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Uniform Domains in Real Banach Spaces

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Abstract

In this paper, we show that certain types of domains in real Banach spaces E with dimension at least two are uniform domains. Our examples incude annulus domains, bounded convex domains C and their complements \(E{\setminus } \overline{C}\), and \(C{\setminus } \alpha \overline{C}\) for all \(0<\alpha <1\) when the zero vector \(o\in C\).

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References

  1. Conway, J.B.: A First Course in Functional Analysis. Graduate Texts in Mathematics, vol. 96, 2nd edn. Springer, New York, xvi+399 pp (1990)

  2. Gehring, F.W., Hag, K.: The Ubiquitous Quasidisk. With Contributions by Ole Jacob Broch. Mathematical Surveys and Monographs, vol. 184. American Mathematical Society, Providence, xii+171 pp (2012)

  3. Guan, T., Zhou, Q., Ponnusamy, S.: A note on \(\partial \)-biLipschitz map**s in quasiconvex metric spaces. Bull. Sci. Math. 176 , Paper No. 103128, 19 pp (2022)

  4. Li, Y., Rasila, A., Zhou, Q.: Removability of uniform metric spaces. Mediterr. J. Math. 19, no. 3, Paper No. 139, 13 pp (2022)

  5. Martio, O.: Definitions of uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 197–205 (1980)

    Article  MathSciNet  Google Scholar 

  6. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 383–401 (1978)

    Article  MathSciNet  Google Scholar 

  7. Väisälä, J.: Uniform domains. Tohoku Math. J. 40, 101–118 (1988)

    Article  MathSciNet  Google Scholar 

  8. Väisälä, J.: Free quasiconformality in Banach spaces. II. Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 255–310 (1991)

    Article  MathSciNet  Google Scholar 

  9. Väisälä, J.: Relatively and inner uniform domains. Conform. Geom. Dyn. 2, 56–88 (1998)

    Article  MathSciNet  Google Scholar 

  10. Väisälä, J.: The Free Quasiworld. Freely Quasiconformal and Related Maps in Banach Spaces, Quasiconformal Geometry and Dynamics (Lublin 1996), vol. 48. Banach Center Publications, pp. 55–118 (1999)

  11. Zhou, Q., Li, Y., Rasila, A.: Gromov hyperbolicity, John spaces and quasihyperbolic geodesic. J. Geom. Anal. 32 , no. 9, Paper No. 228, 15 pp (2022)

  12. Zhou, Q., Ponnusamy, S.: Gromov hyperbolicity in the free quasiworld I. Studia Math. 268(1), 23–49 (2023)

    Article  MathSciNet  Google Scholar 

  13. Zhou, Q., Rasila, A.: Quasimöbius invariance of uniform domains. Studia Math. 261, 1–24 (2021)

    Article  MathSciNet  Google Scholar 

  14. Zhou, Q., Rasila, A.: Teichmüller’s problem on Gromov hyperbolic domains. Isr. J. Math. 252, 399–427 (2022)

    Article  Google Scholar 

  15. Zhou, Q., Rasila, A.: Geometric characterization of Gromov hyperbolic Hölder domains. Forum Math. 34(6), 1621–1640 (2022)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Tiantian Guan was partly supported by NNSF of China (No. 12201115), by the Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515110484), and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212030010).

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Zhengyong Ouyang & Tiantian Guan

  2. School of Information Science and Technology, **amen University Tan Kah Kee College, ZhangZhou, 363105, Fujian, People’s Republic of China

    Bo Jiao

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  1. Zhengyong Ouyang
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  2. Bo Jiao
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  3. Tiantian Guan
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Correspondence to Tiantian Guan.

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Communicated by Saminathan Ponnusamy.

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Ouyang, Z., Jiao, B. & Guan, T. Uniform Domains in Real Banach Spaces. Bull. Malays. Math. Sci. Soc. 47, 13 (2024). https://doi.org/10.1007/s40840-023-01612-0

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  • DOI: https://doi.org/10.1007/s40840-023-01612-0

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