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ISSN 1088-6826(e) ISSN 0002-9939(p)

On uniform properties of doubling measures

Author(s): Michael Ruzhansky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3413-3416.
MSC (2000): Primary 54E35, 54E50, 46A03, 28E15
Posted: May 3, 2001
MathSciNet review: 1845020
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Abstract:

In this paper we prove that if $(X,d,\mu)$ is a metric doubling space with segment property, then $\inf r(E)/r(B)>0$ if and only if $\inf \mu(E)/\mu(B)>0$ , where the infimum is taken over any collection $\mathcal{C}$ of balls such that $E\subset B\subset X$ . As a consequence we show that if is a linear metric doubling space, then it must be finite dimensional.

References:

1.: B. FRANCHI, G. LU and R.L. WHEEDEN, `A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type', Internat. Math. Res. Notices 1 (1996) 1-14. MR 97k:26012
2.: Y. LIU, G. LU and R.L. WHEEDEN, `Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces', preprint.
3.: Y. LIU, G. LU and R.L. WHEEDEN, `Representation formulas and Sobolev spaces of high order on stratified groups and generalizations to metric spaces', Math. Sci. Res. Hot-Line 3 (1999), 35-59. MR 2000f:46044
4.: G. LU, `Local and global interpolation inequalities on the Folland-Stein Sobolev spaces and polynomials on stratified groups', Math. Res. Lett. 4 (1997) 777-790. MR 99i:46024
5.: G. LU, S. PEDERSEN and M. RUZHANSKY, `Distribution theory, high order gradients and polynomials on metric spaces', in preparation.
6.: G. LU and R.L. WHEEDEN, `Poincaré inequalities, isoperimetric estimates, and representation formulas on product spaces', Indiana Univ. Math. J. 47 (1998) 123-151. MR 99e:46043
7.: G. LU and R.L. WHEEDEN, `High order representation formulas and embedding theorems on stratified groups and generalizations', to appear in Studia Math.
8.: N. OBATA, White noise calculus and Fock space (Springer, Berlin, New York, 1994). MR 96e:60061
9.: F. TREVES, Topological vector spaces, distributions and kernels (Academic Press, New York, London, 1967). MR 37:726

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Additional Information:

Michael Ruzhansky
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
Address at time of publication: Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, England
Email: ruzh@maths.ed.ac.uk, ruzh@ic.ac.uk

DOI: 10.1090/S0002-9939-01-05931-7
PII: S 0002-9939(01)05931-7
Received by editor(s): November 23, 1999
Received by editor(s) in revised form: March 24, 2000
Posted: May 3, 2001
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 2001, American Mathematical Society

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