Abstract
The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D pα to D qβ (−1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.
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References
Aleman, A. and Cima, J., An integral operator on Hp and Hardy’s inequality, J. Anal. Math., 85, 2001, 157–176.
Aleman, A. and Siskakis, A., An integral operator on Hp, Complex Variables Theory Appl., 28(2), 1995, 149–158.
Aleman, A. and Siskakis, A., Integration operators on Bergman spaces, Indiana Univ. Math. J., 46(2), 1997, 337–356.
Carleson, L., An interpolation problem for bounded analytic functions, Amer. J. Math., 80, 1958, 921–930.
Constantin, O. and Peláez, J., Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces, J. Geom. Anal., 26(2), 2016, 1109–1154.
Čučković, Ž. and Zhao, R., Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math., 51, 2007, 479–498.
Duren, P., Extension of a theorem of Carleson, Bull. Amer. Math. Soc., 75, 1969, 143–146.
Duren, P. and Schuster, A., Bergman Spaces, Math. Surveys Monogr., 100, Amer. Math. Soc., Providence, RI, 2004.
Galanopoulos, P., Girela, D. and Peláez, J., Multipliers and integration operators on Dirichlet spaces, Trans. Amer. Math. Soc., 363(4), 2011, 1855–1886.
Gao, Y., Kumar, S. and Zhou, Z., Order bounded weighted composition operators mapping into the Dirichlet type spaces, Chin. Ann. Math. Ser. B, 37(4), 2016, 585–594.
Girela, D. and Peláez, J., Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal., 241(1), 2006, 334–358.
Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman Spaces, Grad. Texts in Math., 199, Springer-Verlag, New York, 2000.
Hibschweiler, R., Order Bounded Weighted Composition Operators, Contemp. Math., 454, Amer. Math. Soc., Providence, RI, 2008.
Hunziker, H. and Jarchow, H., Composition operators which improve integrability, Math. Nachr., 152, 1991, 83–99.
Kumar, S., Weighted composition operators between spaces of Dirichlet type, Rev. Mat. Complut., 22(2), 2009, 469–488.
Laitila, J., Miihkinen, S. and Nieminen, P., Essential norms and weak compactness of integration operators, Arch. Math., 97(1), 2011, 39–48.
Li, P., Liu, J. and Lou, Z., Integral operators on analytic Morrey spaces, Sci. China Math., 57(9), 2014, 1961–1974.
Li, S. and Stević, S., Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338(2), 2008, 1282–1295.
Li, S. and Stević, S., Products of Volterra type operator and composition operator from H∞ and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl., 345(1), 2008, 40–52.
Lin, Q., Volterra type operators between Bloch type spaces and weighted Banach spaces, Integral Equations Operator Theory, 91(2), 2019, 91:13.
Lin, Q., Liu, J. and Wu, Y., Volterra type operators on Sp(ⅅ) spaces, J. Math. Anal. Appl., 461, 2018, 1100–1114.
Lin, Q., Liu J. and Wu, Y., Strict singularity of Volterra type operators on Hardy spaces, J. Math. Anal. Appl., 492(1), 2020, 124438, 9 pages.
Luecking, D., Forward and reverse inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math., 107, 1985, 85–111.
Mengestie, T., Product of Volterra type integral and composition operators on weighted Fock spaces, J. Geom. Anal., 24(2), 2014, 740–755.
Mengestie, T., Path connected components of the space of Volterra-type integral operators, Arch. Math., 111(4), 2018, 389–398.
Pau, J. and Zhao, R., Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces, Integr. Equ. Oper. Theory, 78, 2014, 483–514.
Pommerenke, Ch., Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. (German), 52(4), 1977, 591–602.
Sharma, A., On order bounded weighted composition operators between Dirichlet spaces, Positivity, 21(3), 2017, 1213–1221.
Ueki, S., Order bounded weighted composition operators mapping into the Bergman space, Complex Anal. Oper. Theory, 6(2), 2012, 549–560.
Wang, S., Wang, M. and Guo, X., Differences of Stevic-Sharma operators, Banach J. Math. Anal., 14(3), 2020, 1019–1054.
Wu, Z., Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal., 169, 1999, 148–163.
Xiao, J., The Qp Carleson measure problem, Adv. Math., 217(5), 2008, 2075–2088.
Zhao, R., Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces, Ann. Acad. Sci. Fenn. Math., 29(1), 2004, 139–150.
Zhu, K., Operator Theory in Function Spaces, 2nd ed., Mathematical Surveys and Monographs, 138, Amer. Math. Soc., Providence, RI, 2007.
Acknowledgements
The author is grateful to the referee for his (or her) valuable comments and suggestions. Also, he would like to thank the brilliant mathematician, Loo-Keng Hua, for his excellent books which had inspired him into mathematics. At last, he wants to express his gratitude to the great star, Bruce Lee, for inspiring him with the fighting spirit.
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This work was supported by the National Natural Science Foundation of China (No. 11801094).
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Lin, Q. Volterra Type Operators on Weighted Dirichlet Spaces. Chin. Ann. Math. Ser. B 42, 601–612 (2021). https://doi.org/10.1007/s11401-021-0281-6
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DOI: https://doi.org/10.1007/s11401-021-0281-6