Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-27T07:15:43.528Z Has data issue: false hasContentIssue false
  • English
  • Français

Hausdorff operators on some classical spaces of analytic functions

Part of: Integral, integro-differential, and pseudodifferential operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  29 February 2024

Huayou Xie  and
Qingze Lin Open the ORCID record for Qingze Lin [Opens in a new window]
Huayou Xie
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China e-mail: xiehy33@mail2.sysu.edu.cn
Qingze Lin*
Affiliation:
Department of Mathematics, Shantou University, Shantou, 515063, People’s Republic of China
*
e-mail: gdlqz@e.gzhu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators

$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$
on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $-measurable function on the open unit disk $\mathbb {D}$, and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$.

Keywords

Hausdorff operatorweighted Dirichlet spacederivative Hardy spaceBloch type spaceboundedness

MSC classification

Primary: 47G10: Integral operators 30H05: Bounded analytic functions
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 13
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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.)

Footnotes

Qingze Lin is supported by the Natural Science Foundation of Jiaxing (Grant No. 2023AY40003).

References

Bonet, J., Hausdorff operators on weighted Banach spaces of type ${H}^{\infty }$ . Complex Anal. Oper. Theory 16(2022), no. 1, Article no. 12, 14 pp.CrossRefGoogle Scholar
Chen, J., Fan, D., and Wang, S., Hausdorff operators on Euclidean spaces . Appl. Math. J. Chinese Univ. Ser. B 28(2013), no. 4, 548–564.CrossRefGoogle Scholar
Cowen, C. and MacCluer, D., Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995, xii+388 pp.Google Scholar
Čučković, Ž. and Paudyal, B., Invariant subspaces of the shift plus complex Volterra operator . J. Math. Anal. Appl. 426(2015), no. 2, 1174–1181.CrossRefGoogle Scholar
Čučković, Ž. and Paudyal, B., The lattices of invariant subspaces of a class of operators on the Hardy space . Arch. Math. 110(2018), 477–486.CrossRefGoogle Scholar
Čučković, Ž. and Zhao, R., Weighted composition operators between different weighted Bergman spaces and different Hardy spaces . Illinois J. Math. 51(2007), no. 2, 479–498.CrossRefGoogle Scholar
Duren, P. L., Theory of ${H}^p$ spaces, Pure and Applied Mathematics, 38, Academic Press, New York, 1970.Google Scholar
Duren, P. L. and Schuster, A., Bergman spaces, Mathematical Surveys and Monographs, 100, American Mathematical Society, Providence, RI, 2004.CrossRefGoogle Scholar
Galanopoulos, P., Girela, D., and Peláez, J. Á., Multipliers and integration operators on Dirichlet spaces . Trans. Amer. Math. Soc. 363(2011), no. 4, 1855–1886.CrossRefGoogle Scholar
Georgakis, C., The Hausdorff mean of a Fourier–Stieltjes transform . Proc. Amer. Math. Soc. 116(1992), no. 2, 465–471.CrossRefGoogle Scholar
Girela, D. and Peláez, J. Á., Carleson measures, multipliers and integration operators for spaces of Dirichlet type . J. Funct. Anal. 241(2006), no. 1, 334–358.CrossRefGoogle Scholar
Girela, D. and Peláez, J. Á., Carleson measures for spaces of Dirichlet type . Integral Equations Operator Theory 55(2006), no. 3, 415–427.CrossRefGoogle Scholar
Girela, D. and Peláez, J. Á., Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type . J. Aust. Math. Soc. 80(2006), no. 3, 397–418.CrossRefGoogle Scholar
