Densities and harmonic measure. (English) Zbl 1120.31001

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 67-76 (2006).
The author surveys several notions of densities and its relation to some classical problems in function theory. It is shown how some of such densities can be computed using precise estimates of the harmonic measure in certain domains. Despite the fact that no new results are presented, it is a very nice article where the author “walks” along zero sequences, interpolating sequences and sampling sequences for different classes of holomorphic functions analyzing the appropriate density to study these topics in different function spaces. So, for the Paley-Wiener PW\(_{\tau}\) space of the entire functions of exponential type smaller or less than \(\tau\), in a series of deep and very celebrated papers, Beurling and Malliavin gave an almost complete description of the uniqueness sets (or zero sets) for PW-spaces by introducing the so-called Beurling-Malliavin density. By the way, sampling sequences for PW-spaces can be described in terms of the Beurling-Nyquist density [see A. Beurling, Volume 1: Complex analysis. Volume 2: Harmonic analysis. Boston: Birkhäuser (1989; Zbl 0732.01042)].
In [K. Seip Invent. Math. 113, 21–39 (1993; Zbl 0789.30025)] the so-called Seip densities were introduced to characterize interpolating sequences and sampling sequences in Bergman spaces. Previously, B. Korenblum [Acta Math. 135, 187–219 (1975; Zbl 0323.30030)] used his density to study the uniqueness sets in Bergman spaces. A more elementary proof of this Korenblum result was given by J. Bruna and X. Massaneda [J. Anal. Math. 66, 217–252 (1995; Zbl 0858.32009)] using some estimates of the harmonic measure that allows them to work in higher dimensions.
Weighted densities, introduced by N. Marco, X. Massaneda and J. Ortega-Cerdá [Geom. Funct. Anal. 13, 862–914 (2003; Zbl 1097.30041)] are also covered. These are the natural ones to describe interpolating and sampling sequences in the space \(\mathcal{F}_{\varphi}\) of all entire functions \(f\) such that \(fe^{-\varphi} \in L^{\infty}(\mathbb{C})\) where \(\varphi\) is subharmonic in \(\mathbb{C}\). Some useful ideas to consider this kind of problems in Riemann surfaces are also pointed out.
In short, a very concise survey that I would recommend to specialists and young researchers in the topic.
For the entire collection see [Zbl 1102.31001].


31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
32A36 Bergman spaces of functions in several complex variables
30C85 Capacity and harmonic measure in the complex plane
30F15 Harmonic functions on Riemann surfaces