Geodesic
The Range of the Cesàro Operator Acting on $H^{\infty }$
Canadian mathematical bulletin, Tome 63 (2020) no. 3, pp. 633-642

Voir la notice de l'article provenant de la source Cambridge University Press

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. 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 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.
DOI : 10.4153/S0008439519000717
Mots-clés : The Cesàro operator, H∞, Möbius invariant function space, superharmonic function 
Bao, Guanlong; Wulan, Hasi; Ye, Fangqin. The Range of the Cesàro Operator Acting on $H^{\infty }$. Canadian mathematical bulletin, Tome 63 (2020) no. 3, pp. 633-642. doi: 10.4153/S0008439519000717
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