Geodesic
On~measures with the doubling condition
Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 629-638.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_1988_30_3_a9,
     author = {A. L. Vol'berg and S. V. Konyagin},
     title = {On~measures with the doubling condition},
     journal = {Izvestiya. Mathematics },
     pages = {629--638},
     publisher = {mathdoc},
     volume = {30},
     number = {3},
     year = {1988},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a9/}
}
TY  - JOUR
AU  - A. L. Vol'berg
AU  - S. V. Konyagin
TI  - On~measures with the doubling condition
JO  - Izvestiya. Mathematics 
PY  - 1988
SP  - 629
EP  - 638
VL  - 30
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a9/
LA  - en
ID  - IM2_1988_30_3_a9
ER  - 
%0 Journal Article
%A A. L. Vol'berg
%A S. V. Konyagin
%T On~measures with the doubling condition
%J Izvestiya. Mathematics 
%D 1988
%P 629-638
%V 30
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a9/
%G en
%F IM2_1988_30_3_a9
A. L. Vol'berg; S. V. Konyagin. On~measures with the doubling condition. Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 629-638. http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a9/

[1] Coifman R. R., Guzman M., “Singular integrals and multipliers on homogeneous spaces”, Rev. Un. Mat. Argentina, 25 (1970), 137–143 | MR | Zbl

[2] Coifman R. R., Weiss G., Analyse harmonique non-commutative sur certains espaces homogenes, Lect. Notes Math., 242, 1971 | MR | Zbl

[3] Calderon A. P., “Inequalities for the maximal function relative to a metric”, Studia Math., 57:3 (1976), 297–306 | MR | Zbl

[4] Stromberg J. O., Torchinsky A., “Weights, Sharp maximal functions and Hardy spaces”, Bull. Amer. Math. Soc., 3:3 (1980), 1053–1056 | DOI | MR

[5] Coifman R. R., Weiss G., “Extension of Hardy spaces and their use in analysis”, Bull. Amer. Math. Soc., 83:4 (1977), 569–645 | DOI | MR | Zbl

[6] Macias R., Segovia C., “A decomposition into atoms of distributions in spaces of homogeneous type”, Adv. Math., 33:3 (1979), 271–309 | DOI | MR | Zbl

[7] Gusman M., Differentsirovanie integralov v $\mathbf{R}^n$, Mir, M., 1978 | MR

[8] Dynkin E. M., “Svobodnaya interpolyatsiya funktsiyami s proizvodnoi iz $H^1$”, Zap. nauchn. sem. LOMI, 126, 1983, 77–83 | MR

[9] Dyn'kin E. M., “Homogeneous measures on subsets of $\mathbf{R}^n$”, Lect. Notes. Math., 1043, 1983, 698–700

[10] Beurling A., Ahlfors L. V., “The boundary correspondence under quasi-conformal mappings”, Acta Math., 96:1,2 (1956), 125–142 | DOI | MR | Zbl

[11] Aleksandrov A. B., “Suschestvovanie vnutrennikh funktsii v share”, Matem. sb., 118:2 (1982), 147–163 | MR | Zbl

[12] Volberg A. L., Konyagin S. V., “Na lyubom kompakte v $\mathbf{R}^n$ suschestvuet odnorodnaya mera”, Dokl. AN SSSR, 278:4 (1984), 783–786 | MR

[13] Kolmogorov A. N., Tikhomirov V. M., “$\varepsilon$-entropiya i $\varepsilon$-emkost mnozhestv v funktsionalnykh prostranstvakh”, Uspekhi matem. nauk, 14:2 (1959), 3–86 | MR

[14] Larman D. G., “A new theory of dimension”, Proc. Lond Math. Soc., 17 (1967), 178–192 | DOI | MR | Zbl