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Weighted composition operators from Hardy to Zygmund type spaces. (English) Zbl 1273.47050

Summary: This paper aims at studying the boundedness and compactness of weighted composition operators between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator \(uC_{\phi}\) from the Hardy spaces \(H^p\) to the Zygmund type spaces \(\mathcal Z_\alpha = \{f \in H(D) : \text{sup}_{z \in D}(1 - |z|^2)^\alpha |f''(z)| < \infty\}\) and the little Zygmund type spaces \(\mathcal Z_{\alpha, 0}\) in terms of function theoretic properties of the symbols \(u\) and \(\phi\).

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30H10 Hardy spaces
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[1] Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Transactions of the American Mathematical Society, 347, 7, 2679-2687 (1995) · Zbl 0826.47023
[2] Madigan, K. M., Composition operators on analytic Lipschitz spaces, Proceedings of the American Mathematical Society, 119, 2, 465-473 (1993) · Zbl 0793.47037
[3] Smith, W., Composition operators between Bergman and Hardy spaces, Transactions of the American Mathematical Society, 348, 6, 2331-2348 (1996) · Zbl 0857.47020
[4] Yoneda, R., The composition operators on weighted Bloch space, Archiv der Mathematik, 78, 4, 310-317 (2002) · Zbl 1038.47020
[5] Fleming, R. J.; Jamison, J. E., Isometries on Banach Spaces: Function Spaces. Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129, x+197 (2003), Boca Raton, Fla, USA: Chapman & Hall/CRC, Boca Raton, Fla, USA · Zbl 1011.46001
[6] Contreras, M. D.; Hernández-Díaz, A. G., Weighted composition operators on Hardy spaces, Journal of Mathematical Analysis and Applications, 263, 1, 224-233 (2001) · Zbl 1026.47016
[7] Čučković, Z.; Zhao, R., Weighted composition operators on the Bergman space, Journal of the London Mathematical Society, 70, 2, 499-511 (2004) · Zbl 1069.47023
[8] Laitila, J., Weighted composition operators on BMOA, Computational Methods and Function Theory, 9, 1, 27-46 (2009) · Zbl 1163.47018
[9] Ohno, S.; Zhao, R., Weighted composition operators on the Bloch space, Bulletin of the Australian Mathematical Society, 63, 2, 177-185 (2001) · Zbl 0985.47022
[10] Ye, S., A weighted composition operator on the logarithmic Bloch space, Bulletin of the Korean Mathematical Society, 47, 3, 527-540 (2010) · Zbl 1250.47031
[11] Ye, S.; Hu, Q., Weighted composition operators on the Zygmund space, Abstract and Applied Analysis, 2012 (2012) · Zbl 1277.47038
[12] Čučković, Z.; Zhao, R., Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois Journal of Mathematics, 51, 2, 479-498 (2007) · Zbl 1147.47021
[13] Ohno, S.; Stroethoff, K.; Zhao, R., Weighted composition operators between Bloch-type spaces, The Rocky Mountain Journal of Mathematics, 33, 1, 191-215 (2003) · Zbl 1042.47018
[14] Sharma, A. K., Products of multiplication, composition and differentiation between weighted Bergman-Nevanlinna and Bloch-type spaces, Turkish Journal of Mathematics, 35, 2, 275-291 (2011) · Zbl 1236.47025
[15] Sharma, A. K.; Ueki, S.-I., Composition operators from Nevanlinna type spaces to Bloch type spaces, Banach Journal of Mathematical Analysis, 6, 1, 112-123 (2012) · Zbl 1269.47024
[16] Stević, S., Weighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ball, Applied Mathematics and Computation, 217, 5, 1939-1943 (2010) · Zbl 1250.47030
[17] Stević, S.; Sharma, A. K., Essential norm of composition operators between weighted Hardy spaces, Applied Mathematics and Computation, 217, 13, 6192-6197 (2011) · Zbl 1211.30062
[18] Stević, S.; Sharma, A. K., Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk, Applied Mathematics and Computation, 217, 24, 10187-10194 (2011) · Zbl 1220.30072
[19] Ye, S., A weighted composition operator between different weighted Bloch-type spaces, Acta Mathematica Sinica. Chinese Series, 50, 4, 927-942 (2007) · Zbl 1131.30340
[20] Ye, S., Weighted composition operators from \(F(p, q, s)\) into logarithmic Bloch space, Journal of the Korean Mathematical Society, 45, 4, 977-991 (2008) · Zbl 1156.47031
[21] Ye, S., Weighted composition operators between the little logarithmic Bloch space and the \(\alpha \)-Bloch space, Journal of Computational Analysis and Applications, 11, 3, 443-450 (2009) · Zbl 1221.47063
[22] Ye, S., Norm of composition followed by differentiation from logarithmic Bloch space to theweighted-type space
[23] Colonna, F.; Li, S., Weighted composition operators from Hardy spaces into logarithmic Bloch spaces, Journal of Function Spaces and Applications, 2012 (2012) · Zbl 1250.47027
[24] Duren, P. L., Theory of \(H^p\) Spaces. Theory of \(H^p\) Spaces, Pure and Applied Mathematics, 38, xii+258 (1970), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0215.20203
[25] Garnett, J. B., Bounded Analytic Functions. Bounded Analytic Functions, Graduate Texts in Mathematics, 236, xiv+459 (2007), New York, NY, USA: Springer, New York, NY, USA · Zbl 1106.30001
[26] Zygmund, A., Trigonometric Series (1959), London, UK: Cambridge University Press, London, UK · JFM 58.0296.09
[27] Zhu, K. H., Bloch type spaces of analytic functions, The Rocky Mountain Journal of Mathematics, 23, 3, 1143-1177 (1993) · Zbl 0787.30019
[28] Choe, B. R.; Koo, H.; Smith, W., Composition operators on small spaces, Integral Equations and Operator Theory, 56, 3, 357-380 (2006) · Zbl 1114.47028
[29] Li, S.; Stević, S., Generalized composition operators on Zygmund spaces and Bloch type spaces, Journal of Mathematical Analysis and Applications, 338, 2, 1282-1295 (2008) · Zbl 1135.47021
[30] Li, S.; Stević, S., Products of Volterra type operator and composition operator from \(H^\infty\) and Bloch spaces to Zygmund spaces, Journal of Mathematical Analysis and Applications, 345, 1, 40-52 (2008) · Zbl 1145.47022
[31] Li, S.; Stević, S., Weighted composition operators from Zygmund spaces into Bloch spaces, Applied Mathematics and Computation, 206, 2, 825-831 (2008) · Zbl 1215.47022
[32] Cowen, C. C.; MacCluer, B. D., Composition Operators on Spaces of Analytic Functions. Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, xii+388 (1995), Boca Raton, Fla, USA: CRC Press, Boca Raton, Fla, USA · Zbl 0873.47017
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