GROWTH OF SIBONY METRIC AND BERGMAN KERNEL FOR DOMAINS WITH LOW REGULARITY

Nikolai Nikolov (1) , Pascal J. Thomas (2)

1 Institute of Mathematics and Informatics
2 IMT - Institut de Mathématiques de Toulouse UMR5219

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with C 1 boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general C 1 boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the C 2 case, is extended to bounded pseudoconvex domains with Lipschitz boundaries. 1. Results Let D be a domain in C n , z ∈ D, X ∈ C n. Define the Bergman kernel, the Azukawa metric, and the Sibony metric, respectively, as follows (see e.g. [JP]): K D (z) := sup{|f (z)| 2 : f ∈ O(D), ||f || L 2 (D) ≤ 1}; A D (z; X) := lim sup λ→0 exp g D (z, z + λX) |λ| , where g D (z, w) := sup{u(w) : u ∈ PSH(D), u < 0, u < log ||·−z||+C} is the pluricomplex Green function of D with pole at z; S D (z; X) := sup v [L v (z; X)] 1/2 , where L v is the Levi form of v, and the supremum is taken over all functions v : D → [0, 1) such that v(z) = 0, log v is plurisubharmonic on D, and v is of class C 2 near z. Let M D ∈ {A D , S D } and V M D (z) be the volume of the indicatrix I M D (z) := {X ∈ C n : M D (z; X) < 1}. 2020 Mathematics Subject Classification. 32F45.

Mots clés: Suita conjecture; Bergman kernel; Azukawa and Sibony metrics

Domaines: Variables complexes [math.CV]

Nikolai Nikolov, Pascal J. Thomas. GROWTH OF SIBONY METRIC AND BERGMAN KERNEL FOR DOMAINS WITH LOW REGULARITY. Journal of Mathematical Analysis and Applications, 2021, 499 (1), pp.125018. ⟨10.1016/j.jmaa.2021.125018⟩. ⟨hal-02627955⟩