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\(n\)-th derivative characterisations, mean growth of derivatives and \(F(p,q,s)\). (English) Zbl 1064.30027

For \(0< p<\infty\), \(-2< q<\infty\) and \(0\leq s<\infty\). \(F(p,q,s)\) is defined as the set of all analytic functions, on \(D\) for which \[ \sup_{a\in D}\,\iint_D|f'(z)|^p(1-|z|^2)^qg^s(z, a)\,d\sigma_z<\infty, \] where \(g(z,a)= \log|1- \overline az|/|z- a|\). \(F(2,1,0)\) is the Hardy space \(H^2\), \(F(p,p,0)\) is the Bergman space \(L^p_a\), \(F(2,0,s)= Q_s\) and \(F(2,0,1)= \text{BMOA}\). The author generalizes known results of \(F(p,q,0)\) to \(F(p,q,s)\) for \(0\leq s<\infty\). In particular, he generalizes results of R. Yoneda [Characterizations of Bloch and Besov spaces by oscillations, Hokkaido Math. J. 29, No. 2, 409–451 (2000; Zbl 0984.46019)]. Moreover, he answers partially the question in the paper of R. Yoneda.

MSC:

30D45 Normal functions of one complex variable, normal families
30D50 Blaschke products, etc. (MSC2000)

Citations:

Zbl 0984.46019
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References:

[1] Duren, Theory of H (1970)
[2] DOI: 10.2307/2043708 · Zbl 0524.30023
[3] Aulaskari, Bull. Austral. Math. Soc. 58 pp 43– (1998)
[4] DOI: 10.1006/jmaa.2000.7259 · Zbl 0980.30024
[5] Zygmund, Trigonometric series (1959) · JFM 58.0296.09
[6] Zhu, Operator theory in function spaces 139 (1990) · Zbl 0706.47019
[7] Yoneda, Hokkaido Math. J. 29 pp 409– (2000) · Zbl 0984.46019
[8] DOI: 10.1111/j.1749-6632.1996.tb33169.x
[9] Stroethoff, Bull. Austral. Math. Soc. 54 pp 211– (1996)
[10] Stroethoff, Bull. Austral. Math. Soc. 39 pp 405– (1989)
[11] Rättyä, Ann. Acad. Sci. Fenn. Math. Dissertationes 124 pp 1– (2001)
[12] Heittokangas, Ann. Acad. Sci. Fenn. Math. Dissertationes 122 pp 1– (2000)
[13] Garnett, Bounded analytic functions 96 (1981)
[14] DOI: 10.1016/0022-247X(72)90081-9 · Zbl 0246.30031
[15] Fàbrega, Ann. Inst. Fourier. (Grenoble) 46 pp 111– (1996) · Zbl 0840.32001
[16] Danikas, Function spaces and complex analysis (Joensuu, 1997) pp 9– (1999)
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