# One dimensional estimates for the Bergman kernel and logarithmic capacity

## TL;DR

This paper provides quantitative estimates relating the Bergman kernel, logarithmic capacity, and Green function sublevel sets, extending classical results and exploring their geometric implications for complex domains.

## Contribution

It offers new bounds connecting the Bergman kernel and logarithmic capacity, and generalizes examples of non-convexity of Green function sublevel sets.

## Key findings

- Quantitative bounds for the Bergman kernel in terms of logarithmic capacity.
- Counterexamples showing non-convexity of Green function sublevel sets.
- Extension of the Suita conjecture to broader classes of domains.

## Abstract

Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non simply connected domains, generalizing a recent example of Forn\ae ss.