# Bergman kernel and oscillation theory of plurisubharmonic functions

## TL;DR

This paper investigates the oscillation properties of plurisubharmonic functions and their implications for Bergman kernel approximation, providing new estimates, identities, and extensions in complex analysis.

## Contribution

It introduces local and global BUO properties for plurisubharmonic functions, derives a dimension-free BUO estimate for complex polynomials, and offers a novel approximation formula for the Bergman kernel.

## Key findings

- Plurisubharmonic functions are locally BUO with respect to finite type polydiscs.
- Functions in the Lelong class are globally BUO for all polydiscs.
- A new asymptotic identity for the Bergman kernel improves extension theorems.

## Abstract

Based on Harnack's inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic identity for the Bergman kernel from Berndtsson's complex Brunn--Minkowski theory, which also yields a slightly better version of the sharp Ohsawa--Takegoshi extension theorem in some special cases.