×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baranov, A.; Fedorovskiy, K., Boundary regularity of Nevanlinna domains and univalent functions in model subspaces, Sb. Math., 202, 1723-1740, (2011) · Zbl 1254.30039
[2] Baranov, A.; Fedorovskiy, K., On \(L^1\)-estimates of derivatives of univalent rational functions, J. Anal. Math., 132, 63-80, (2017) · Zbl 1395.30022
[3] Baranov, A.; Carmona, J.; Fedorovskiy, K., Density of certain polynomial modules, J. Approx. Theory, 206, 1-16, (2016) · Zbl 1354.30024
[4] Beliaev, D.; Smirnov, S., Harmonic measure on fractal sets, (Proceedings of the 4th European Congress of Mathematics, (2005), European Mathematical Society: European Mathematical Society Zürich), 41-59 · Zbl 1079.30026
[5] Belov, Yu.; Fedorovskiy, K., Model spaces containing univalent functions, Russian Math. Surveys, 73, 172-174, (2018) · Zbl 1409.30048
[6] Boivin, A.; Gauthier, P.; Paramonov, P., On uniform approximation by n-analytic functions on closed sets in \(\mathbb{C}\), Izv. Math., 68, 447-459, (2004) · Zbl 1069.30076
[7] Carmona, J., Mergelyan’s approximation theorem for rational modules, J. Approx. Theory, 44, 113-126, (1985) · Zbl 0574.30041
[8] Carmona, J.; Fedorovskiy, K., Conformal maps and uniform approximation by polyanalytic functions, (Selected Topics in Complex Analysis. Selected Topics in Complex Analysis, Oper. Theory Adv. Appl., vol. 158, (2005), Birkhäuser: Birkhäuser Basel), 109-130 · Zbl 1089.30035
[9] Carmona, J.; Fedorovskiy, K., On the Dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity, Math. Notes, 83, 31-36, (2008) · Zbl 1175.30004
[10] Carmona, J.; Paramonov, P.; Fedorovskiy, K., On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Sb. Math., 193, 1469-1492, (2002) · Zbl 1053.30028
[11] Davis, P., The Schwarz Function and Its Applications, Carus Math. Monogr., vol. 17, (1974), Math. Ass. of America: Math. Ass. of America Buffalo, NY · Zbl 0293.30001
[12] Dyakonov, K.; Khavinson, D., Smooth functions in star-invariant subspaces, (Recent Advances in Operator-Related Function Theory. Recent Advances in Operator-Related Function Theory, Contemp. Math., vol. 393, (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 59-66 · Zbl 1104.30035
[13] (Ebenfelt, P.; Gustafsson, B.; Khavinson, D.; Putinar, M., Quadrature Domains and Their Applications. Quadrature Domains and Their Applications, Oper. Theory Adv. Appl., vol. 156, (2005), Birkhauser: Birkhauser Basel) · Zbl 1062.30001
[14] Falconer, K., Fractal Geometry. Mathematical Foundations and Applications, (2003), John Wiley & Sons: John Wiley & Sons Hoboken, NJ · Zbl 1060.28005
[15] Fedorovskiy, K., On uniform approximations of functions by n-analytic polynomials on rectifiable contours in \(\mathbb{C}\), Math. Notes, 59, 435-439, (1996) · Zbl 0879.30021
[16] Fedorovskiy, K., Approximation and boundary properties of polyanalytic functions, Proc. Steklov Inst. Math., 235, 4, 251-260, (2001)
[17] Fedorovskiy, K., On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math., 253, 2, 186-194, (2006) · Zbl 1351.30021
[18] Fedorovskiy, K., Two problems on approximation by solutions of elliptic systems on compact sets in the plane, Complex Var. Elliptic Equ., 63, 961-975, (2018) · Zbl 1391.30051
[19] Garnett, J.; Marshall, D., Harmonic Measure, New Math. Monogr., vol. 2, (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1077.31001
[20] Hedenmalm, H.; Shimorin, S., Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J., 127, 341-393, (2005) · Zbl 1075.30005
[21] Kayumov, I., On an inequality for the universal spectrum of integral means, Math. Notes, 84, 137-141, (2008) · Zbl 1219.30006
[22] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Adv. Math., vol. 44, (1995), Cambridge University Press: Cambridge University Press Cambridge
[23] Mazalov, M., An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary, Math. Notes, 62, 524-526, (1997) · Zbl 0919.30037
[24] Mazalov, M., On uniform approximations by bi-analytic functions on arbitrary compact sets in \(\mathbb{C}\), Sb. Math., 195, 687-709, (2004) · Zbl 1075.30020
[25] Mazalov, M., A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations, Sb. Math., 199, 13-44, (2008) · Zbl 1171.41008
[26] Mazalov, M., An example of a nonrectifiable Nevanlinna contour, St. Petersburg Math. J., 27, 625-630, (2016) · Zbl 1345.30006
[27] Mazalov, M., On Nevanlinna domains with fractal boundaries, St. Petersburg Math. J., 29, 777-791, (2018) · Zbl 1400.30012
[28] Mazalov, M.; Paramonov, P.; Fedorovskiy, K., Conditions for the \(C^m\)-approximability of functions by solutions of elliptic equations, Russian Math. Surveys, 67, 1023-1068, (2012) · Zbl 1262.30027
[29] Nikolskiĭ, N., Treatise on the Shift Operator, (1986), Springer-Verlag: Springer-Verlag Berlin
[30] Pommerenke, Ch., Boundary Behaviour of Conformal Maps, (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0762.30001
[31] Sakai, M., Regularity of a boundary having a Schwarz function, Acta Math., 166, 263-297, (1991) · Zbl 0728.30007
[32] Shapiro, H., The Schwarz Function and Its Generalization to Higher Dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, (1992), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0784.30036
[33] Trent, T.; Wang, J., Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc., 81, 62-64, (1981) · Zbl 0472.30032
[34] Verdera, J., On the uniform approximation problem for the square of the Cauchy-Riemann operator, Pacific J. Math., 159, 379-396, (1993) · Zbl 0822.30029
[35] Zaitsev, A., On the uniform approximability of functions by polynomial solutions of second-order elliptic equations on planar compact sets, Izv. Math., 68, 1143-1156, (2004) · Zbl 1073.41016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.