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Nevanlinna domains with large boundaries. (English) Zbl 1437.30003
In this paper the authors give a complete solution to the following problem posed in the early 2000s: how large, in the sense of dimension, can be the boundaries of Nevanlinna domains?
We recall that a bounded simply connected domain $$G\subset\mathbb{C}$$ is a Nevanlinna domain if there exist two functions $$u,v\in H^{\infty}(G)$$ with $$v\ne 0$$ such that the equality $\bar{z}=\frac{u(z)}{v(z)}$ holds on $$\partial G$$ in the sense of conformal mappings. This means the equality of angular boundary values $\overline{f(\zeta)}= \frac{(u\circ f)(\zeta)} {(v\circ f)(\zeta)}$ for almost all $$\zeta\in \mathbb{T}$$, where $$f$$ is a conformal mapping from $$\mathbb{D}$$ to $$G$$, and $$f(\zeta)$$ is the angular boundary value of $$f\in H^{\infty}(\mathbb{D})$$ at the point $$\zeta\in \mathbb{T}$$.
Nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in $$\mathbb{C}$$ by polynomial solutions of elliptic equations with constant complex coefficients.
The answer to the posed question is the following:
For every $$\beta\in [1,2]$$ there exists a Nevanlinna domain $$G$$ such that $$\dim_{H}(\partial G)=\beta$$, where $$\dim_{H}$$ stands for the Hausdorff dimension of sets.

Constructing Nevanlinna domains with irregular boundaries is a rather difficult and delicate problem. It is of interest to consider a quantitative version of the problem of the existence of Nevanlinna domains with non-rectifiable boundaries. In particular one studies the question of how the length of the boundary of the Nevanlinna domain $$R(\mathbb{D})$$ grows in relation to the degree of the rational function $$R$$.
Given a positive integer $$n$$, let denote by $$\mathcal{R}_{n}$$ the set of all rational functions of degree at most $$n$$ and by $$\mathcal{R}\mathcal{U}_{n}$$ the set of all functions in $$\mathcal{R}_{n}$$ without poles in $$\overline{\mathbb{D}}$$ and univalent on $$\mathbb{D}$$. Finally, let $$\mathcal{R}\mathcal{U}_{n,1}$$ be the set of all functions $$R\in\mathcal{R}\mathcal{U}_{n}$$ such that $$\|R\|_{\infty,\mathbb{T}}\le 1$$. Set $\gamma_{0}=\limsup_{n\to\infty}\sup_{R\in\mathcal{R}\mathcal{U}_{n,1}}\frac{\log\ell(R)}{\log n}\text{ where }\ell(R)=\frac{1}{2\pi}\int_{\mathbb{T}} |R'(\zeta)||d\zeta|.$
With these notations the following result is proved:
For some absolute constant $$\alpha>0$$ and for every $$n\ge 1$$ we have $\alpha\sqrt{n}\le \sup_{R\in\mathcal{R}\mathcal{U}_{n,1}}\ell (R)\le 6\pi\sqrt{n},$ so that $$\gamma_{0}=1/2$$.

The new result here is the lower estimate that is obtained by constructing a snake-like domain $$R(\mathbb{D})$$ with long boundary.
Given a bounded simply connected domain $$G$$ consider the set $$\partial_{a}G\subset \partial G$$ which consists of all points of $$\partial G$$ being accessible from $$G$$ by some curve. It is known that the equality $\partial_{a}G=\{f(\zeta)\cdot \zeta \in \mathcal{F}(f)\}$ holds, where $$f$$ is some conformal mapping from the unit disc $$\mathbb{D}$$ onto $$G$$ and $$\mathcal{F}(f)$$ it is the set of all points $$\zeta\in\mathbb{T}$$ where the angular boundary value $$f(\zeta)$$ exist. The definition of Nevanlinna domains imposes conditions only on the accessible part $$\partial_{a}G$$ of their boundaries. So it is more accurate to pose the question about the existence of Nevanlinna domains with large accessible boundaries.
Let $$f$$ be a map defined on the unit disc by $f(z)=\sum^{\infty}_{n=1}\frac{c_{n}}{1-\bar{a}_{n}z}, \tag{1}$ where $$(a_{n})_{n\ge 1}$$ is some interpolating Blaschke sequence, that is, satisfying the condition $\inf_{n\in \mathbb{N}}\prod^{\infty}_{\begin{subarray}{c} k=1\\ k\ne n\end{subarray}}\left| \frac{a_{n}-a_{k}}{1-a_{n}\bar{a}_{k}}\right|>0,$ and $$(c_{n})_{n\ge 1}$$ is an appropriately chosen sequence of coefficients.
The main result of the paper is the following:
For every $$\beta\in [1,2]$$ there exists a function $$f$$ of the form (1) univalent in $$\mathbb{D}$$ and such that the Nevanlinna domain $$G=f(\mathbb{D})$$ satisfies the property $$\dim_{H}(\partial_{a}G)=\beta$$.
If $$\theta\in H^{\infty}(\mathbb{D})$$ is an inner function and $$H^{2}=H^{2}(\mathbb{D})$$ is the standard Hardy space, define the so-called model space $K_{\theta}:=(\theta H^{2})^{\perp}=H^{2}\ominus \theta H^{2}.$ The function $$f$$ in the main result belongs to the space $$K_{B}$$ for some appropriately chosen Blachke product $$B$$. It is possible to construct similar examples working with univalent functions from the space $$K_{S}$$, where $$S$$ is some singular inner function. The simplest example of such a space $$K_{S}$$ is the Paley-Wiener space $$\mathcal{P}W^{\infty}_{[0,1]}$$, the Fourier image of $$L^{2}[0,1]$$, considered as a space of functions analytic in the upper half-plane $$\mathbb{C}_{+}$$. Then the authors prove the following result:
For every $$\beta\in [1,2]$$ there exists a univalent function $$f$$ belonging to the space $$\mathcal{P}W^{\infty}_{[0,1]}$$ such that the Nevanlinna domain $$G=f(\mathbb{C}_{+})$$ satisfies the property $$\dim_{H}(\partial G)=\beta$$.
##### MSC:
 30C20 Conformal mappings of special domains 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30J99 Function theory on the disc 30E10 Approximation in the complex plane
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