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Bloch Mappings on Bounded Symmetric Domains

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Advances in Real and Complex Analysis with Applications

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Abstract

We introduce Bloch mappings on bounded symmetric domains which can be infinite dimensional and generalize Bonk’s distortion theorem on \(\mathbb {C}\) to locally biholomorphic Bloch mappings on finite dimensional bounded symmetric domains. As an application, we give a lower bound of the Bloch constant for these locally biholomorphic Bloch mappings. Finally, we show that there exist no isometric composition operators from the space \(H^{\infty }(\mathbb {B}_X)\) of bounded and holomorphic functions on \(\mathbb {B}_X\) into the \(\alpha \)-Bloch space \(\mathcal {B}^\alpha (\mathbb {B}_X)\) on \(\mathbb {B}_X\).

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References

  1. L.V. Ahlfors, An extension of Schwarz’s lemma. Trans. Amer. Math. Soc. 43, 359–364 (1938)

    MathSciNet  MATH  Google Scholar 

  2. L.V. Ahlfors, Complex Analysis (McGraw-Hill, New York, 1966)

    MATH  Google Scholar 

  3. R.F. Allen, F. Colonna, Weighted composition operators from \(H^{\infty }\) to the Bloch space of a bounded homogeneous domain. Integral Equ. Oper. Theory 66, 21–40 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.M. Anderson, J.G. Clunie, Ch. Pommerenke, On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)

    MathSciNet  MATH  Google Scholar 

  5. O. Blasco, P. Galindo, A. Miralles, Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Funct. Anal. 267, 1188–1204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bonk, On Bloch’s constant. Proc. Amer. Math. Soc. 110, 889–894 (1990)

    MathSciNet  MATH  Google Scholar 

  7. M. Bonk, D. Minda, H. Yanagihara, Distortion theorems for locally univalent Bloch functions. J. Anal. Math. 69, 73–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bonk, D. Minda, H. Yanagihara, Distortion theorem for Bloch functions. Pacific. J. Math. 179, 241–262 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. É. Cartan, Sur les domaines bornés homogènes de l’espace den variables complexes Abh. Math. Sem. Univ. Hamburg 11, 116–162 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Chen, P.M. Gauthier, On Bloch’s constant. J. Anal. Math. 69, 275–291 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Chen, P.M. Gauthier, Bloch constants in several variables. Trans. Am. Math. Soc. 353, 1371–1386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Chen, P.M. Gauthier, The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings. Proc. Amer. Math. Soc. 139, 583–595 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Chen, S. Ponnusamy, X. Wang, Landau-Bloch constants for functions in \(\alpha \)-Bloch spaces and Hardy spaces. Complex Anal. Oper. Theory. 6, 1025–1036 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. C.-H. Chu, Jordan structures in geometry and analysis, Cambridge Tracts in Mathematics, vol. 6 (Cambridge University Press, Cambridge, 2012)

    Google Scholar 

  15. C.-H. Chu, H. Hamada, T. Honda, G. Kohr, Distortion theorems for convex mappings on homogeneous balls. J. Math. Anal. Appl. 369, 437–442 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. C.-H. Chu, H. Hamada, T. Honda, G. Kohr, Distortion of locally biholomorphic Bloch mappings on bounded symmetric domains. J. Math. Anal. Appl. 441, 830–843 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. C.H. Chu, H. Hamada, T. Honda, G. Kohr, Bloch functions on bounded symmetric domains. J. Funct. Anal. 272, 2412–2441 (2017)

    Google Scholar 

  18. F. Colonna, G.R. Easley, D. Singman, Norm of the multiplication operators from \(H^{\infty }\) to the Bloch space of a bounded symmetric domain. J. Math. Anal. Appl. 382, 621–630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. C.H. FitzGerald, S. Gong, The Bloch theorem in several complex variables. J. Geom. Anal. 4, 35–58 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Graham, D. Varolin, Bloch constants in one and several complex variables. Pac. J. Math. 174, 347–357 (1996)

    Article  MATH  Google Scholar 

  21. K.T. Hahn, Higher dimensional generalizations of the Bloch constant and their lower bounds. Trans. Amer. Math. Soc. 179, 263–274 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  22. K.T. Hahn, Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Canad. J. Math. 27, 446–458 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. D.J. Hallenbeck, T.H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory (Pitman, Boston, 1984)

    MATH  Google Scholar 

  24. H. Hamada, A distortion theorem and the Bloch constant for Bloch mappings in \({\mathbb{C}}^{n}\), J. Anal. Math. (to appear)

    Google Scholar 

  25. H. Hamada, Weighted composition operators from \(H^{\infty }\) to the Bloch space of infinite dimensional bounded symmetric domains. Complex Anal. Oper. Theory (to appear)

    Google Scholar 

  26. H. Hamada, T. Honda, G. Kohr, Bohr’s theorem for holomorphic mappings with values in homogeneous balls. Isr. J. Math. 173, 177–187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Hamada, T. Honda, G. Kohr, Linear invariance of locally biholomorphic mappings in the unit ball of a JB\(^*\)-triple. J. Math. Anal. Appl. 385, 326–339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Hamada, T. Honda, G. Kohr, Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB\(^*\)-triple. J. Math. Anal. Appl. 396, 829–843 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Hamada, T. Honda, G. Kohr, Growth and distortion theorems for linearly invariant families on homogeneous unit balls in \(\mathbb{C}^n\). J. Math. Anal. Appl. 407, 398–412 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Hamada, G. Kohr, Pluriharmonic mappings in \(\mathbb{C}^n\) and complex Banach spaces. J. Math. Anal. Appl. 426, 635–658 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Hamada, G. Kohr, \(\alpha \)-Bloch mappings on bounded symmetric domains in \({\mathbb{C}}^{n}\) (preprint)

