The Kobayashi balls of ( $${\mathbb {C}}$$ C -)convex domains

Nikolov, Nikolai; Trybuła, Maria

doi:10.1007/s00605-015-0746-3

Cited by 16 publications

(3 citation statements)

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“…Remark. After the first version of this paper was finished, the author was kindly informed by Nikolai Nikolov that Proposition 7.2 follows also from Proposition 3/(ii) of [45].…”

Section: Appendix: Examples Of Domains With Positive Hyperconvexity Imentioning

confidence: 99%

Bergman kernel and hyperconvexity index

Chen

2017

Analysis & PDE

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Let Ω ⊂ C n be a bounded domain with the hyperconvexity index α(Ω) > 0. Let ̺ be the relative extremal function of a fixed closed ball in Ω and set µ := |̺|(1 + | log |̺||) −1 , ν := |̺|(1 + | log |̺||) n . We obtain the following estimates for the Bergman kernel: (1) For every 0 < α < α(Ω) and 2 ≤ p < 2+ 2α(Ω) 2n−α(Ω) , there exists a constant C > 0 such that Ω | K Ω (·,w) √K Ω (w)

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“…Remark. After the first version of this paper was finished, the author was kindly informed by Nikolai Nikolov that Proposition 7.2 follows also from Proposition 3/(ii) of [45].…”

Section: Appendix: Examples Of Domains With Positive Hyperconvexity Imentioning

confidence: 99%

Bergman kernel and hyperconvexity index

Chen

2017

Analysis & PDE

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“…By Theorem 2 in [37], for convex domains that contain no complex lines, the Kobayashi metric and the Bergman metric are comparable. It follows by [38] Corollary 2 that if Ω is a convex domain with no complex lines, then for every ǫ > 0 there exists constants C 1 and C 2 such that for any a, D(a; w,…”

Section: Bergman Kernel and Metricmentioning

confidence: 99%

Sharp pointwise and uniform estimates for $\bar\partial$

Dong,

Li,

Treuer

2019

Preprint

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We use weighted L 2 -methods initiated by Berndtsson to obtain sharp pointwise estimates for the canonical solution to the equation ∂u = f on bounded convex homogeneous domains and on smoothly bounded strictly convex domains Ω in C n when |f | g is bounded in Ω with the Bergman type metric g. Additionally, we obtain uniform estimates for the canonical solutions on polydiscs, strictly pseudoconvex domains and Cartan classical domains under stronger conditions than the boundedness of |f | g . We provide examples to show our pointwise estimates are sharp. In particular, we show that if Ω is a Cartan classical domain of rank 2 then the maximum blow up order is greater than that of the unit ball, − log δ Ω (z), which was obtained by Berndtsson. For example, for IV(n) with n ≥ 3, the maximum blow up order is δ(z) 1− n 2 because of the contribution from the Bergman kernel.

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“…Theorem 2.4 (Theorem 1, [11]). If D is a bounded convex domain with smooth boundary of finite type, there exists C 11 > 0 such that for any r ∈ (0, 1)…”

Section: Define the Kobayashi Length Of Any Curvementioning

confidence: 99%