On non-uniqueness sets for spaces of holomorphic functions

B. N. Khabibullin ; F. B. Khabibullin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2016), pp. 108-115.

Résumé

Problems of description of zero subsequences for weight spaces of holomorphic functions are reduced, according to a general scheme, to solving certain problems in weight classes of subharmonic functions. Let $D$ be a domain in the complex plane $\mathbb C$. We associate with every at most countable sequence $\Lambda = \{\lambda_k\}_{k=1,2, \dots} \subset D$, without accumulation points in $D$, the counting measure $n_{\Lambda}(S) := \sum_{\lambda_k\in S} 1$. We denote by $\mathrm{Hol} (D)$ the vector space of all holomorphic functions in $D$. For $0\neq f\in \mathrm{Hol} (D)$, denote by $\mathrm{Zero}_f$ zero sequence of $f$ with account of multiplicities. A sequence $\Lambda\subset D$ is called the non-uniqueness sequence for a subspace $H\subset \mathrm{Hol} (D)$, if there exists a nonzero function $f\in H$ such that $\Lambda \subset \mathrm{Zero}_f$, i. e. $n_\Lambda (\lambda)\leq n_{\mathrm{Zero}_f}(\lambda)$ for all $\lambda \in D$. We denote by $\mathrm{sbh} (D)$ the convex cone of all subharmonic functions in $D\subset \mathbb{C}$. For $-\infty\not\equiv s\in \mathrm{sbh} (D)$ we denote by $\nu_s$ the Riesz measure of $s$. A Borel positive measure $\nu$ is called the submeasure for a subset $S\subset \mathrm{sbh} (D)$, if there exists a function $s\in S$, $s\not\equiv -\infty$, with the Riesz measure $\nu_s\geq \nu$ on $D$. For a (weight) function $M\colon D\to [-\infty,+\infty]$ we define the weight classes $\mathrm{sbh}(D;M]:=\{s \in \mathrm{sbh} (D) \colon s\leq M +\mathrm{const} \; \text{on } D \}$ and $\mathrm{Hol}(D;\exp M]:=\{f\in \mathrm{Hol} (D)\colon |f|\leq \mathrm{const} \cdot \exp M \; \text{on } D \}$, where $\mathrm{const}$ is a constant. Let $S$ be a subset of the extended complex plane $\mathbb{C}_{\infty}:=\mathbb{C}\cup \{\infty\}$. Denote by $\mathrm{clos} S$ and $\mathrm{bd} S$ the closure and the boundary of $S$ in $\mathbb{C}_{\infty}$ resp. Let $\mathrm{dist} (\cdot , \cdot)$ be the Euclidean distance between two objects (points or subsets) in $\mathbb{C}$. Let $d\colon D\to (0,1]$ be a continuous function such that $0$, $z\in D$. We will juxtapose to a weight function $N \colon D \to [-\infty,+\infty]$ its average value of $N$ over the disk $\{z'\in \mathbb{C} \colon |z'-z|$: \begin{equation*} B (z,r;N):=\frac{1}{\pi r^2}\int_0^{2\pi}\int_0^r N(z+te^{i\theta}) t\, d t \,d \theta, \end{equation*} and some its “lifting” $N^{\uparrow}\colon D\to [-\infty,+\infty]$ so that $$ \begin{aligned} N^{\uparrow}(z)= B(z,d(z);N)+\ln\frac{1}{d(z)}, \quad \text{if} \; \mathbb{C}_{\infty}\setminus \mathrm{clos} D\neq \varnothing; \\ ^{\uparrow}(z):= B\Bigl(z,\frac{1}{(1+|z|)^P};N\Bigr), \quad \text{if} \; D=\mathbb{C}, \end{aligned} $$ where $P\geq 0$ is an arbitrary fixed number. Theorem 1. Let $N,M, M-N\in \mathrm{sbh} (D)$, $N,M\neq \boldsymbol{-\infty}$, and $\Lambda$ be a sequence in $D$. If $\Lambda$ is the non-uniqueness sequence for $\mathrm{Hol}(D;\exp N]$, then $n_\Lambda+\nu_{M-N}$ is submeasure for $\mathrm{sbh}(D;M]$. Conversely, if $n_\Lambda+\nu_{M-N}$ is a submeasure for $\mathrm{sbh}(D;M]$ and $N$ is a continuous function on $D$, then $\Lambda$ is a non-uniqueness sequence for $\mathrm{Hol}(D;\exp N^{\uparrow}]$ with a suitable lifting $N^{\uparrow}$ (see above cases $D=\mathbb{C}$ with an arbitrary fixed $P\geq 0$ and $\mathbb{C}_{\infty}\setminus \mathrm{clos} D\neq \varnothing$). We also consider an important special case of subharmonic positively homogeneous of degree $\rho>0$ weight functions $N, M$ on $\mathbb{C}$ (see Section 2, Theorem 2).

Détail

Keywords: holomorphic function, zero sequence, subharmonic function, Riesz measure, non-uniqueness sequence.

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