Abstract
Motivated by a problem in approximation theory, we find a necessary and sufficient condition for a model (backward shift invariant) subspace $K_\varTheta = H^2\ominus \varTheta H^2$ of the Hardy space $H^2$ to contain a bounded univalent function.
Results & Lemmas (7)
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Theorem 1.1.
Theorem 1.1. Let S be a singular inner function and let µ be the corresponding (positive singular) measure on T. The following conditions…
Theorem 1.1. Let S be a singular inner function and let µ be the corresponding (positive singular) measure on T. The following conditions are equivalent. (i) The space KS contains bounded univalent functions. (ii) There exists a Carleson set E ⊂T such that µ(E) > 0. An immediate corollary of Theorem 1.1 is
Corollary 1.2.
Corollary 1.2. A model space KΘ contains a bounded univalent function if and only if either Θ has a zero in D or Θ is a singular inner…
Corollary 1.2. A model space KΘ contains a bounded univalent function if and only if either Θ has a zero in D or Θ is a singular inner function such that the associated singular measure satisfies condition (ii) of Theorem 1.1. We give two different proofs of Theorem 1.1. The first one is based on delicate esti- mates of entropy, which seem to be of independent interest. The second proof is more straightforward.
Lemma 2.1.
Lemma 2.1. There exist absolute constants β > 0, ε ∈(0, 1/e) and M ∈N such that for every singular probability measure µ supported by a…
Lemma 2.1. There exist absolute constants β > 0, ε ∈(0, 1/e) and M ∈N such that for every singular probability measure µ supported by a closed set E ⊂I for an arc I with EntI(E) ≤ε, there exists a function f ∈KSµ ∩C3(D) such that f(z) = P n≥0 cnzn, |c1| ≥β (2.2) and ∞ X j=1 |cMj+1|(Mj + 1) < β. (2.3)
Lemma 2.2.
Lemma 2.2. Let µ be a non-trivial continuous singular measure supported by a closed set E of finite entropy. Then for any ε, δ > 0 there…
Lemma 2.2. Let µ be a non-trivial continuous singular measure supported by a closed set E of finite entropy. Then for any ε, δ > 0 there exists an arc I such that 0 < µ(I) < δ and EntI(E)/µ(I) < ε.
Lemma 2.3.
Lemma 2.3. Let S = Sµ be a singular inner function with supp(µ) = E ⊂I, where I is an arc with endpoint 1 and |I| < 1/100. Let r ∈(9/10, 1)…
Lemma 2.3. Let S = Sµ be a singular inner function with supp(µ) = E ⊂I, where I is an arc with endpoint 1 and |I| < 1/100. Let r ∈(9/10, 1) be such that 1 −r > 10|I|. Put eS = S ◦ϕ−r, the composition of S with the M¨obius transformation ϕ−r. Then (i) eS is a singular inner function and the corresponding singular measure eµ satisfies µ(T) 1 −r ≤eµ(T) = Z I 1 −r2 |ζ −r|2 dµ(ζ) ≤3µ(T) 1 −r . (2.10) (ii) There exists an arc eI with endpoint 1 such that eE := supp(eµ) ⊂eI and |eI| ≤4|I| 1 −r,
Lemma 2.4.
Lemma 2.4. Let Θ be an inner function and let a ∈D, a ̸= 0, be such that Θ(−a) ̸= 0. Define eΘ = Θ ◦ϕa. Let f ∈KΘ and let g = f ◦ϕa. Then…
Lemma 2.4. Let Θ be an inner function and let a ∈D, a ̸= 0, be such that Θ(−a) ̸= 0. Define eΘ = Θ ◦ϕa. Let f ∈KΘ and let g = f ◦ϕa. Then there exists cf ∈C such that g −cf ∈K e Θ.
Lemma 3
Lemma 3]). Moreover, it is easy to see that S5(z) = exp −M2 Z T ζM + z ζM −z dµ3(ζ) = exp −M2 Z T ξ + z
Lemma 3]). Moreover, it is easy to see that S5(z) = exp −M2 Z T ζM + z ζM −z dµ3(ζ) = exp −M2 Z T ξ + z