On uniform properties of doubling measures

Ruzhansky, Michael

doi:10.1090/S0002-9939-01-05931-7

On uniform properties of doubling measures
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Proc. Amer. Math. Soc. 129 (2001), 3413-3416 Request permission

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

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, DOI 10.1155/S1073792896000013
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.
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
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, DOI 10.4310/MRL.1997.v4.n6.a1
G. Lu, S. Pedersen and M. Ruzhansky, ‘Distribution theory, high order gradients and polynomials on metric spaces’, in preparation.
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, DOI 10.1512/iumj.1998.47.1494
G. Lu and R.L. Wheeden, ‘High order representation formulas and embedding theorems on stratified groups and generalizations’, to appear in Studia Math.
Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MR 1301775, DOI 10.1007/BFb0073952
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131