Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-04-30T18:04:49.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Corona and Wolff theorems for the multiplier algebra of Dirichlet–Morrey spaces

Part of: Spaces and algebras of analytic functions

Published online by Cambridge University Press:  13 January 2022

Lian Hu ,
Songxiao Li Open the ORCID record for Songxiao Li [Opens in a new window]  and
Rong Yang
Lian Hu
Affiliation:
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054 Sichuan, P.R. China e-mail: 201921210220@std.uestc.edu.cn 201921210221@std.uestc.edu.cn
Songxiao Li*
Affiliation:
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054 Sichuan, P.R. China e-mail: 201921210220@std.uestc.edu.cn 201921210221@std.uestc.edu.cn
Rong Yang
Affiliation:
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054 Sichuan, P.R. China e-mail: 201921210220@std.uestc.edu.cn 201921210221@std.uestc.edu.cn
*
e-mail: lisongxiao@uestc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

For $0<\lambda ,p<1$ , the Dirichlet–Morrey space $\mathcal {D}_p^{\lambda } $ is the space of all analytic function on the unit disc such that the measure $ |f'(z)|^2(1-|z|^2)^pdA(z)$ is a $p\lambda $ -Carleson measure. In this paper, we show that the corona theorem and the Wolff theorem hold for the multiplier algebra of Dirichlet–Morrey spaces.

Keywords

Dirichlet–Morrey spacecorona theoremWolff theoremCarleson measure

MSC classification

Primary: 30H80: Corona theorems
Secondary: 30H99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 4 , December 2022 , pp. 963 - 975
Check for updates
Copyright
© Canadian Mathematical Society, 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banjade, D. P., Wolff’s ideal theorem on Qp spaces. Rocky Mountain J. Math. 49(2019), no. 7, 2121–2133.CrossRefGoogle Scholar
Banjade, D. P. and Trent, T. T., Wolff’s problem of ideals in the multiplier algebra on Dirichlet space. Complex Anal. Oper. Theory 8(2014), 1707–1721.10.1007/s11785-014-0356-4CrossRefGoogle Scholar
Banjade, D. P. and Trent, T. T., Wolff’s problem of ideals in the multiplier algebra on weighted Dirichlet space. Houston J. Math. 41(2015), no. 3, 915–932.Google Scholar
Bao, G., Lou, Z., Qian, R., and Wulan, H., On multipliers of Dirichlet type spaces. Complex Anal. Oper. Theory 9(2015), no. 8, 1701–1732.CrossRefGoogle Scholar
Bao, G. and Pau, J., Boundary multipliers of a family of Möbius invariant function spaces. Ann. Acad. Sci. Fenn. Math. 41(2016), 199–220.CrossRefGoogle Scholar
Carleson, L., Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(1962), no. 3, 547–559.CrossRefGoogle Scholar
Essén, M. and Wulan, H., On analytic and meromorphic functions and spaces of QK-type. Illinois J. Math. 46(2002), no. 4, 1233–1258.CrossRefGoogle Scholar
Galanopoulos, P., Merchán, N., and Siskakis, A. G., A family of Dirichlet–Morrey spaces. Complex Var. Elliptic Equ. 64(2019), no. 10, 1686–1702.CrossRefGoogle Scholar
Garnett, J., Bounded analytic functions, Academic Press, New York, 1981.Google Scholar
Girela, D., Analytic functions of bounded mean oscillation. In: Complex function spaces, Univ. Joensuu Dept. Math. Rep. Ser., 4, University of Joensuu, Joensuu, 2001, pp. 61–170.Google Scholar
Jones, P. W., L∞ estimates for the $\bar{\partial }$ problem in a half-plane. Acta Math. 150(1983), 137–152.CrossRefGoogle Scholar
Li, D. and Wulan, H., Corona and Wolff theorems for the multiplier algebras of QK spaces (in Chinese). Sci. China Math. 51(2021), no. 2, 301–314.Google Scholar
Li, P., Liu, J., and Lou, Z., Integral operators on analytic Morrey spaces. Sci. China Math. 57(2014), 1961–1974.CrossRefGoogle Scholar
Liu, J. and Lou, Z., Carleson measure for analytic Morrey spaces. Nonlinear Anal. 125(2015), 423–432.CrossRefGoogle Scholar
Nicolau, A. and Xiao, J., Bounded functions in Möbius invariant Dirichlet spaces. J. Funct. Anal. 150(1997), no. 2, 383–425.CrossRefGoogle Scholar
Pau, J., Multipliers of Qp spaces and the corona theorem. Bull. Lond. Math. Soc. 40(2008), no. 2, 327–336.CrossRefGoogle Scholar
Pau, J., Bounded Möbius invariant QK spaces. J. Math. Anal. Appl. 338(2008), no. 2, 1029–1042.CrossRefGoogle Scholar
Pau, J. and Peláez, J., Multipliers of Möbius invariant Qs spaces. Math. Z. 261(2009), no. 3, 545–555.CrossRefGoogle Scholar
Pau, J. and Zhao, R., Carleson measures, Riemann–Stieltjes and multiplication operators on a general family of function spaces. Integr. Equ. Oper. Theory 78(2014), 483–514.CrossRefGoogle Scholar
Pommerenke, C., Schlichte Funktionen und analytische Funktionen von beschr $\ddot{a}$ nkter mittlerer Oszillation. Comment. Math. Helv. 52(1997), 591–602.CrossRefGoogle Scholar
Treil, S., Estimates in the corona theorem and ideal of H∞: a problem of T. Wolff. J. Anal. Math. 87(2002), 481–495.CrossRefGoogle Scholar
Trent, T. T., A corona theorem for the multipliers on Dirichlet space. Integr. Equ. Oper. Theory 49(2004), 123–139.CrossRefGoogle Scholar
Wu, Z. and Xie, C., Q spaces and Morrey spaces. J. Funct. Anal. 201(2003), no. 1, 282–297.CrossRefGoogle Scholar
Wulan, H. and Zhou, J., QK and Morrey type spaces. Ann. Acad. Sci. Fenn. Math. 38(2013), 193–207.CrossRefGoogle Scholar
Wulan, H. and Zhu, K., Möbius invariant QK spaces, Springer, Cham, 2017, 253 pp.CrossRefGoogle Scholar
Xiao, J., The Qp corona theorem. Pac. J. Math. 194(2000), no. 2, 491–509.CrossRefGoogle Scholar
Xiao, J., Holomorphic Q classes, Springer, Berlin, 2001.10.1007/b87877CrossRefGoogle Scholar
Zhu, K., Operator theory in function spaces. 2nd ed., American Mathematical Society, Providence, RI, 2007.CrossRefGoogle Scholar