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Products of unbounded Bloch functions. (English) Zbl 1479.30024

Bastos, M. Amélia (ed.) et al., Operator theory, functional analysis and applications. Proceedings of the 30th international workshop on operator theory and its applications, IWOTA 2019, Lisbon, Portugal, July 22–26, 2019. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 282, 283-292 (2021).
The author provides a method to generate for each unbounded Bloch function \(f\) another function \(g\) in the minimal Besov space \(B_1\), that is satisfying the condition \(\int_{\mathbb D}|f''(z)|dA(z)<\infty\), such that the product \(g\cdot f\) is not a Bloch function. The procedure uses the description of \(B_1\) in terms of Moebious transformations.
For the entire collection see [Zbl 1471.47002].

MSC:

30D45 Normal functions of one complex variable, normal families
30H30 Bloch spaces
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References:

[1] Anderson, JM; Clunie, J.; Pommerenke, Ch, On Bloch functions and normal functions, J. Reine Angew. Math., 270, 12-37 (1974) · Zbl 0292.30030
[2] Arazy, J.; Fisher, SD; Peetre, J., Möbius invariant function spaces, J. Reine Angew. Math., 363, 110-145 (1985) · Zbl 0566.30042
[3] Blasco, O.; Girela, D.; Márquez, MA, Mean growth of the derivative of analytic functions, bounded mean oscillation, and normal functions, Indiana Univ. Math. J., 47, 893-912 (1998) · Zbl 0943.30021
[4] Brown, L.; Shields, AL, Multipliers and cyclic vectors in the Bloch space, Michigan Math. J., 38, 141-146 (1991) · Zbl 0749.30023
[5] Campbell, DM, Nonnormal sums and products of unbounded normal functions, II. Proc. Amer. Math. Soc., 74, 202-203 (1979) · Zbl 0406.30021
[6] Donaire, JJ; Girela, D.; Vukotić, D., On univalent functions in some Möbius invariant spaces, J. Reine Angew. Math., 553, 43-72 (2002) · Zbl 1006.30031
[7] P.L. Duren, Theory ofH^pSpaces (Academic, New York, 1970; Reprint: Dover, Mineola-New York, 2000) · Zbl 0215.20203
[8] Galanopoulos, P.; Girela, D.; Hernández, R., Univalent functions, VMOA and related spaces. J. Geom. Anal., 21, 665-682 (2011) · Zbl 1220.30010
[9] Garnett, JB, Bounded Analytic Functions (1981), New York: Academic, New York · Zbl 0469.30024
[10] Girela, D., On a theorem of Privalov and normal funcions, Proc. Amer. Math. Soc., 125, 433-442 (1997) · Zbl 0861.30034
[11] Girela, D., Mean Lipschitz spaces and bounded mean oscillation, Illinois J. Math., 41, 214-230 (1997) · Zbl 0878.30028
[12] Girela, D.; Suárez, D., On Blaschke products, Bloch functions and normal functions. Rev. Mat. Complut., 24, 49-57 (2011) · Zbl 1219.30015
[13] Girela, D.; González, C.; Peláez, JA, Multiplication and division by inner functions in the space of Bloch functions, Proc. Amer. Math. Soc., 134, 1309-1314 (2006) · Zbl 1094.30039
[14] Lappan, P., Non-normal sums and products of unbounded normal function, Michigan Math. J., 8, 187-192 (1961) · Zbl 0133.03603
[15] Lehto, O.; Virtanen, KI, Boundary behaviour and normal meromorphic functions, Acta Math., 97, 47-65 (1957) · Zbl 0077.07702
[16] Ch. Pommerenke, Univalent Functions (Vandenhoeck und Ruprecht, Göttingen, 1975)
[17] Yamashita, S., A nonnormal function whose derivative has finite area integral of order 0 < p < 2, Ann. Acad. Sci. Fenn. Ser. A I Math., 4, 2, 293-298 (1979) · Zbl 0433.30026
[18] Yamashita, S., A nonnormal function whose derivative is of Hardy class H^p, 0 < p < 1, Canad. Math. Bull., 23, 499-500 (1980) · Zbl 0438.30033
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