INNER UNIFORM DOMAINS, THE QUASIHYPERBOLIC METRIC AND WEAK BLOCH FUNCTIONS

• Kim, Ki-Won (Department of Mathematics Education Silla University)
• Published : 2012.01.31

Abstract

We characterize the class of inner uniform domains in terms of the quasihyperbolic metric and the quasihyperbolic geodesic. We also characterize uniform domains and inner uniform domains in terms of weak Bloch functions.

References

1. Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihyperbolic 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 semihyperbolic dynamics, Ark. Mat. 34 (1996), no. 1, 21-49. https://doi.org/10.1007/BF02559505
3. M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Asteroque 270 (2001), viii+99 pp.
4. O. J. Broch, Geometry of John disks, Norwegian University of Science and Technology Doctoral Thesis, 2005.
5. F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand. 65 (1989), no. 1, 75-92. https://doi.org/10.7146/math.scand.a-12267
6. F. W. Gehring and W. F. Hayman, An inequality in the theory of conformal mapping, J. Math. Pures Appl. (9) 41 (1962), 353-361.
7. 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
8. F. W. Gehring and B. P. Palka, Quasiconformal homogeneous domains, J. Analyse Math. 30 (1976), 172-199. https://doi.org/10.1007/BF02786713
9. J. Heinonen and S. Rohde, The Gehring and Hayman inequality for quasihyperbolic geodesics, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 3, 393-405. https://doi.org/10.1017/S0305004100071681
10. K. Kim, The quasihyperbolic metric and analogues of the Hardy-Littlewood property for $\alpha$ = 0 in uniformly John domains, Bull. Korean Math. Soc. 43 (2006), no. 2, 395-410. https://doi.org/10.4134/BKMS.2006.43.2.395
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. N. Langmeyer, The quasihyperbolic metric, growth and John domains, University of Michigan Ph.D. Thesis, 1996.
13. N. Langmeyer, The quasihyperbolic metric, growth and John domains, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 205-224.
14. J. Vaisala, Relatively and inner uniform domains, Conform. Geom. Dyn. 2 (1998), 56-88. https://doi.org/10.1090/S1088-4173-98-00022-8

Cited by

1. WEAK BLOCH FUNCTIONS, ∅-UNIFORM AND ∅-JOHN DOMAINS vol.19, pp.4, 2012, https://doi.org/10.7468/jksmeb.2012.19.4.423
2. INNER UNIFORM DOMAINS AND THE APOLLONIAN INNER METRIC vol.50, pp.6, 2013, https://doi.org/10.4134/BKMS.2013.50.6.1873