Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 695-711, 2019


Bloch type spaces on the unit ball of a Hilbert space

Zhenghua Xu

Received October 26, 2017.   Published online November 9, 2018.

Abstract:  We initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.
Keywords:  Bloch type space; Lipschitz space; Hardy-Littlewood theorem; Hilbert space
Classification MSC:  32A18, 46E15
DOI:  10.21136/CMJ.2018.0495-17


References:
[1] O. Blasco, P. Galindo, A. Miralles: Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Funct. Anal. 267 (2014), 1188-1204. DOI 10.1016/j.jfa.2014.04.018 | MR 3217061 | Zbl 1293.32010
[2] H. Chen: Characterizations of $\alpha$-Bloch functions on the unit ball without use of derivative. Sci. China Ser. A 51 (2008), 1965-1981. DOI 10.1007/s11425-008-0104-1 | MR 2447421 | Zbl 1187.32004
[3] S. Chen, S. Ponnusamy, A. Rasila: On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces. Math. Z. 279 (2015), 163-183. DOI 10.1007/s00209-014-1361-z | MR 3299847 | Zbl 1314.30116
[4] J. Dai, B. Wang: Characterizations of some function spaces associated with Bloch type spaces on the unit ball of $\mathbb{C}^n$. J. Inequal. Appl. 2015 (2015), 10 pages. DOI 10.1186/s13660-015-0846-6 | MR 3411320 | Zbl 1336.32006
[5] F. Deng, C. Ouyang: Bloch spaces on bounded symmetric domains in complex Banach spaces. Sci. China Ser. A 49 (2006), 1625-1632. DOI 10.1007/s11425-006-2050-0 | MR 2288219 | Zbl 1114.32002
[6] I. Graham, G. Kohr: Geometric Function Theory in One and Higher Dimensions. Monographs and Textbooks in Pure and Applied Mathematics 255, Marcel Dekker, New York (2003). DOI 10.1201/9780203911624 | MR 2017933 | Zbl 1042.30001
[7] K. T. Hahn: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Canad. J. Math 27 (1975), 446-458. DOI 10.4153/CJM-1975-053-0 | MR 0466641 | Zbl 0269.32014
[8] G. H. Hardy, J. E. Littlewood: Some properties of fractional integrals II. Math. Z. 34 (1932), 403-439. DOI 10.1007/BF01180596 | MR 1545260 | Zbl 0003.15601
[9] F. Holland, D. Walsh: Criteria for membership of Bloch space and its subspace, BMOA. Math. Ann. 273 (1986), 317-335. DOI 10.1007/BF01451410 | MR 0817885 | Zbl 0561.30025
[10] S. G. Krantz: Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math. 1 (1983), 193-260. MR 0782608 | Zbl 0518.46018
[11] S. G. Krantz, D. Ma: Bloch functions on strongly pseudoconvex domains. Indiana Univ. Math. J. 37 (1988), 145-163. DOI 10.1512/iumj.1988.37.37007 | MR 0942099 | Zbl 0628.32006
[12] O. Lehto, K. I. Virtanen: Boundary behaviour and normal meromorphic functions. Acta Math. 97 (1957), 47-65. DOI 10.1007/BF02392392 | MR 0087746 | Zbl 0077.07702
[13] S. Li, H. Wulan: Characterizations of $\alpha$-Bloch spaces on the unit ball. J. Math. Anal. Appl. 343 (2008), 58-63. DOI 10.1016/j.jmaa.2008.01.023 | MR 2409457 | Zbl 1204.32006
[14] M. Nowak: Bloch space and Möbius invariant Besov spaces on the unit ball of $\mathbb{C}^n$. Complex Variables, Theory Appl. 44 (2001), 1-12. DOI 10.1080/17476930108815339 | MR 1826712 | Zbl 1026.32011
[15] M. Pavlović: On the Holland-Walsh characterization of Bloch functions. Proc. Edinb. Math. Soc., II. Ser. 51 (2008), 439-441. DOI 10.1017/S0013091506001076 | MR 2465917 | Zbl 1165.30016
[16] G. Ren, C. Tu: Bloch space in the unit ball of $\mathbb{C}^n$. Proc. Am. Math. Soc. 133 (2005), 719-726. DOI 10.1090/S0002-9939-04-07617-8 | MR 2113920 | Zbl 1056.32005
[17] G. Ren, Z. Xu: Slice Lebesgue measure of quaternions. Adv. Appl. Clifford Algebr. 26 (2016), 399-416. DOI 10.1007/s00006-015-0578-1 | MR 3460007 | Zbl 1337.30061
[18] W. Rudin: Function Theory in the Unit Ball of $\mathbb{C}^n$. Classics in Mathematics, Springer, Berlin (2008). DOI 10.1007/978-3-540-68276-9 | MR 2446682 | Zbl 1139.32001
[19] R. M. Timoney: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12 (1980), 241-267. DOI 10.1112/blms/12.4.241 | MR 0576974 | Zbl 0416.32010
[20] R. M. Timoney: Bloch functions in several complex variables II. J. Reine Angew. Math. 319 (1980), 1-22. DOI 10.1515/crll.1980.319.1 | MR 0586111 | Zbl 0425.32008
[21] W. Yang, C. Ouyang: Exact location of $\alpha$-Bloch spaces in $L^p_a$ and $H^p$ of a complex unit ball. Rocky Mountain J. Math. 30 (2000), 1151-1169. DOI 10.1216/rmjm/1021477265 | MR 1797836 | Zbl 0978.32002
[22] M. Zhang, H. Chen: Equivalent characterizations of $\alpha$-Bloch functions on the unit ball. Acta Math., Sin. Engl. Ser. 27 (2011), 2395-2408. DOI 10.1007/s10114-011-9391-5 | MR 2853797 | Zbl 1262.32002
[23] R. Zhao: A characterization of Bloch-type spaces on the unit ball of $\mathbb{C}^n$. J. Math. Anal. Appl. 330 (2007), 291-297. DOI 10.1016/j.jmaa.2006.06.100 | MR 2302923 | Zbl 1118.32006
[24] K. Zhu: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23 (1993), 1143-1177. DOI 10.1216/rmjm/1181072549 | MR 1245472 | Zbl 0787.30019
[25] K. Zhu: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226, Springer, New York (2005). DOI 10.1007/0-387-27539-8 | MR 2115155 | Zbl 1067.32005

Affiliations:   Zhenghua Xu, School of Mathematics, Hefei University of Technology, No. 420 Feicui Road, Hefei 230601, Anhui, P. R. China, e-mail: zhxu@hfut.edu.cn


 
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