A lower bound for the Bergman kernel and the Bourgain-Milman inequality. (English) Zbl 1321.32003
Klartag, Bo’az (ed.) et al., Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 2011–2013. Cham: Springer (ISBN 978-3-319-09476-2/pbk; 978-3-319-09477-9/ebook). Lecture Notes in Mathematics 2116, 53-63 (2014).
The main result of the paper under review is the following. If \(\Omega\) is a pseudoconvex domain in \(\mathbb C^n\), then for \(w\in\Omega\) and \(a\geq0\) we have
\[
K_{\Omega}(w)\geq\frac{1}{e^{2na}\lambda_{2n}(\{G_{\Omega,w}<-a\})},
\]
where
\[
K_{\Omega}(w):=\sup\bigg\{|f(w)|^2:f\in\mathcal O(\Omega),\;\int_{\Omega}|f|^2\,d\lambda_{2n}\leq1\bigg\}
\]
is the Bergman kernel and
\[
G_{\Omega,w}:=\sup\Big\{u\in\mathcal P\mathcal S\mathcal H(\Omega):\limsup_{z\to w}(u(z)-\log|z-w|<\infty),\;u<0\Big\}
\]
is the pluricomplex Green function.
The above estimate is nontrivial already for \(n=1\), since it gives the inequality conjectured by N. Suita in [Arch. Ration. Mech. Anal. 46, 212–217 (1972; Zbl 0245.30014)] and proved by Z. Błocki in [Invent. Math. 193, No. 1, 149–158 (2013; Zbl 1282.32014)].
The above estimate together with Lempert’s theory gives the following result. If \(\Omega\) is a convex domain in \(\mathbb C^n\), then \[ K_{\Omega}(w)\geq\frac{1}{\lambda_{2n}(I_{\Omega}(w))},\quad w\in\Omega, \] where \[ I_{\Omega}(w):=\big\{\varphi'(0):\varphi\in\mathcal O(\mathbb D,\Omega),\;\varphi(0)=w\big\} \] is the Kobayashi indicatrix (here \(\mathbb D\) denotes the unit disc). One can use the above lower bound to simplify F. Nazarov’s approach in [Lect. Notes Math. 2050, 335–343 (2012; Zbl 1291.52014)] to the Bourgain-Milman inequality.
For the entire collection see [Zbl 1300.46002].
The above estimate is nontrivial already for \(n=1\), since it gives the inequality conjectured by N. Suita in [Arch. Ration. Mech. Anal. 46, 212–217 (1972; Zbl 0245.30014)] and proved by Z. Błocki in [Invent. Math. 193, No. 1, 149–158 (2013; Zbl 1282.32014)].
The above estimate together with Lempert’s theory gives the following result. If \(\Omega\) is a convex domain in \(\mathbb C^n\), then \[ K_{\Omega}(w)\geq\frac{1}{\lambda_{2n}(I_{\Omega}(w))},\quad w\in\Omega, \] where \[ I_{\Omega}(w):=\big\{\varphi'(0):\varphi\in\mathcal O(\mathbb D,\Omega),\;\varphi(0)=w\big\} \] is the Kobayashi indicatrix (here \(\mathbb D\) denotes the unit disc). One can use the above lower bound to simplify F. Nazarov’s approach in [Lect. Notes Math. 2050, 335–343 (2012; Zbl 1291.52014)] to the Bourgain-Milman inequality.
For the entire collection see [Zbl 1300.46002].
Reviewer: Paweł Zapałowski (Kraków)
MSC:
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
Keywords:
Bergman kernel; pluricomplex Green function; logarithmic capacity; Suita conjecture; Bourgain-Milman inequalityReferences:
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