    Google Scholar 

  32. L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, Proceedings on Infinite Dimensional Holomorphy, Internat. Conference University of Kentucky, Lexington, KY,1973, Lecture Notes in Mathematics (Springer, Berlin, 1974), pp. 13–40

    Google Scholar 

  33. L.K. Hua, Harmonic analysis of functions of several complex variables in the classical domains. Translations of Mathematical Monographs, vol. 6, American Mathematical Society,Providence (1963)

    Google Scholar 

  34. T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  35. W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Kaup, Hermitian Jordan, triple systems and the automorphisms of bounded symmetric domains. Math. Appl. 303, 204–214 (1994)

    MathSciNet  MATH  Google Scholar 

  37. W. Kaup, H. Upmeier, Jordan algebras and symmetric Siegel domains in complex Banach spaces. Math. Z. 157, 179–200 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Li, S. Stević, Weighted composition operators from \(H^{\infty }\) to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, Art. ID 48478, 13 pp (2007)

    Google Scholar 

  39. S. Li, S. Stević, Weighted composition operators between \(H^{\infty }\) and \(\alpha \)-Bloch spaces in the unit ball. Taiwan. J. Math. 12, 1625–1639 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. X.Y. Liu, Bloch functions of several complex variables. Pac. J. Math. 152, 347–363 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. O. Loos, Bounded Symmetric Domains and Jordan Pairs (University of California, Irvine, 1977)

    Google Scholar 

  42. K.M. Madigan, Composition operators on analytic Lipschitz spaces. Proc. Amer. Math. Soc. 119, 465–473 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  43. K.M. Madigan, A. Matheson, Compact composition operators on the Bloch space. Trans. Amer. Math. Soc. 347, 2679–2687 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. M.T. Malavé Ramírez, J. Ramos Ferenández, On a criterion for continuity and compactness of composition operators acting on \(\alpha \)-Bloch spaces, C. R. Acad. Sci. Paris, Ser. I. 351, 23–26 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Ohno, Weighted composition operators between \(H^{\infty }\) and the Bloch space. Taiwan. J. Math. 5, 555–563 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Ohno, K. Stroethoff, R. Zhao, Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33, 191–215 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ch. Pommerenke, On Bloch functions. J. London Math. Soc. 2, 689–695 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  48. R.C. Roan, Composition operators on a space of Lipschitz functions. Rocky Mt. J. Math. 10, 371–379 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Roos, Jordan triple systems, pp. 425–534, in J. Faraut, S. Kaneyuki, A. Koranyi, Q.-K. Lu, G. Roos, Analysis and geometry on complex homogeneous domains, Progress in Mathematics, 185, Birkhauser Boston (Inc, Boston, MA, 2000)

    Google Scholar 

  50. J. Shi, L. Luo, Composition operators on the Bloch space. Acta Math Sinica, Engl. ser. 16, 85–98 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  51. R.M. Timoney, Bloch functions in several complex variables, I. Bull. Lond. Math. Soc. 12, 241–267 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  52. R.M. Timoney, Bloch functions in several complex variables, II. J. Reine Angew. Math. 319, 1–22 (1980)

    MathSciNet  MATH  Google Scholar 

  53. J. Wang, T. Liu, Bloch constant of holomorphic mappings on the unit polydisk of \(\mathbb{C}^n\). Sci. China Ser. A. 51, 652–659 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  54. J. Wang, T. Liu, Distortion theorem for Bloch mappings on the unit ball \(\cal{B}^n\). Acta Math. Sin. (Engl. Ser.) 25, 1583–1590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. F.D. Wicker, Generalized Bloch mappings in complex Hilbert space. Can. J. Math. 29, 299–306 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  56. Z.M. Yan, S. Gong, Bloch constant of holomorphic mappings on bounded symmetric domains. Sci. China Ser. A. 36, 285–299 (1993)

    MathSciNet  MATH  Google Scholar 

  57. M.Z. Zhang, W. Xu, Composition operators on \(\alpha \)-Bloch spaces on the unit ball. Acta Math Sinica, Engl. ser. 23, 1991–2002 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. M. Zhang, H. Chen, Weighted composition operators of \(H^{\infty }\) into \(\alpha \)-Bloch spaces on the unit ball. Acta Math. Sin. (Engl. Ser.) 25, 265–278 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Z. Zhou, J. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains. Michigan Math. J. 50, 381–405 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  60. K. Zhu, Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23, 1143–1177 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  61. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball (Springer, New York, 2005)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Hiroshima Institute of Technology, Hiroshima, 731-5193, Japan

    Tatsuhiro Honda

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  1. Tatsuhiro Honda
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Correspondence to Tatsuhiro Honda .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College London, London, United Kingdom

    Michael Ruzhansky

  2. Department of Mathematics Education and the RINS, Gyeonsang National University, Jinju, Korea (Republic of)

    Yeol Je Cho

  3. Department of Mathematics, Anand International College of Engineering, Jaipur, Rajasthan, India

    Praveen Agarwal

  4. Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Ourense, Spain

    Iván Area

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Honda, T. (2017). Bloch Mappings on Bounded Symmetric Domains. In: Ruzhansky, M., Cho, Y., Agarwal, P., Area, I. (eds) Advances in Real and Complex Analysis with Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-10-4337-6_3

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