Nevanlinna domains with large boundaries.

*(English)*Zbl 1437.30003In 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:

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:

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:

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 \(\dim_{H}(\partial_{a}G)=\beta\).

- ●
- 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 \(\dim_{H}(\partial G)=\beta\).

Reviewer: Julià Cufí (Bellaterra)

##### 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 |

##### Keywords:

Nevanlinna domain; Hausdorff dimension; model space \(K_\Theta\); univalent rational function##### References:

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