Growth of Sibony metric and Bergman kernel for domains with low regularity

摘要

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. (C) 2021 Elsevier Inc. All rights reserved.