×

Actions of the Möbius group on analytic functions. (English) Zbl 1475.30119

Summary: In his PhD dissertation, R. Zhao [On a general family of function spaces. Helsinki: Suomalainen Tiedeakatemia. (1996; Zbl 0851.30017)] introduced a new notion of weighted actions by the Möbius group on analytic functions on the unit disk indexed by a positive parameter \(\alpha\) and proved that the so-called \(\alpha\)-Bloch space is maximal among all \(\alpha\)-Möbius invariant function spaces. In this paper we continue the study of \(\alpha\)-Möbius invariant function spaces. In particular, we identify the minimal non-trivial \(\alpha\)-Möbius invariant function space and prove the existence and uniqueness of a non-trivial \(\alpha\)-Möbius invariant semi-Hilbert space of analytic functions on the unit disk, thus answering two questions left open by Zhao [loc. cit.].

MSC:

30H20 Bergman spaces and Fock spaces
30H25 Besov spaces and \(Q_p\)-spaces
30H30 Bloch spaces
30H35 BMO-spaces
30B10 Power series (including lacunary series) in one complex variable
46E15 Banach spaces of continuous, differentiable or analytic functions

Citations:

Zbl 0851.30017
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. · Zbl 0292.30030
[2] J. Arazy, Realization of the invariant inner products on the highest quotients of the composition series, Ark. Mat. 30 (1992), 1-24. · Zbl 0767.46015
[3] J. Arazy and S. Fisher, Some aspects of the minimal, Möbius-invariant space of analytic functions on the unit disc, in: Lecture Notes in Math. 1070, Springer, Berlin, 1984, 24-44. · Zbl 0553.46021
[4] J. Arazy and S. Fisher, The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math. 29 (1985), 449-462. · Zbl 0555.30021
[5] J. Arazy and S. Fisher, Invariant Hilbert spaces of analytic functions on bounded symmetric domains, in: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, L. de Branges et al. (eds.), Oper. Theory Adv. Appl. 48, Birkhäuser, Basel, 1990, 67-91. · Zbl 0733.46011
[6] J. Arazy, S. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. · Zbl 0566.30042
[7] R. Aulaskari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their characterization in terms of Carleson measures, Rocky Mountain J. Math. 26 (1996), 485-506. · Zbl 0861.30033
[8] R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis 15 (1995), 101-121. · Zbl 0835.30027
[9] J. Donaire, D. Girela and D. Vukotić, On the growth and range of functions in Möbius invariant spaces, J. Anal. Math. 112 (2010), 237-260. · Zbl 1211.30063
[10] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. · Zbl 0469.30024
[11] F. Holland and D. Walsh, Growth estimates for functions in the Besov spaces Ap, Proc. Roy. Irish Acad. Sect. A 88 (1988), 1-18. · Zbl 0629.30036
[12] J. Ortega and J. Fàbrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier (Grenoble) 46 (1996), 111-137. · Zbl 0840.32001
[13] L. Rubel and R. Timoney, An extremal property of the Bloch space, Proc. Amer. Math. Soc. 75 (1979), 45-49. · Zbl 0405.46020
[14] R. Timoney, Natural function spaces, J. London Math. Soc. 41 (1990), 78-88. · Zbl 0661.46021
[15] H. Wulan and K. Zhu, Möbius Invariant QK Spaces, Springer, 2017.
[16] J. Xiao, Holomorphic Q Classes, Lecture Notes in Math. 1767, Springer, Berlin, 2001. · Zbl 0983.30001
[17] J. Xiao, Geometric Qp Functions, Birkhäuser, Basel, 2006. · Zbl 1104.30036
[18] R. Zhao, On a general family of function spaces, Ann. Acad. Sci. Fenn. Math. Diss. 105 (1996), 56 pp. · Zbl 0851.30017
[19] R. Zhao, Distances from Bloch functions to some Möbius invariant spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 303-313. · Zbl 1147.30024
[20] R. Zhao and K. Zhu, Theory of Bergman spaces in the unit ball of C n , Mém. Soc. Math. France 115 (2008), vi+103 pp. · Zbl 1176.32001
[21] K. Zhu, Möbius invariant Hilbert spaces of holomorphic functions on the unit ball of C n , Trans. Amer. Math. Soc. 323 (1991), 823-842. · Zbl 0739.46009
[22] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), 1143-1177. · Zbl 0787.30019
[23] K. Zhu, The Möbius invariance of Besov spaces on the unit ball of C n , in: Com-plex Analysis and Applications, Y. Wang et al. (eds.), World Sci., Hackensack, NJ, 2006, 328-337. · Zbl 1112.32006
[24] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Amer. Math. Soc., Prov-idence, RI, 2007.
[25] K. Zhu, A class of Möbius invariant function spaces, Illinois J. Math. 51 (2007), 977-1002. · Zbl 1154.30040
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.