Abstract
We study Q-spaces and Q #-classes which are Möbius-invariant, weighted Dirichlet spaces or classes of analytic or meromorphic functions in the unit disc in the plane. Generalizations to half-spaces in higher dimensions are also possible. In the analytic case, they are subspaces of BMOA or of the Bloch space B. In the meromorphic case, they are subclasses of the class of normal functions N or of the class of spherical Bloch functions B #. In the first part of the survey, we discuss concrete examples where different kinds of p-Carleson measures (0 < p < 1) are important. In the last section, we discuss a more general theory which gives both new results and new proofs of several results from the first part.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arazy, J., Fisher, S., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363 (1985), 110–145.
Aulaskari, R., Lappan, P.: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. Complex analysis and its applications Pitman Research Notes in Mathematics 305. Longman Scientific Technical Harlow 1994, 136–146.
Aulaskari, R., Stegenga, D., Xiao, J.: Some subclasses of BMOA and their characterization in terms of Carleson measures. Rocky Mountain J. Math. 26 (1996), 485–506.
Aulaskari, R., Wulan, H., Zhao, R.: Carleson measure and some classes of meromorphic functions. Proc. Amer. Math. Soc. 128 (2000), 2329–2335.
Aulaskari, R., Xiao, J., Zhao,R.: On subspaces and subsets of BMOA and UBC. Analysis 15 (1995), 101–121.
Baernstein, A. II: Analytic functions of bounded mean oscillation. Aspects of contemporary complex analysis. Academic Press, London, 1980, 3–36.
Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76 (1962), 547–559.
Duren, P.: Theory of H“ spaces. Academic Press, New York and London, 1970.
Essén, M.: Qp-spaces. “Complex Function Spaces”, Mekrijärvi 1999 (ed.Aulaskari), Department of Mathematics, University of Joensuu, Finland, Report no 4, 2001, 9–40.
Essén, M., Janson, S., Peng, L., Xiao, J.: Q-spaces of several real variables. Ind. Univ. Math. J. 49 (2000), 575–615.
Essén, M., Wulan, H.: Carleson type measures and their applications. Complex Variables, Theory Appl. 42 (2000), 67–88.
Essén, M., Wulan, H.: On analytic and meromorphic functions and spaces of Qx-type. Department of Mathematics, Uppsala University, Report 2000: 32.
Essén, M., Xiao, J.: Some results on Qp-spaces, 0 p 1. J. Reine Angew. Math. 485 (1997), 173–195.
Essén, M., Xiao, J.: QP-spaces–a survey. Complex Function Spaces, Mekrijärvi 1999, (ed. Aulaskari), Department of Mathematics, University of Joensuu, Finland, Report no 4, 2001, 41–60.
Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77 (1971), 587–588.
Garnett, J.: Bounded analytic functions. Academic Press, New York 1981.
Lappan, P.: A non-normal locally uniformly univalent function. Bull. London Math. Soc. 5 (1973), 291–294.
Nicolau, A.: The corona property for bounded analytic functions in some Besov spaces. Proc. Amer. Math. Soc. 110 (1990), 135–140.
Ortega, J.M., Fàbrega, J.: The corona type decomposition in some Besov spaces. Math. Scand. 78 (1996), 93–111.
Pommerenke, Ch.: Boundary behaviour of conformal maps. Springer, Berlin 1992.
Tolokonnikov, V.A.: The corona theorem in algebras of bounded analytic functions. Amer. Math. Soc. Trans. 149 (1991), 61–93.
Wulan, H.: On some classes of meromorphic functions. Ann. Acad. Sci. Fenn. Math. Diss. 116 (1998), 1–57.
Wu, P., Wulan, H.: Characterizations of QT spaces. J. Math. Anal. Appl. 254 (2001), 484–497.
Xiao, J.: The Qp corona theorem. Pac. J. of Math. 194 (2000), 491–509.
Xiao, J.: Some essential properties of Qp(ÔO)-spaces. J. of Fourier Analysis and Applications 6 (2000), 311–323.
Xiao, J.: Outer functions in Q p and Qp,o. Preprint 1999.
Xiao, J.: Holomorphic Q classes. Lecture Notes in Mathematics 1767. Springer, Berlin 2001.
Yamashita, S.: Functions of uniformly bounded characteristic. Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 349–367.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Essén, M. (2003). A Survey of Q—Spaces and Q #-Classes. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3741-7_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5247-9
Online ISBN: 978-1-4757-3741-7
eBook Packages: Springer Book Archive