A Counterexample to a Conjecture by Błocki–Zwonek

Åhag, Per; Czyż, Rafał; Lundow, Per-Håkan

doi:10.1080/10586458.2016.1230913

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2018

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A counterexample was found by Fornaess [10]. It was also shown numerically in [1] that the conjecture does not hold in an annulus. Here we will generalize and simplify both results proving the following:…”

Section: Introductionmentioning

confidence: 84%

One dimensional estimates for the Bergman kernel and logarithmic capacity

Błocki

Zwonek

2018

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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 Fornaess. IntroductionFor w ∈ Ω, where Ω is a domain in C, Carleson (1)gives a quantitative version of one of the implications. Hereis the logarithmic capacity of C \ Ω with respect to w,is the (negative) Green function,is the Bergman kernel on the diagonal and ||f || = ||f || L 2 (Ω) .Our first result is the following upper bound for the Bergman kernel:2010 Mathematics Subject Classification. 30H20, 30C85, 32A36.

show abstract

Section: Introductionmentioning

confidence: 84%