Regularity of the p-Bergman kernel

We show that the p - Bergman kernel K p ( z ) on a bounded domain Ω is of locally C 1 , 1 for p ≥ 1 .The proof is based on the locally Lipschitz continuity of the off-diagonal p - Bergman kernel K p ( ζ , z ) for fixed ζ ∈ Ω . Global irregularity of K p ( ζ , z ) is presented for some smooth strongl...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 63; no. 2
Main Authors Chen, Bo-Yong, Xiong, Yuanpu
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Springer Nature B.V
Subjects
Analysis
Calculus of Variations and Optimal Control; Optimization
Continuity
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Regularity
Systems Theory
Theoretical
32A26
58E15
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Summary:We show that the p - Bergman kernel K p ( z ) on a bounded domain Ω is of locally C 1 , 1 for p ≥ 1 .The proof is based on the locally Lipschitz continuity of the off-diagonal p - Bergman kernel K p ( ζ , z ) for fixed ζ ∈ Ω . Global irregularity of K p ( ζ , z ) is presented for some smooth strongly pseudoconvex domains when p ≫ 1 . As an application of the local C 1 , 1 - regularity, an upper estimate for the Levi form of log K p ( z ) for 1 < p < 2 is provided. Under the condition that the hyperconvexity index of Ω is positive, we obtain the log-Lipschitz continuity of p ↦ K p ( z ) for 1 ≤ p ≤ 2 .
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-023-02643-y