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Embedding of Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ into tent spaces and Volterra operators

Part of: Spaces and algebras of analytic functions Special classes of linear operators

Published online by Cambridge University Press:  08 September 2020

Ruishen Qian  and
Xiangling Zhu
Ruishen Qian
Affiliation:
School of Mathematics and Statistics, Lingnan Normal University, 524048, Zhanjiang, Guangdong, P. R. Chinae-mail:qianruishen@sina.cn
Xiangling Zhu*
Affiliation:
University of Electronic Science and Technology of China, Zhongshan Institute, 528402, Zhongshan, Guangdong, P. R. China
*
e-mail: jyuzxl@163.com
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Abstract

In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

Keywords

Dirichlet type spaceVolterra integral operatorCarleson measuretent spaceessential norm

MSC classification

Primary: 30H99: None of the above, but in this section
Secondary: 47B38: Operators on function spaces (general)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 697 - 708
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

This work was supported by NNSF of China (Nos. 11801250, 11871257, and 11701222), Overseas Scholarship Program for Elite Young and Middle-aged Teachers of Lingnan Normal University, Yanling Youqing Program of Lingnan Normal University, the Key Program of Lingnan Normal University (No. LZ1905), and Department of Education of Guangdong Province (No. 2018KTSCX133). Xiangling Zhu is the corresponding author.

References

Aleman, A. and Cima, J., An integral operator on ${H}^p$ and Hardy’s inequality. J. Anal. Math. 85(2001), 157–176. http://dx.doi.org/10.1007/BF02788078 CrossRefGoogle Scholar
Aleman, A. and Siskakis, A., An integral operator on ${H}^p$ . Complex Variables Theory Appl. 28(1995), 149–158. http://dx.doi.org/10.1080/1746939508814844 CrossRefGoogle Scholar
Aleman, A. and Siskakis, A., Integration operators on Bergman spaces. Indiana Univ. Math. J. 46(1997), 337–356. http://dx.doi.org/10.1512/iumj.1997.46.1373 CrossRefGoogle Scholar
Attele, K., Interpolating sequences for the derivatives of Bloch functions. Glasgow Math. J. 34(1992), 35–41. http://dx.doi.org/10.1017/S0017089500008521 CrossRefGoogle Scholar
Duren, P., Theory of ${H}^p$ spaces. Academic Press, New York-London, 1970.Google Scholar
Flett, T., The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 38(1972), 756–765. http://dx.doi.org/10.1016/0022-247X(72)90081-9 CrossRefGoogle Scholar
Girela, D. and Peláez, J., Carleson measures, multipliers and integration operators for spaces of Dirichlet type. J. Funct. Anal. 241(2006), 334–358. http://dx.doi.org/10.1016/j.jjfa.2006.04.025 CrossRefGoogle Scholar
Li, P., Liu, J., and Lou, Z., Integral operators on analytic Morrey spaces. Sci. China Math. 57(2014), 1961–1974. http://dx.doi.org/10.1007/s11425-014-4811-5 CrossRefGoogle Scholar
Li, S., Liu, J., and Yuan, C., Embedding theorems for Dirichlet type spaces. Canad. Math. Bull. 63(2020), 106–117. http://dx.doi.org/10.4153/S0008439519000201 CrossRefGoogle Scholar
Li, S. and Stević, S., Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 117(2007), 371–385. http://dx.doi.org/10.1007/s12044-007-0032-y CrossRefGoogle Scholar
Littlewood, J. and Paley, R., Theorems on Fourier series and power series II.. Proc. London Math. Soc. 42(1936), 52–89. http://dx.doi.org/10.1112/plms/s2-42.1.52 Google Scholar
Pau, J. and Zhao, R., Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces. Integral Equat. Operator Theory 78(2014), 483–514. http://dx.doi.org/10.1007/s00020-014-2124-2 CrossRefGoogle Scholar
Pommerenke, C., Schlichte funktionen und analytische funktionen von beschränkten mittlerer Oszillation. Comm. Math. Helv. 52(1977), 591–602. http://dx.doi.org/10.1007/BF02567392 CrossRefGoogle Scholar
Shi, Y. and Li, S., Essential norm of integral operators on Morrey type spaces. Math. Inequal. Appl. 19(2016), 385–393. http://dx.doi.org/10.7153/mia-19-30 Google Scholar
Siskakis, A. and Zhao, R., A Volterra type operator on spaces of analytic functions. Contemp. Math. 232(1999), 299–311.CrossRefGoogle Scholar
Tjani, M., Compact composition operators on some Möbius invariant Banach spaces. PhD dissertation, Michigan State University, 1996.Google Scholar
Wu, Z., Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169(1999), 148–163. http://dx.doi.org/10.1006/jfan.1999.3490 CrossRefGoogle Scholar
Zhao, R., On a general family of function spaces. Ann. Acad. Sci. Fenn. Math. Diss. 105(1996), 56 pp.Google Scholar
Zhu, K., Operator theory in function spaces. 2nd ed., Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. http://dx.doi.org/10.1090/surv/138 Google Scholar