×

Distance of a Bloch function to the little Bloch space. (English) Zbl 1101.30035

The Bloch space \({\mathcal B}\) is the space of all functions \(f\) holomorphic on the unit disc \(U\) such that \[ \| f\| _{{\mathcal B}} = \sup_{z \in U}{ (1 - | z| ^{2} | f'(z)| } < \infty \;. \] The Bloch space is a Banach space under the norm \(\| f\| _{B} = \| f\| _{{\mathcal B}} + | f(0)| \). The little Bloch space \({\mathcal B}_{0}\) is the closed subspace of \({\mathcal B}\) of those functions \(f\) for which \[ \lim_{| z| \to 1} (1 - | z| ^{2}) | f'(z)| = 0 \;. \] For \(f \in {\mathcal B}\) and \(\epsilon > 0\), let \[ \Omega_{\epsilon}(f) = \{z \in U: (1 - | z| ^{2}) | f'(z)| \geq \epsilon \} \;, \] and let both \[ A(f) = \{\epsilon > 0: \Omega_{\epsilon}(f) \text{ is a compact subset of } U \} \;, \] and, for \(p > 0\), \[ A_{p}(f) = \{\epsilon > 0: \frac {\chi_{\Omega_{\epsilon}(f)}(z)} {(1 - | z| ^{2})^{p}} dA(z) \text{ is a finite measure on} U \} \;, \] where \(dA(z)\) is the element of Euclidean area and \(\chi_{G}(z)\) denotes the characteristic function for the set \(G\), that is, \(\chi_{G}(z) = 1\) if \(z \in G\), and \(\chi_{G}(z) = 0\) if \(z \not\in G\). The author proves a number of results relating to the distance \[ \text{dist}_{{\mathcal B}}(f, {\mathcal B}_{0}) = \inf_{g \in {\mathcal B}_{0}} {\| f - g\| _{B}} \] from a Bloch function \(f\) to the supspace \({\mathcal B}_{0}\). For example, it is shown that for \(f \in {\mathcal B}\) and \(p > 0\), both \[ \inf{A_{p}(f)} \leq \text{dist}_{{\mathcal B}}(f, {\mathcal B}_{0}) \leq 6 \inf{A_{p}(f)} \;, \] and \[ \inf{A(f)} \leq \text{dist}_{{\mathcal B}}(f, {\mathcal B}_{0}) \leq 6 \inf{A(f)} \;. \] Another result is that for \(f \in {\mathcal B}\), \[ \limsup_{| z| \to 1} {| f'(z)| (1 - | z| ^{2})} \leq \text{ dist}_{{\mathcal B}}(f, {\mathcal B}_{0}) \leq 2 \limsup_{| z| \to 1} {| f'(z)| (1 - | z| ^{2})} \;. \] Some characterizations of both \({\mathcal B}\) and \({\mathcal B}_{0}\) are given, and some examples are given computing the distance between a specific Bloch function and \({\mathcal B}_{0}\). Some analogous results are given for some analogues of the Bloch space, such as \({\mathcal B}_{\alpha}\), the set of all functions \(f\) holomorphic in \(U\) for which \[ \sup_{z \in U}{(1 - | z| ^{2})^{\alpha} | f'(z)| } \;. \] Also, some applications are made to the norm of the composition operator \(C_{\phi}(f) = f \circ \phi\), where \(\phi\) is a holomorphic self-map of \(U\).

MSC:

30D45 Normal functions of one complex variable, normal families
46E15 Banach spaces of continuous, differentiable or analytic functions
47B33 Linear composition operators
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1007/BF01200391 · Zbl 0796.46011
[2] DOI: 10.1090/S0002-9947-99-02387-9 · Zbl 0920.47029
[3] Arazy, J. Reine Angew. Math. 363 pp 110– (1985)
[4] Anderson, Operators and Function Theory 153 pp 1– (1985)
[5] DOI: 10.1216/rmjm/1181072549 · Zbl 0787.30019
[6] Zhu, Operator theory on function spaces (1990) · Zbl 0706.47019
[7] Makhmutov, Bull. Austral. Math. Soc. 62 pp 1– (2000)
[8] Shapiro, Composition operators and classical faction theory (1993)
[9] Stegenga, Contemp. Math. 137 pp 421– (1992)
[10] DOI: 10.1007/BF01445229 · Zbl 0727.32002
[11] Pommerenke, Boundary behaviour of conformal maps (1992)
[12] Montes-Rodriguez, Pacific J. Math. 188 pp 339– (1999)
[13] DOI: 10.1007/b87877 · Zbl 0983.30001
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.