Grudsky, S., Karapetyants, A., and Mirotin, A., Estimates for singular numbers of Hausdorff–Zhu operators and applications . Math. Methods Appl. Sci. 46(2023), no. 8, 9676–9693.CrossRefGoogle Scholar
Gu, C. and Luo, S., Composition and multiplication operators on the derivative Hardy space ${S}^2(D)$ . Complex Var. Elliptic Equ. 63(2018), no. 5, 599–624.CrossRefGoogle Scholar
Karapetyants, A. and Mirotin, A., A class of Hausdorff–Zhu operators . Anal. Math. Phys. 12(2022), no. 3, Article no. 79, 19 pp.CrossRefGoogle Scholar
Karapetyants, A., Samko, S., and Zhu, K., A class of Hausdorff–Berezin operators on the unit disc . Complex Anal. Oper. Theory 13(2019), no. 8, 3853–3870.CrossRefGoogle Scholar
Lerner, A. and Liflyand, E., Multidimensional Hausdorff operators on the real Hardy space . J. Aust. Math. Soc. 83(2007), 79–86.CrossRefGoogle Scholar
Liflyand, E., Boundedness of multidimensional Hausdorff operators on ${H}^1\left({\mathbb{R}}^n\right)$ . Acta Sci. Math. (Szeged) 74(2008), nos. 3–4, 845–851.Google Scholar
Liflyand, E., Hausdorff operators on Hardy spaces . Eurasian Math. J. 4(2013), no. 4, 101–141.Google Scholar
Liflyand, E. and Miyachi, A., Boundedness of the Hausdorff operators in ${H}^p$ spaces, $0<p<1$ . Studia Math. 194(2009), no. 3, 279–292.CrossRefGoogle Scholar
Liflyand, E. and Miyachi, A., Boundedness of multidimensional Hausdorff operators in ${H}^p$ spaces, $0<p<1$ . Trans. Amer. Math. Soc. 371(2019), no. 7, 4793–4814.CrossRefGoogle Scholar
Lin, Q., The invariant subspaces of the shift plus integer multiple of Volterra operator on Hardy spaces . Arch. Math. 111(2018), no. 5, 513–522.CrossRefGoogle Scholar
Lin, Q., Volterra type operators on weighted Dirichlet spaces . Chinese Ann. Math. Ser. B 42(2021), no. 4, 601–612.CrossRefGoogle Scholar
Lin, Q., Liu, J., and Wu, Y., Volterra type operators on ${S}^p(D)$ spaces . J. Math. Anal. Appl. 461(2018), no. 2, 1100–1114.CrossRefGoogle Scholar
Lin, Q., Liu, J., and Wu, Y., Order boundedness of weighted composition operators on weighted Dirichlet spaces and derivative Hardy spaces . Bull. Belg. Math. Soc. Simon Stevin 27(2020), no. 4, 627–637.CrossRefGoogle Scholar
MacCluer, B. D., Composition operators on ${S}^p$ . Houston J. Math. 13(1987), no. 2, 245–254.Google Scholar
Mirotin, A. R., Boundedness of Hausdorff operators on real Hardy spaces ${H}^1$ over locally compact groups . J. Math. Anal. Appl. 473(2019), no. 1, 519–533.CrossRefGoogle Scholar
Mirotin, A. R., Hausdorff operators on some spaces of holomorphic functions on the unit disc . Complex Anal. Oper. Theory 15(2021), no. 5, Article no. 85, 16 pp.CrossRefGoogle Scholar
Mirotin, A. R., Generalized Hausdorff-Zhu operators on Möbius invariant spaces . Complex Anal. Oper. Theory 16(2022), no. 7, Article no. 100, 15 pp.CrossRefGoogle Scholar
Ohno, S., Stroethoff, K., and Zhao, R., Weighted composition operators between Bloch-type spaces . Rocky Mountain J. Math. 33(2003), 191–215.CrossRefGoogle Scholar
Roan, R. C., Composition operators on the space of functions with ${H}^p$ -derivative . Houston J. Math. 4(1978), no. 3, 423–438.Google Scholar
Wu, Z., Carleson measures and multipliers for Dirichlet spaces . J. Funct. Anal. 169(1999), no. 1, 148–163.CrossRefGoogle Scholar
Yosida, K., Functional analysis, Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. xii+501 pp.Google Scholar
Zhu, K., Bloch type spaces of analytic functions . Rocky Mountain J. Math. 23(1993), no. 3, 1143–1177.CrossRefGoogle Scholar
Zhu, K., Operator theory in function spaces. 2nd ed., Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.Google Scholar