Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-19T01:53:41.562Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear Differential Equations with Solutions in the Dirichlet Type Subspace of the Hardy Space

Published online by Cambridge University Press:  11 January 2016

J. Heittokangas ,
R. Korhonen  and
J. Rättyä
J. Heittokangas
Affiliation:
University of Joensuu, Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland, janne.heittokangas@joensuu.fi
R. Korhonen
Affiliation:
University of Joensuu, Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland, risto.korhonen@joensuu.fi
J. Rättyä
Affiliation:
University of Joensuu, Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland, jouni.rattya@joensuu.fi
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Sufficient conditions for the analytic coefficients of the linear differential equation

are found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp, where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2)q is uniformly bounded in D, and f ∈ Dp if the integral ∫D |f′(z)|p(1 – |z|2)p–1dσz converges.

Keywords

34M1030D5030D55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 187 , September 2007 , pp. 91 - 113
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Aulaskari, R., Stegenga, D. and Xiao, J., Some subclasses of BMOA and their characterization in terms of Carleson measures, Rocky Mountain J. Math., 26 (1996), 485–506.Google Scholar
[2] Aulaskari, R., Xiao, J. and Zhao, R., On subspaces and subsets of BMOA and UBC, Analysis, 15 (1995), no. 2, 101–121.CrossRefGoogle Scholar
[3] Benbourenane, D. and Sons, L., On global solutions of complex differential equations in the unit disk, Complex Var. Theory Appl., 49 (2004), no. 13, 913–925.Google Scholar
[4] Cao, T.-B. and Yi, H.-X., The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl., 319 (2006), no. 1, 278–294.Google Scholar
[5] Carleson, L., An interpolation problem for bounded analytic functions, Amer. J. Math., 80 (1958), 921–930.Google Scholar
[6] Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math., 76 (1962), no. 3, 547–559.Google Scholar
[7] Chen, Z.-X. and Shon, K. H., The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal. Appl., 297 (2004), no. 1, 285–304.CrossRefGoogle Scholar
[8] Chyzhykov, I., Gundersen, G. G. and Heittokangas, J., Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc., 86 (2003), no. 3, 735–754.Google Scholar
[9] Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.Google Scholar
[10] Duren, P., Theory of Hp Spaces, Academic Press, New York-London, 1970.Google Scholar
[11] Flett, T. M., The dual of an inequality of Hardy and Littlewood and some related inequalitites, J. Math. Anal. Appl., 38 (1972), 746–765.CrossRefGoogle Scholar
[12] Garnett, J. B., Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc., New York-London, 1981.Google Scholar
[13] Girela, D., Growth of the derivative of bounded analytic functions, Complex Variables, 20 (1992), no. 1–4, 221–227.Google Scholar
[14] Pérez-Gonzáalez, F. and Rättyä, J., Forelli-Rudin estimates, Carleson measures and F(p,q, s)-functions, J. Math. Anal. Appl., 315 (2006), no. 2, 394–414.Google Scholar
[15] Gundersen, G. G., Steinbart, E. M. and Wang, S., The possible orders of solutions of linear differential equations with polynomial coefficients, Trans. Amer. Math. Soc., 350 (1998), no. 3, 1225–1247.Google Scholar
[16] Heittokangas, J., On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1–54.Google Scholar
[17] Heittokangas, J., Korhonen, R. and Rättyä, J., Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn., 29 (2004), 233–246.Google Scholar
[18] Heittokangas, J., Korhonen, R. and Rättyä, J., Fast growing solutions of linear differential equations in the unit disc, Results Math., 49 (2006), 265–278.CrossRefGoogle Scholar
[19] Heittokangas, J., Korhonen, R. and Rättyä, J., Linear differential equations with coefficients in weighted Bergman and Hardy spaces, to appear in Trans. Amer. Math. Soc.Google Scholar
[20] Hille, E., Ordinary Differential Equations in the Complex Domain, Pure and Applied Mathematics, Wiley-Interscience (John Wiley & Sons), New York-London-Sydney, 1976.Google Scholar
[21] Korhonen, R. and Rättyä, J., Linear differential equations in the unit disc with analytic solutions of finite order, Proc. Amer. Math. Soc, 135 (2007), no. 5, 1355–1363.Google Scholar
[22] Korhonen, R. and Rättyä, J., Finite order solutions of linear differential equations in the unit disc, preprint.Google Scholar
[23] Laine, I., Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.Google Scholar
[24] Littlewood, J. E. and Paley, R. E. A. C., Theorems on Fourier series and power series (II), Proc. London Math. Soc, 42 (1936), 52–89.Google Scholar
[25] Luecking, D., A new proof of an inequality of Littlewood and Paley, Proc. Amer. Math. Soc, 103 (1988), no. 3, 887–893.Google Scholar
[26] Pommerenke, Chr., On the mean growth of the solutions of complex linear differential equations in the disk, Complex Variables, 1 (1982), no. 1, 23–38.Google Scholar
[27] Rudin, W., The radial variation of analytic functions, Duke Math. J., 22 (1955), 235–242.Google Scholar
[28] Rättyä, J., On some complex function spaces and classes, Ann. Acad. Sci. Fenn. Math. Diss., 124 (2001), 1–73.Google Scholar
[29] Rättyä, J., n-th derivative characterizations, mean growth of derivatives and F(p,q,s), Bull. Australian Math. Soc, 68 (2003), 405–421.Google Scholar
[30] Rättyä, J., Linear differential equations with solutions in function spaces, to appear in Complex Var. Elliptic Equ.Google Scholar
[31] Vinogradov, S. A., Multiplication and division in the space of analytic functions with an area-integrable derivative, and in some related spaces (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 222 (1995), Issled. po Linein. Oper. i Teor. Funktsii, no. 23, 45–77, 308; Translation in J. Math. Sci. (New York), 87 (1997), no. 5, 3806–3827.Google Scholar
[32] Wittich, H., Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A. I., 379 (1966), 1–18.Google Scholar
[33] Zhao, R., On a general family of function spaces, Ann. Acad. Sci. Fenn. Diss., 105 (1996), 1–56.Google Scholar
[34] Zhu, K., Operator theory in function spaces, Marcel Dekker, New York, 1990.Google Scholar
[35] Zhu, K., Bloch type spaces of analytic functions, Rocky Mountain J. Math., 23 (1993), no. 3, 1143–1177.Google Scholar