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One dimensional estimates for the Bergman kernel and logarithmic capacity


Authors: Zbigniew Błocki and Włodzimierz Zwonek
Journal: Proc. Amer. Math. Soc. 146 (2018), 2489-2495
MSC (2010): Primary 30H20, 30C85, 32A36
DOI: https://doi.org/10.1090/proc/13916
Published electronically: February 16, 2018
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Abstract: Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, and the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non-simply connected domains, generalizing a recent example of Fornæss.


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Zbigniew Błocki
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Kraków, Poland
Email: zbigniew.blocki@im.uj.edu.pl

Włodzimierz Zwonek
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Krawów, Poland
Email: wlodzimierz.zwonek@im.uj.edu.pl

DOI: https://doi.org/10.1090/proc/13916
Keywords: Bergman kernel, logarithmic capacity, Suita conjecture
Received by editor(s): March 27, 2017
Received by editor(s) in revised form: July 19, 2017
Published electronically: February 16, 2018
Additional Notes: The first-named author was supported by the Ideas Plus grant no. 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education and the second-named author by the Polish National Science Centre (NCN) Opus grant no. 2015/17/B/ST1/00996
Communicated by: Filippo Bracci
Article copyright: © Copyright 2018 American Mathematical Society

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