomorphic functions. Similar results hold in several complex variables for circular, strictly convex domains with smooth boundary. About the proofs. 1. The restrictions on the admissible weight functions 2. Constructive part: the desired holomorphic functions are obtained as appropriate lacunary series. Namely, given an admissible weight function Applications. The test functions obtained are useful in the studies of Carleson measures, weighted composition operators, extended Cesaro operators and other concrete linear operators. The talk is based on a joint work with E.Abakumov. |