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This paper introduces the logarithmic Bloch space $B_{\log }^{q}\left(D^{n}\right)$, a new space of analytic functions on the unit polydisc, investigates the composition operator $C_{\varphi}$ from weighted Bergman space to logarithmic Bloch space on the unit polydisc, and provides the sufficient and necessary conditions to ensure the boundedness and compactness of the composition operator $C_{\varphi}$ from weighted Bergman space to logarithmic Bloch space.
Composition operator, Weighted Bergman space, Logarithmic Bloch space, Boundedness, Compactness.
The author wishes to thank the science project of Higher Education of GuangXi in China for contract KY2015LX778, under which the present work was possible.
1. H.Y. Li, P.D. Liu, M.F. Wang, Composition operators between generally weighted Bloch spaces of polydisk, 2007, Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 3, pp. 85-90.
2. K. Madigan, A. Matheson, Compact composition operators on the Bloch space, 1995, Trans. Amer. Math. Soc., vol. 347, pp. 2679-2687.
3. Z.H. Zhou, J.H. Shi, Compact composition operators on the Bloch space in polydisc, 2001, compact composition operators on the Bloch space in polydisc, vol. 44, pp. 286-291.
4. L. Luo, J.H. Shi, Composition operators between the weighted Bergman space on bounded symmetric domains of , 2000, Chinese Ann. Math, vol. 21, pp. 45-52.
5. K.H. Zhu, Composition Operators induced by symbols defined on a polydisk, 2006, Math. Anal. Appl., vol. 319, pp. 815-829.
6. X.M. Tang, Z.J. Hu, Composition operators between Bergman spaces and q-Bloch spaces, 2006, Chinese Ann. Math., vol. 27, no. 1, pp. 109-118.
7. X.J. Zhang, Composition type operator from Bergman space to -Bloch type space in , 2004, J. Math. Anal. Appl, vol. 298, pp. 710-721.
8. W.W. Wu, M.X. Hu, Weighted composition operators from weighted Bergman space to Bloch-type space on the polydisc, 2014, Journal of Zhejiang Normal University, vol. 37, no. 2, pp. 151-157.
9. Q.H. Huang, Composition operator from weighted Bergman space to q-Bloch space in polydisc, 2014, Journal of Chemical and Pharmaceutical Research, vol. 6, no. 6, pp. 1973-1979.
10. R. Yoneda, The Composition operator on weighted Bloch space, 2002, Arch Math, vol. 78, pp. 310-317.
11. E. Rene, Castillo, Composition operators from logarithmic Bloch spaces to weighted Bloch spaces, 2013, Applied Mathematics and Computation, vol. 219, pp. 6692-6706.
12. C. Julio, Ramos-Fernandez, Logarithmic Bloch spaces and their weighted composition operators, 2016, Rend. Circ. Mat. Palermo, vol. 65, pp. 159-174.
13. C.C. Cowen, B.D. MacCluer, Coposition operators on spaces of analytic functions, Boca Roton, 1995.
14. K.H. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, 2005.
15. K.H. Zhu, Operator Theory in Function spaces, Marcel Dekker, New York, 1990.