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On the existence of doubling measures with certain regularity properties. (English) Zbl 0957.28006

Let \(X\) be a compact pseudo-metric space. Mainly, it is constructed a doubling measure for which the measure of a dilated ball is closely related to the upper and lower dimensions associated to \(X\), extending (Theorem 2) a result stated by A. L. Vol’berg and S. V. Konyagin [Math. USSR, Izv. 30, No. 3, 629-638 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 666-675 (1987; Zbl 0649.42010)]. The main result of this paper was already formulated for Euclidean spaces by P. Bylund [Doctoral Thesis. Umeå: Univ. of Umeå, Dept. of Mathematics (1994; Zbl 0860.46021)], and for metric spaces by J. Gudayol [Doktoral Dissertation, Univ. of Barcelona (1997)].

MSC:

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
54F45 Dimension theory in general topology
54E45 Compact (locally compact) metric spaces
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[1] Per Bylund, Besov spaces and measures on arbitrary closed sets, Doctoral Thesis, vol. 8, University of Umeå, Department of Mathematics, Umeå, 1994. Dissertation, University of Umeå, Umeå, 1994. · Zbl 0860.46021
[2] E. M. Dynkin, Free interpolation by functions with derivatives in \(H^1\), J. Soviet Math. 27 (1984), 2475-2481. · Zbl 0541.41001
[3] Jaume Gudayol, Boundary behaviour of functions in Hardy-Sobolev spaces, doctoral dissertation, Universitat de Barcelona, 1997.
[4] Alf Jonsson, Besov spaces on closed subsets of \?\(^{n}\), Trans. Amer. Math. Soc. 341 (1994), no. 1, 355 – 370. · Zbl 0803.46035
[5] Alf Jonsson and Hans Wallin, Function spaces on subsets of \(\mathbb{R}^n\), Math. Reports Volume 2, Part 1, Harwood Academics Publ. GmbH, 1984. · Zbl 0875.46003
[6] D. G. Larman, A new theory of dimension, Proc. London Math. Soc. (3) 17 (1967), 178 – 192. · Zbl 0152.24502 · doi:10.1112/plms/s3-17.1.178
[7] Jouni Luukkainen and Eero Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), no. 2, 531 – 534. · Zbl 0897.28007
[8] A. L. Vol\(^{\prime}\)berg and S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 666 – 675 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 3, 629 – 638. · Zbl 0649.42010
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