scholarly journals Localization lemmas for the Bergman metric at plurisubharmonic peak points

2003 ◽  
Vol 171 ◽  
pp. 107-125 ◽  
Author(s):  
Gregor Herbort
Keyword(s):  
Green Function ◽  
Positive Constant ◽  
Bergman Kernel ◽  
Extremal Function ◽  
Additional Assumption ◽  
Bergman Metric ◽  
Pluricomplex Green Function ◽  
Mass Property ◽  
Peak Points ◽  
The Extremal Function

AbstractLet D be a bounded pseudoconvex domain in ℂn and ζ ∈ D. By KD and BD we denote the Bergman kernel and metric of D, respectively. Given a ball B = B(ζ, R), we study the behavior of the ratio KD/KD∩B(w) when w ∈ D ∩ B tends towards ζ. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to 1, as w tends to ζ, under an additional assumption on the pluricomplex Green function D(·, w) of D with pole at w, namely that the diameter of the sublevel sets Aw :={z ∈ D | D(z, w) < −1} tends to zero, as w → ζ. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in [DiOh1], where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set Aw is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

scholarly journals Boundary behavior of the Bergman metric

2002 ◽  
Vol 168 ◽  
pp. 27-40 ◽  
Author(s):  
Bo-Yong Chen
Keyword(s):  
Green Function ◽  
Pseudoconvex Domain ◽  
Boundary Point ◽  
Boundary Behavior ◽  
Sufficient Conditions ◽  
Bergman Metric ◽  
Hölder Continuous ◽  
Pluricomplex Green Function ◽  
Peak Function

AbstractLet Ω be a bounded pseudoconvex domain in Cn. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property (P) (in the sense of Coman) which is characterized by the pluricomplex Green function.

THE PLURICOMPLEX GREEN FUNCTION ON PSEUDOCONVEX DOMAINS WITH A SMOOTH BOUNDARY

2000 ◽  
Vol 11 (04) ◽  
pp. 509-522 ◽  
Author(s):  
GREGOR HERBORT
Keyword(s):  
Green Function ◽  
Boundary Point ◽  
Additional Assumption ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Exceptional Set ◽  
Exhaustion Function ◽  
Pluricomplex Green Function ◽  
Pluripolar Set ◽  
Hyperconvex Domain

In this article we deal with the behavior of the pluricomplex Green function GD(·;w), of a pseudoconvex domain D in [Formula: see text], when the pole tends to a boundary point. In [7], it was shown that, given a boundary point w0 of a hyperconvex domain D, then there is a pluripolar set E⊂D, such that lim sup w→w0 GD(z;w)=0 for z∈D\E. Under an additional assumption on D, that can be viewed as natural, one can avoid the pluripolar exceptional set. Our main result is that on a bounded domain [Formula: see text] that admits a Hoelder continuous plurisubharmonic exhaustion function ρ:D→[-1,0), the pluricomplex Green function GD(·,w) tends to zero uniformly on compact subsets of D, if the pole w tends to a boundary point w0 of D.

scholarly journals Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

2007 ◽  
Vol 185 ◽  
pp. 143-150 ◽  
Author(s):  
Zbigniew Błocki
Keyword(s):  
Green Function ◽  
Bounded Domain ◽  
Bergman Kernel ◽  
Main Tool ◽  
Logarithmic Capacity ◽  
Bergman Metric ◽  
Similar Estimate ◽  
Several Variables ◽  
The Green Function ◽  
Complex Laplacian

AbstractFor a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

scholarly journals Gaussian curvature of the Bergman metric with weighted Bergman kernel on the unit disc

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
Marzena Szajewska
Keyword(s):  
Weight Function ◽  
Bergman Kernel ◽  
Gaussian Curvature ◽  
Unit Disc ◽  
Bergman Metric ◽  
Weighted Bergman Kernel

AbstractIn the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied.

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

scholarly journals The extremal function for partial bipartite tilings

2012 ◽  
Vol 33 (5) ◽  
pp. 807-815 ◽  
Author(s):  
Codruţ Grosu ◽  
Jan Hladký
Keyword(s):  
Extremal Function ◽  
The Extremal Function

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

Sign in / Sign up

Export Citation Format

Share Document