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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 3, Pages 666–675 (Mi izv1314)  

This article is cited in 21 scientific papers (total in 22 papers)

On measures with the doubling condition

A. L. Vol'berg, S. V. Konyagin

Abstract: The author characterizes metric spaces $(X,\rho)$ that carry a nontrivial measure $\mu$ with the doubling condition
$$ \forall x\in X,\quad R>0\qquad \mu(B(x,2R))\leqslant C\mu(B(x,R)), $$
where $B(x,R)=\{y:\rho(x,y)\leqslant R\}$. In particular, such a measure exists on any compact set in $\mathbf R^n$.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 30:3, 629–638

Bibliographic databases:

UDC: 519.53
MSC: Primary 28C15; Secondary 54E53, 54E45
Received: 07.05.1985

Citation: A. L. Vol'berg, S. V. Konyagin, “On measures with the doubling condition”, Izv. Akad. Nauk SSSR Ser. Mat., 51:3 (1987), 666–675; Math. USSR-Izv., 30:3 (1988), 629–638

Citation in format AMSBIB
\by A.~L.~Vol'berg, S.~V.~Konyagin
\paper On~measures with the doubling condition
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 3
\pages 666--675
\jour Math. USSR-Izv.
\yr 1988
\vol 30
\issue 3
\pages 629--638

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    2. Tord Sjödin, “Ons-sets and mutual absolute continuity of measures on homogeneous spaces”, manuscripta math, 94:1 (1997), 169  crossref  mathscinet  zmath  isi
    3. N. O. Belova, “Ob isklyuchitelnykh mnozhestvakh v $L_p$-teorii potentsiala”, Vladikavk. matem. zhurn., 1:1 (1999), 5–22  mathnet  mathscinet  zmath  elib
    4. H. Triebel, “Spaces on sets”, Russian Math. Surveys, 60:6 (2005), 1195–1215  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Manli Lou, Shengyou Wen, Min Wu, “Two examples on atomic doubling measures”, Journal of Mathematical Analysis and Applications, 333:2 (2007), 1111  crossref
    6. P. Shvartsman, “On extensions of Sobolev functions defined on regular subsets of metric measure spaces”, Journal of Approximation Theory, 144:2 (2007), 139  crossref
    7. Xiaohua Wang, Shengyou Wen, Zhixiong Wen, “Doubling measures on Whitney modification sets”, Journal of Mathematical Analysis and Applications, 342:2 (2008), 761  crossref
    8. Dan Han, Lisha Wang, Shengyou Wen, “Thickness and thinness of uniform Cantor sets for doubling measures”, Nonlinearity, 22:3 (2009), 545  crossref  isi
    9. P. Shvartsman, “Sobolev -spaces on closed subsets of”, Advances in Mathematics, 220:6 (2009), 1842  crossref
    10. FengJi Peng, ShengYou Wen, “Fatness and thinness of uniform Cantor sets for doubling measures”, Sci. China Math, 54:1 (2011), 75  crossref
    11. Wen Wang, Shengyou Wen, Zhixiong Wen, “Note on atomic doubling measures, quasisymmetrically thin sets and thick sets”, Journal of Mathematical Analysis and Applications, 2011  crossref
    12. Zhong Wang, Wei Wang, Joon-Mo Kim, Bhavani Thuraisingham, Weili Wu, “PTAS for the minimum weighted dominating set in growth bounded graphs”, J Glob Optim, 2011  crossref
    13. Yu Xia Dai, Sheng You Wen, Zhi Xiong Wen, “A 1 weights and d-homogeneous measures on Ahlfors d-regular space”, Acta. Math. Sin.-English Ser, 2011  crossref
    14. Xiaohua Wang, Shengyou Wen, “Doubling measures on Cantor sets and their extensions”, Acta Math Hung, 2011  crossref
    15. Marianna Csörnyei, Ville Suomala, “On Cantor sets and doubling measures”, Journal of Mathematical Analysis and Applications, 393:2 (2012), 680  crossref
    16. M. DAI, L. OLSEN, Y. SHI, “MULTIFRACTAL q RÉNYI DIMENSIONS OF POLISH SPACES FOR q < 1”, Stoch. Dyn, 12:03 (2012), 1150027  crossref
    17. Manli Lou, Wen Wang, Lifeng Xi, “Thickness and thinness of λ-Moran sets for doubling measures”, Chaos, Solitons & Fractals, 52 (2013), 53  crossref
    18. Peng SUN, Xiaohua WANG, “Doubling Measures on Generalized Cantor Sets”, Acta Mathematica Scientia, 33:6 (2013), 1551  crossref
    19. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. Chun Wei, Shengyou Wen, Zhixiong Wen, “Doubling measures on uniform Cantor sets”, Journal of Mathematical Analysis and Applications, 2015  crossref
    21. Peng F., Wen Sh., “On Distribution of Local Dimensions of Doubling Measures on Euclidean Space”, J. Math. Anal. Appl., 453:1 (2017), 502–508  crossref  isi
    22. B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, K. S. Ryutin, I. D. Shkredov, “Sergei Vladimirovich Konyagin turns 60”, Proc. Steklov Inst. Math., 303 (2018), 1–9  mathnet  crossref  crossref  mathscinet  isi  elib
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