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The Range of the Cesàro Operator Acting on $H^{\infty }$

Published online by Cambridge University Press:  04 December 2019

Guanlong Bao
Affiliation:
Department of Mathematics, Shantou University, Shantou515063, Guangdong, China Email: [email protected]@stu.edu.cn
Hasi Wulan
Affiliation:
Department of Mathematics, Shantou University, Shantou515063, Guangdong, China Email: [email protected]@stu.edu.cn
Fangqin Ye
Affiliation:
Business School, Shantou University, Shantou515063, Guangdong, China Email: [email protected]

Abstract

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

Type
Article
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The work was supported by NNSF of China (No. 11801347, No. 11720101003 and No. 11571217), NSF of Guangdong Province (No. 2018A030313512), Department of Education of Guangdong Province (No. 2017KQNCX078), Key projects of fundamental research in universities of Guangdong Province (No. 2018KZDXM034), and STU SRFT (No. NTF17020 and No. STF17005). F. Ye is the corresponding author.

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