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The n-th derivative characterisation of Möbius invariant Dirichlet space

Published online by Cambridge University Press:  17 April 2009

Rauno Aulaskari ,
Maria Nowak  and
Ruhan Zhao
Rauno Aulaskari
Affiliation:
University of Joensuu, PO Box 111, FIN-8010 Joensuu, Finland e-mail: aulaskari@joyl.joensuu.fi
Maria Nowak
Affiliation:
Institute of Mathematics, Maria Curie-Sklodowska University, pl. M. Curie-Skodowskiej 1, 20–031 Lublin, Poland
Ruhan Zhao
Affiliation:
University of Joensuu, PO Box 111, FIN-8010 Joensuu, Finland
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Abstract

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In this paper we give the n-th derivative criterion for functions belonging to recently defined function spaces Qp and Qp, 0. For a special parameter value p = 1 this criterion is applied to BMOA and VMOA, and for p > 1 it is applied to the Bloch space and the little Bloch space . Further, a Carleson measure characterisation is given to Qp, and in the last section the multiplier space from Hq into Qp is considered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 43 - 56
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Anderson, J. M., Clunie, J. and Pommerenke, Ch., ‘On Bloch functions and normal functions’, J. Reine Angew. Math. 240 (1974), 12–37.Google Scholar
[2]Arazy, J., Fisher, S.D. and Peetre, J., ‘Möbius invariant function spaces’, J. Reine Angew. Math. 363 (1985), 110–145.Google Scholar
[3]Aulaskari, R., He, Y. and Zhao, R., ‘On entire functions, Bloch and normal functions’, Chinese Ann. Math. Ser. B 17 (1996), 139–148.Google Scholar
[4]Aulaskari, R. and Lappan, P., ‘Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal’, in Complex analysis and its applications, Pitman Research Notes in Mathematics Series 305 (Longman Scientific and Technical, Harlow, 1994), pp. 136–146.Google Scholar
[5]Aulaskari, R., Stegenga, D. A. and Xiao, J., ‘Some subclasses of BMOA and their characterization in terms of Carleson measures’, Rocky Mountain J. Math. 26 (1996), 485–506.CrossRefGoogle Scholar
[6]Aulaskari, R., Xiao, J. and Zhao, R., ‘On subspaces and subsets of BMOA and UBC’, Analysis 15 (1995), 101–121.CrossRefGoogle Scholar
[7]Axler, S., ‘The Bergman space, the Bloch space and commutators of multiplication operators’, Duke Math. J. 53 (1986), 315–332.CrossRefGoogle Scholar
[8]Baernstein, A. II, ‘Analytic functions of bounded mean oscillation’, in Aspects of contemporary complex analysis (Academic Press, London, New York, 1980), pp. 2–26.Google Scholar
[9]Cima, J.A. and Wogen, W., ‘A Carleson measure theorem for the Bergman space on the ball’, J. Operator Theory 7 (1982), 157–165.Google Scholar
[10]Essén, M. and Xiao, J., ‘Some results on Qp spaces, 0 < p < 1’, J. Reine Angew. Math. 485 (1997), 173–195.Google Scholar
[11]Mateljevic, M. and Pavlovic, M., ‘Multipliers of Hp and BMOA’, Pacific J. Math. 146 (1990), 71–84.CrossRefGoogle Scholar
[12]Sarason, D., ‘Functions of vanishing mean oscillation’, Trans. Amer. Math. Soc. 205 (1975), 391–405.CrossRefGoogle Scholar
[13]Stegenga, D.A., ‘Multipliers of Dirichlet space’, Illinois J. Math. 24 (1980), 113–139.CrossRefGoogle Scholar
[14]Stroethoff, K., ‘Besov-type characterizations for the Bloch space’, Bull. Austral. Math. Soc. 39 (1989), 405–420.CrossRefGoogle Scholar
[15]Xiao, J., ‘Carleson measure, atomic decomposition and free interpolation from Bloch space’, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 35–46.Google Scholar
[16]Xiao, J. and Zhong, L., ‘On little Bloch space, atomic decomposition and free interpolation’, Complex Variables Theory Appl. 27 (1995), 175–184.Google Scholar
[17]Zhao, R., ‘On a general family of function spaces’, Ann. Acad. Sci. Fenn. Math. Dissertationes 105 (1996), 1–56.Google Scholar
[18]Zhu, K.H., Operator Theory in Function Spaces (Marcel Dekker, Inc., New York and Basel, 1990).Google Scholar