Abstract
In this paper, we show that Loewner spaces introduced by Heinonen and Koskela (Acta Math., 1998) are preserved under quasimöbius mappings between Ahlfors regular spaces.
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Balogh, Z., Buckley, S.: Sphericalization and flattening. Conform. Geom. Dyn. 9, 76–101 (2005)
Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150, 127–183 (2002)
Bonk, M., Kleiner, B.: Rigidity for quasi-Möbius group actions. J. Differential Geom. 61, 81–106 (2002)
Brania, A., Yang, S.: Domains with controlled modulus and quasiconformal mappings. Nonlinear Stud. 9, 57–74 (2002)
Buckley, S.M., Herron, D., Xie, X.: Metric space inversions, quasihyperbolic distance, and uniform spaces. Indiana Univ. Math. J. 57, 837–890 (2008)
David, G., Semmes, S.: Fractured fractals and broken dreams: self-similar geometry through metric and measure. Oxford lecture series in mathematics and its applifications, 7. Clarendon Press, Oxford (1997)
Heinonen, J.: Lectures on analysis on metric spaces. Springer-Verlag, Berlin-Heidelberg-New York (2001)
Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)
Li, X., Shanmugalingam, N.: Preservation of bounded geometry under sphericalization and flattening. Indiana. Math. J. 64, 1303–1341 (2015)
Tyson, J.: Quasiconformality and quasisymmetry in metric measure spaces. Ann. Acad. Sci. Fenn. Math. 23, 525–548 (1998)
Väisälä, J., Quasi-Möbius maps, J. Anal. Math. 44, 218–234 (1984/85)
Wang, X., Zhou, Q.: Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces. Ann. Acad. Sci. Fenn. Ser. AI Math. 42, 257–284 (2017)
Zhou, Q., Li, X., Li, Y.: Sphericalizations and applications in Gromov hyperbolic spaces. J. Math. Anal. Appl. 509, 125948 (2022)
Zhou, Q., Li, X., Li, Y.: Deformations on symbolic Cantor sets and ultrametric spaces. Bull. Malays. Math. Sci. Soc. 43(4), 3259–3270 (2020)
Zhou, Q. and Ponnusamy, S.: Gromov hyperbolic type metrics and quasimöbius invariance of uniform domains, Submitted
Zhou, Q., Rasila, A.: Quasimöbius invariance of uniform domains. Stud. Math. 261(1), 1–24 (2021)
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Communicated by Saminathan Ponnusamy.
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The first author was supported by Department of Education of Guangdong Province, China (No. 2021KTSCX116), by Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515012289), and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212030010). The second author was supported by National Natural Science Foundation of Hunan Province (No. 2021JJ3016), by Scientific Research Fund of Hunan Provincial Education Department (Nos. 20B118, 18C0253), by NNSF of China (No. 11971124) and by NSF of Guangdong Province (No. 2021A1515010326).
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Guan, T., Li, Y. Quasimöbius invariance of Loewner spaces. Bull. Malays. Math. Sci. Soc. 45, 1903–1912 (2022). https://doi.org/10.1007/s40840-022-01296-y
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DOI: https://doi.org/10.1007/s40840-022-01296-y