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THE QUASIHYPERBOLIC METRIC AND ANALOGUES OF THE HARDY-LITTLEWOOD PROPERTY FOR α = 0 IN UNIFORMLY JOHN DOMAINS

  • Kim, Ki-Won (DEPARTMENT OF MATHEMATICS EDUCATION, SILLA UNIVERSITY)
  • Published : 2006.05.01

Abstract

We characterize the class of uniformly John domains in terms of the quasihyperbolic metric and from the result we get some analogues of the Hardy-Littlewood property for ${\alpha} = 0$ in uniformly John domains.

Keywords

References

  1. Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihy- perbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana 12 (1996), no. 2, 299-336
  2. Z. Balogh and A. Volberg, Geometric localization, uniformly John property and separated semihy- perbolic dynamics, Ark. Mat. 34 (1996), no. 1, 21-49 https://doi.org/10.1007/BF02559505
  3. F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John do- mains, Math. Scand. 65 (1989), no. 1, 75-92 https://doi.org/10.7146/math.scand.a-12267
  4. F. W. Gehring and O. Martio, Quasidisks and the Hardy-Littlewood property, Complex Variables Theory Appl. 2 (1983), no. 1, 67-78 https://doi.org/10.1080/17476938308814032
  5. F. W. Gehring and O. Martio, Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 203-219 https://doi.org/10.5186/aasfm.1985.1022
  6. F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50-74 https://doi.org/10.1007/BF02798768
  7. F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172-199 https://doi.org/10.1007/BF02786713
  8. G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403-439 https://doi.org/10.1007/BF01180596
  9. K. Kim, Lipschitz class, growth of derivative and uniformly John domains, East Asian Math. J. 19 (2003), 291-303
  10. K. Kim, Hardy-Littlewood property with the inner length metric, Commun. Ko- rean Math. Soc. 19 (2004), no. 1, 53-62 https://doi.org/10.4134/CKMS.2004.19.1.053
  11. K. Kim and N. Langmeyer, Harmonic measure and hyperbolic distance in John disks, Math. Scand. 83 (1998), no. 2, 283-299 https://doi.org/10.7146/math.scand.a-13857
  12. R. Kaufman and J. -M. Wu, Distances and the Hardy-Littlewood property, Com- plex Variables Theory Appl. 4 (1984), no. 1, 1-5 https://doi.org/10.1080/17476938408814086
  13. N. Langmeyer, The quasihyperbolic metric, growth, and John domains, Univer- sity of Michigan Ph.D. Thesis (1996)
  14. N. Langmeyer, The quasihyperbolic metric, growth, and John domains, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 205-224
  15. V. Lappalainen, Lip$_{h}$-extension domains, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 56 (1985), 52pp
  16. R. Nakki and J. Vaisala, John disks, Exposition. Math. 9 (1991), no. 1, 3-43

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  1. INNER UNIFORM DOMAINS, THE QUASIHYPERBOLIC METRIC AND WEAK BLOCH FUNCTIONS vol.49, pp.1, 2012, https://doi.org/10.4134/BKMS.2012.49.1.011
  2. INNER UNIFORM DOMAINS AND THE APOLLONIAN INNER METRIC vol.50, pp.6, 2013, https://doi.org/10.4134/BKMS.2013.50.6.1873