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)

 

 

On uniform properties of doubling measures


Author: Michael Ruzhansky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3413-3416
MSC (2000): Primary 54E35, 54E50, 46A03, 28E15
Published electronically: May 3, 2001
MathSciNet review: 1845020
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $E, B$ such that $E\subset B\subset X$. As a consequence we show that if $X$ is a linear metric doubling space, then it must be finite dimensional.


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

  • 1. Bruno Franchi, Guozhen Lu, and Richard 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 1383947, 10.1155/S1073792896000013
  • 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. Yongping Liu, Guozhen Lu, and Richard 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), no. 7, 35–59. MR 1702600
  • 4. Guozhen Lu, Local and global interpolation inequalities on the Folland-Stein Sobolev spaces and polynomials on stratified groups, Math. Res. Lett. 4 (1997), no. 6, 777–790. MR 1492120, 10.4310/MRL.1997.v4.n6.a1
  • 5. G. LU, S. PEDERSEN and M. RUZHANSKY, `Distribution theory, high order gradients and polynomials on metric spaces', in preparation.
  • 6. Guozhen Lu and Richard L. Wheeden, Poincaré inequalities, isoperimetric estimates, and representation formulas on product spaces, Indiana Univ. Math. J. 47 (1998), no. 1, 123–151. MR 1631545, 10.1512/iumj.1998.47.1494
  • 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. Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MR 1301775
  • 9. François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54E35, 54E50, 46A03, 28E15

Retrieve articles in all journals with MSC (2000): 54E35, 54E50, 46A03, 28E15


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: http://dx.doi.org/10.1090/S0002-9939-01-05931-7
Received by editor(s): November 23, 1999
Received by editor(s) in revised form: March 24, 2000
Published electronically: May 3, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society