The essential norm of the integral type operators

  • Xiaoman Liu
  • Yongmin LiuEmail author
  • Lina Xia
  • Yanyan Yu
Original Paper


On the basis of the characterizations of the boundedness and compactness of the Volterra type operator \(I_{g, \varphi }\) from mixed-norm spaces \(H(p,\, q,\, \phi )\) to Zygmund spaces \( \mathcal {Z}\), the authors provide a function-theoretic estimate for the essential norm of Volterra type operator \(I_{g, \varphi }\) by means of the definition of the essential norm of an operator and the properties of the analytic function. An estimate for the essential norm of the generalized integration operator
$$\begin{aligned} I^{(n)}_{g, \varphi }f(z)=\int ^{z}_{0}f^{(n)}(\varphi (\xi ))g(\xi )d\xi , \ z\in \mathbb {D}, \end{aligned}$$
from Bloch-type spaces to F(pqs) spaces is also obtained.


Essential norm Mixed-norm space Zygmund space Volterra type operator Generalized integral operator 

Mathematics Subject Classification

47B38 47B33 30H10 30D45 



The authors are very grateful to anonymous referees and editors for their valuable and detailed suggestions and insightful comments to improve the original manuscript. The project is supported by the Natural Science Foundation of China (Grant nos. 11771184, 11771188) and the Foundation Research Project of Jiangsu Province of China (no. BK20161158).

Author Contributions

All authors contributed equally to the writing of this paper. They also read and approved the final manuscript.


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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.College of SciencesNanjing Agricultural UniversityNanjingChina
  2. 2.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina
  3. 3.School of Mathematics and Physics ScienceXuzhou Institute of TechnologyXuzhouChina

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