Year: 2025
Type: article
Abstract: For plurisubharmonic $$\varphi $$ φ in a bounded pseudoconvex $$\Omega $$ Ω in $$\mathbb C^n$$ C n , we analyze the lower bound for the Bergman kernel (on the diagonal, with weight $$e^{-\varphi }$$ e - φ ) of the form $$K_{\Omega ,\varphi }\ge e^{\varphi }/C$$ K Ω , φ ≥ e φ / C . By the Ohsawa–Takegoshi extension theorem, there exists such a constant depending on n and the diameter of $$\Omega $$ Ω . We give a proof of this using Hörmander’s $$L^2$$ L 2 estimate for $$\bar{\partial }$$ ∂ ¯ directly. We also show that one can improve the constant to $$C=|\Omega |$$ C = | Ω | if $$n=1$$ n = 1 and to $$C=2\,|\Omega |$$ C = 2 | Ω | if $$\Omega $$ Ω is convex, where $$|\Omega |$$ | Ω | denotes the volume (or area) of $$\Omega $$ Ω . We conjecture that the former holds in arbitrary dimension (it trivially does in the unweighted case).
Author: Zbigniew Błocki
Institution: Jagiellonian University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: Narodowe Centrum Nauki
Award: G8607339911
OpenAlex ID: W4414485064