Zeros of functions in Bergman class

Abstract

[en] These notes are a survey of the results known of zero sets of functions in Bergman space. Characterization of zero sets for functions in Bergman space remains being an open problem, which is closed in the case of $H^p(\mathbb{D})$ spaces, for $0 \leq p \leq \infty$ where $H^0$ corresponds to the Nevanlinna class $\mathcal{N}$, where we have an indistinguishable geometric characterization of zero sets of functions in terms of Blaschke products, as we will see it in Chapter 1.

In Chapter 2 we will introduce the weighted Bergman space $\mathcal{B}_\alpha^p$, which is a parametrization of the usual weighted Bergman space $A_\alpha^p$. Such parametrization let us estimate a function $f \in \mathcal{B}_\alpha^p$ as $$ |f(z)| \leq \frac{C}{\left(1-|z|^2\right)^\alpha}, \quad z \in \mathbb{D} $$ for some $C>0$ constant. Notice that the estimation (1) does not depends on $p$. Chapter 3 is devoted to study the basic properties of zero sets of functions in $\mathcal{B}_\alpha^p$ and a probabilistic model of random zero sets, by Gregory Bomash and apparently initiated by Emile Leblanc.

Furthermore, Chapter 3 will show us that characterization of zero sets of function in Bergman space is a hard problem. It is because by using Blaschke-type products, which involves only the modulus of the zeros, we can obtain necessary conditions that are far from being a sufficient condition or sufficient conditions that are far from being a necessary condition, which is the case of the sharp sufficient condition obtained by Bomash. Moreover, zero sets of the Bergman space $\mathcal{B}_\alpha^p$ are not necessary to be a zero set of a different Bergman space $\mathcal{B}_\gamma^q$, and union of zeros sets of $\mathcal{B}_\alpha^p$ are not necessary a zero set of $\mathcal{B}_\alpha^p$, which contrasts with the case of the spaces $H^p$.

Since working only with the modulus of the zeros is insufficient in order to obtain a characterization of zeros sets of functions in $\mathcal{B}_\alpha^p$ (i.e., a necessary and sufficient condition), in Chapter 4 we will introduce some notions of density, which join with the growth spaces $\mathcal{A}^{-\alpha}$ are the framework considered by Korenblum, who obtained the latest results about characterization of zero sets of functions in Bergman space, whose necessary condition and whose sufficient condition are very close to be a characterization, as we will see it in Chapter 5.