Document Type : Research Paper


1 Department of Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran.

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.

3 Technical faculty of Khoy, Urmia University, Urmia, Iran.


Let $H(\mathbb{D})$ be the space of all analytic functions on the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. In this paper, we investigate the boundedness and compactness of the generalized integration operator
$$I_{g,\varphi}^{(n)}(f)(z)=\int_0^z f^{(n)}(\varphi(\xi))g(\xi)\ d\xi,\quad z\in\mathbb{D},$$ from Zygmund space into weighted Dirichlet type space, where $\varphi$ is an analytic self-map of $\mathbb{D}$, $n\in\mathbb{N}$ and $g\in H(\mathbb{D})$. Also we give an estimate for the essential norm of the above operator.


[1] E. Abbasi, H. Vaezi and S. Li, Essential norm of weighted composition operators from $H^{infty}$ to $n$th weighted type spaces, Mediterr. J. Math., 16(5) (2019), 14 pages.
[2] F. Alighadr-Ardebili, H. Vaezi and M. Hassanlou, Generalized Integration Operator between the Bloch-type Space and Weighted Dirichlet-type Spaces, KYUNGPOOK Math. J., 60 (2020), pp. 519-534.
[3] H.-B. Bai and Z.J. Jiang, Generalized weighted composition operators from Zygmund spaces to Bloch-Orlicz type spaces, Appl. Math. Comput., 273 (2016), pp. 89-97.
[4] D.C. Chang, S. Li and S. Stevic, On some integral operators on the unit polydisc and unit ball, Taiwanes J. Math., 11(5) (2007), pp. 1251-1286.
[5] D.C. Chang and S. Stevic, The generalized cesaro operator on the unit polydsic, Taiwanese J. Math., 7(2) (2003), pp. 293--308.
[6] F. Colonna and S. Li, Weighted composition operators from the Lipschitz space into the Zygmund space, Math. Ineq. Appl., 17(3) (2014), pp. 963-975.
[7] C.C. Cowen, and B.D. Maccluer, Compostion Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
[8] P. L. Duren, Theory of $H^p$ spaces, Academic Press, New York and London, 1970.
[9] Z.H. He and G. Cao, Generalized integration operators between Bloch-type spaces and $F(s,p,q)$ spaces, Taiwanese J. Math., 17(4) (2013), pp. 1211-1225.
[10] S. Kummar, Weighted composition operators between spaces of Dirichlet-type, Rev. Math. Complut., 22(2) (2009), pp. 469-488.
[11] S. Li and S. Stevic, Generalized composition operators on Zygmund spaces and Bloch-Type spaces, J. Math. Anal. Appl., 338(2) (2008), pp. 1282-1295.
[12] S. Li and S. Stevic, Products of volterra type operator and composition operator from $H^{infty}$ and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl., 345(1) (2008), pp. 40-52.
[13] S. Li and S. Stevic, Products of integral type operators and composition operators between Bloch-Type spaces, J. Math. Anal. Appl., 349(12) (2009), pp. 596-610.
[14] J. Liu, Z. Lou and C. Xiong, Essential norms of integral operators on spaces of analytic functions, Nonlinear Analysis, 75(2) (2012), pp. 5145-5156.
[15] Y. Liu Y. Yu, Riemann-Stieltjes operator from mixed norm spaces to Zygmund-Type spaces on the unit ball, Taiwanes J. Math., 17(5) (2013), pp. 1751-1764.
[16] A. Salaryan and H. Vaezi, Adjoints of rationally induced weighted composition operators on the Hardy, Bergman and Dirichlet spaces, Period. Math. Hung., 72 (2016), pp. 76-89.
[17] J.H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New York, 1993.
[18] S.D. Sharma and A. Sharma, Generalized integration operators from Bloch type spaces to weighted BMOA spaces, Demonstratio Math., 44(2) (2011), pp. 373-390.
[19] S. Stevic, Boundedness and Compactness of an integral type operator from Bloch-Type spaces with normal weights to $F(p,q,s)$ space, Appl. Math. Comput., 218(9) (2012), pp. 5414-5421.
[20] M. Tjani, Compact Composition Operators on Some Moebius Invariant Banach Spaces, Thesis (PhD.), Michigan State University, 1996.
[21] D. Vukotic, A sharp estimate for $A_beta^p$ functions in ${C}^n$, Proc. Amer. Math. Soc., 117 (1993), pp. 753-756.
[22] Y. Yu and Y. Liu, Integral-type operators from weighted Bloch spaces into Bergman-type spaces, Integral Transforms Spec. Funct., 20(6) (2009), pp. 419-428.
[23] X. Zhu, An integral type operator from $H^{infty}$ to Zygmund-Type spaces, Bull. Malays. Math. Sci. Soc., 35(2) (2012), pp. 679-686.