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Proceedings of the American Mathematical Society

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Every complete doubling metric space
carries a doubling measure


Authors: Jouni Luukkainen and Eero Saksman
Journal: Proc. Amer. Math. Soc. 126 (1998), 531-534
MSC (1991): Primary 28A12; Secondary 54F45
DOI: https://doi.org/10.1090/S0002-9939-98-04201-4
MathSciNet review: 1443161
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a complete metric space $X$ carries a doubling measure if and only if $X$ is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on $X$ and of the homogeneity exponents of $X$ are equal. We also show that a closed subset $X$ of $\mathbf{R}^{n}$ carries a measure of homogeneity exponent $n$. These results are based on the case of compact $X$ due to Vol$^{\prime }$berg and Konyagin.


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

  • [A1] P. Assouad, Étude d'une dimension métrique liée à la possibilité de plongements dans $\mathbf R^n$, C. R. Acad. Sci. Paris Sér. A 288 (1979), 731-734. MR 80f:54030
  • [A2] -, Pseudodistances, facteurs et dimension métrique, Séminaire d'Analyse Harmonique 1979-1980, Publ. Math. Orsay 80, 7, Univ. Paris XI, Orsay, 1980, pp. 1-33. MR 84h:54027
  • [CW] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin, 1971. MR 58:17690
  • [D1] E. M. Dyn$^{\prime }$kin, Free interpolation by functions with derivatives in $H^{1}$, J. Soviet Math. 27 (1984), 2475-2481. (Russian) MR 84h:30052
  • [D2] -, Homogeneous measures on subsets of $\mathbf R^n$, Linear and complex analysis problem book (V. P. Havin, S. V. Hru\v{s}\v{c}ëv, and N. K. Nikol$^{\prime }$skii, eds.), Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 698-699.
  • [HR] E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, Springer-Verlag, Berlin, 1963. MR 28:158
  • [J1] A. Jonsson, Besov spaces on closed subsets of $\mathbf R^n$, Trans. Amer. Math. Soc. 341 (1994), 355-370. MR 94c:46065
  • [J2] -, Measures satisfying a refined doubling condition and absolute continuity, Proc. Amer. Math. Soc. 123 (1995), 2441-2446. MR 95j:28004
  • [Lu] J. Luukkainen, Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures, manuscript, 1996.
  • [VK1] A. L. Vol$^{\prime }$berg and S. V. Konyagin, There is a homogeneous measure on any compact subset in $\mathbf R^n$, Soviet Math. Dokl. 30 (1984), 453-456. (Russian) MR 86d:28018
  • [VK2] -, On measures with the doubling condition, Math. USSR-Izv. 30 (1988), 629-638. (Russian) MR 88i:28006

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Additional Information

Jouni Luukkainen
Affiliation: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland
Email: luukkain@cc.helsinki.fi

Eero Saksman
Affiliation: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland
Email: saksman@cc.helsinki.fi

DOI: https://doi.org/10.1090/S0002-9939-98-04201-4
Keywords: Doubling metric space, homogeneous metric space, Assouad dimension, doubling measure, homogeneous measure
Received by editor(s): August 20, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

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