Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hausdorff dimension and doubling measures
on metric spaces

Author: Jang-Mei Wu
Journal: Proc. Amer. Math. Soc. 126 (1998), 1453-1459
MSC (1991): Primary 28C15; Secondary 54E35, 54E45
MathSciNet review: 1443418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Vol$'$berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any $\alpha > 0$, the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most $\alpha $.

References [Enhancements On Off] (What's this?)

  • [BA] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mapping, Acta Math. 96 (1956), 125-142. MR 19:258c
  • [FKP] R. Fefferman, C. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65-124. MR 93h:31010
  • [KW] R. Kaufman and J.-M. Wu, Two problems on doubling measures, Revista Mat. Iberoamericana 11 (1995), 527-545. CMP 96:05
  • [T] P. Tukia, Hausdorff dimension and quasisymmetric mappings, Math. Scand. 65 (1989), 152-160. MR 92b:30026
  • [VK] A.L. Vol'berg and S.V. Konyagin, On measures with the doubling condition, Math. USSR Izvestiya 30 (1988), 629-638. MR 88i:28006

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28C15, 54E35, 54E45

Retrieve articles in all journals with MSC (1991): 28C15, 54E35, 54E45

Additional Information

Jang-Mei Wu
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Keywords: Doubling measure, metric space, Hausdorff dimension
Received by editor(s): October 24, 1996
Additional Notes: Partially supported by the National Science Foundation
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society