Model spaces containing univalent functions
Model spaces containing univalent functions
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包含单价函数的模型空间
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
K. Fedorovskiy
中科院分区：
文献类型：
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作者：
Y. Belov;K. Fedorovskiy
where Θ is an arbitrary inner function on D. Such spaces are often called model spaces in view of their role in the functional model of Sz.-Nagy and Foias [1]. In connection with problems of approximating functions by polyanalytic polynomials and with related properties of Nevanlinna domains (see [3]–[6]), the question of describing model spaces KΘ containing bounded univalent functions was posed in [2], § 2.4. In this paper we resolve this question. It is clear that if an inner function Θ has at least one zero a in D, then the corresponding space KΘ contains the bounded univalent function 1/(1−az). Therefore, in what follows we are dealing with singular inner functions only. Recall that every singular inner function S has the form S(z) = Sμ(z) = exp { − ∫ T ζ+z ζ−z dμ(ζ) } , where μ is a positive measure on the unit circle T and is singular with respect to Lebesgue measure. The first example of a singular function S for which the corresponding space KS contains univalent functions was given in [7]. In [6] it was proved that if a measure μ has an atom, then the space KSμ contains bounded univalent functions. Independently, another (implicit) construction of a bounded univalent function in the Paley–Wiener space PW[0,1] (an analogue of the model space generated by the inner function exp { z+1 z−1 } in the upper half-plane) was given by N. A. Shirokov. On the other hand, it was shown in [6] that if KS contains bounded univalent functions, then the measure μ possesses the following property: there exists a Beurling–Carleson set E with μ(E) > 0. A set E ⊂ T is called a Beurling–Carleson set (or a Carelson set, or a set with finite entropy) if ∫ T log dist(ζ, E) dm(ζ) > −∞, where m( · ) denotes Lebesgue measure on T. In particular, m(E) = 0. We also recall that if T E = ⊔ l Il is a disjoint union of open arcs, then E is a Carleson set if and only if ∑ l m(Il) log m(Il) −1 < ∞.
where Θ is an arbitrary inner function on D. Such spaces are often called model spaces in view of their role in the functional model of Sz.-Nagy and Foias [1]. In connection with problems of approximating functions by polyanalytic polynomials and with related properties of Nevanlinna domains (see [3]–[6]), the question of describing model spaces KΘ containing bounded univalent functions was posed in [2], § 2.4. In this paper we resolve this question. It is clear that if an inner function Θ has at least one zero a in D, then the corresponding space KΘ contains the bounded univalent function 1/(1−az). Therefore, in what follows we are dealing with singular inner functions only. Recall that every singular inner function S has the form S(z) = Sμ(z) = exp { − ∫ T ζ+z ζ−z dμ(ζ) } , where μ is a positive measure on the unit circle T and is singular with respect to Lebesgue measure. The first example of a singular function S for which the corresponding space KS contains univalent functions was given in [7]. In [6] it was proved that if a measure μ has an atom, then the space KSμ contains bounded univalent functions. Independently, another (implicit) construction of a bounded univalent function in the Paley–Wiener space PW[0,1] (an analogue of the model space generated by the inner function exp { z+1 z−1 } in the upper half-plane) was given by N. A. Shirokov. On the other hand, it was shown in [6] that if KS contains bounded univalent functions, then the measure μ possesses the following property: there exists a Beurling–Carleson set E with μ(E) > 0. A set E ⊂ T is called a Beurling–Carleson set (or a Carelson set, or a set with finite entropy) if ∫ T log dist(ζ, E) dm(ζ) > −∞, where m( · ) denotes Lebesgue measure on T. In particular, m(E) = 0. We also recall that if T E = ⊔ l Il is a disjoint union of open arcs, then E is a Carleson set if and only if ∑ l m(Il) log m(Il) −1 < ∞.