One dimensional estimates for the Bergman kernel and logarithmic capacity

Błocki, Zbigniew; Zwonek, Włodzimierz

doi:10.1090/proc/13916

One dimensional estimates for the Bergman kernel and logarithmic capacity
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by Zbigniew Błocki and Włodzimierz Zwonek PDF

Proc. Amer. Math. Soc. 146 (2018), 2489-2495 Request permission

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.

References

P. Åhag, R. Czyż, and P. H. Lundow, A counterexample to a conjecture by Błocki-Zwonek, Exp. Math. 27 (2018), no. 1, 119–124. MR 3750933, DOI 10.1080/10586458.2016.1230913
Bo Berndtsson, Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1633–1662 (English, with English and French summaries). MR 2282671
Bo Berndtsson and László Lempert, A proof of the Ohsawa-Takegoshi theorem with sharp estimates, J. Math. Soc. Japan 68 (2016), no. 4, 1461–1472. MR 3564439, DOI 10.2969/jmsj/06841461
Zbigniew Błocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math. 193 (2013), no. 1, 149–158. MR 3069114, DOI 10.1007/s00222-012-0423-2
Zbigniew Blocki, A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 53–63. MR 3364678, DOI 10.1007/978-3-319-09477-9_{4}
Zbigniew Błocki and Włodzimierz Zwonek, Estimates for the Bergman kernel and the multidimensional Suita conjecture, New York J. Math. 21 (2015), 151–161. MR 3318425
Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449, DOI 10.1007/978-1-4612-0817-4
Harold Donnelly and Charles Fefferman, $L^{2}$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), no. 3, 593–618. MR 727705, DOI 10.2307/2006983
John Erik Fornæss, On a conjecture by Blocki and Zwonek, Sci. China Math. 60 (2017), no. 6, 963–966. MR 3647125, DOI 10.1007/s11425-016-5124-7
Fumio Maitani and Hiroshi Yamaguchi, Variation of Bergman metrics on Riemann surfaces, Math. Ann. 330 (2004), no. 3, 477–489. MR 2099190, DOI 10.1007/s00208-004-0556-8
Nobuyuki Suita, Capacities and kernels on Riemann surfaces, Arch. Rational Mech. Anal. 46 (1972), 212–217. MR 367181, DOI 10.1007/BF00252460
Jan J. O. O. Wiegerinck, Domains with finite-dimensional Bergman space, Math. Z. 187 (1984), no. 4, 559–562. MR 760055, DOI 10.1007/BF01174190