One dimensional estimates for the Bergman kernel and logarithmic capacity

Abstract

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.

Rigidity theorems by capacities and kernels

“…and essentially gives the equality conditions between πv −1 (Ω) and the other quantities in (1.3) for a hyperbolic domain Ω ⊂ C. As a corollary (see Corollary 3.1), we can reprove the main theorem of [12], a rigidity theorem which established the equality conditions of K(z) ≥ v −1 (Ω). Moreover, let K (j) , j ∈ N, denote the higher-order Bergman kernels of a domain Ω ⊂ C. Combining Theorem 1.2 with a result of Błocki and Zwonek in [7], we get the inequalities…”

“…The equality part of Suita's Theorem (Theorem 1.1) implies additionally that P is polar. The case when v(Ω) = ∞ is already known (see [7,Theorem 4]).…”

“…denote the Bergman kernels for higher order derivatives, and set K (0) = K. Błocki and Zwonek [7] established that for z ∈ Ω ⊂ C,…”

Rigidity theorems by the logarithmic capacity

“…Following [3,4], we see from the Cauchy-Schwarz inequality that for almost every t ∈ (−∞, 0), According to the classical isoperimetric inequality (see [5] and the references therein), D t 0 is equivalent (two sets E 1 and E 2 are equivalent if and only if Vol(E 1 ∪ E 2 \ E 1 ∩ E 2 ) = 0 ) to a disc centered at a point a ∈ D with radius re t 0 for some r > 0, and in particular, the involved Cauchy-Schwarz inequality (3.1) attains the equality. This means…”

Generalizations of the Higher Dimensional Suita Conjecture and Its Relation with a Problem of Wiegerinck

“…The generalization of the Suita conjecture requires the definition of the higher order Bergman kernels. The introduced objects as well as analoguous inequalities have been recently presented in the case of one dimensional domains in the paper [8].…”