Geodesic
A result on differentiable measures on a linear space
Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 217-222 Cet article a éte moissonné depuis la source Math-Net.Ru

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The basic content of this note is the proof of the following result. Let $X$ be a linear space, let $L$ be a subspace of it with $\dim L=m<\infty$, let $R$ be a ring of subsets of $X$ which is invariant with respect to shifts by vectors in $L$, and let $\sigma$ be a finitely additive bounded quasi-content on $R$ which is differentiable $n$ times with respect to the subspace $L$. Then, for any bounded set $W\subset L$, $$ \lim_{r\to0}\sup_{L^c}\frac{|\sigma|(rW+L^c)}{r^{mn/(m+n)}}=0, $$ where $L^c$ is a linear complement to $L$ with respect to $X$, and $|\sigma|$ is the total variation of the quasi-content $\sigma$. Bibliography: 2 titles. 
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     author = {A. V. Uglanov},
     title = {A~result on differentiable measures on a~linear space},
     journal = {Sbornik. Mathematics},
     pages = {217--222},
     year = {1976},
     volume = {29},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_29_2_a6/}
}
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%F SM_1976_29_2_a6
A. V. Uglanov. A result on differentiable measures on a linear space. Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 217-222. http://geodesic.mathdoc.fr/item/SM_1976_29_2_a6/

[1] V. I. Averbukh, O. G. Smolyanov, S. V. Fomin, “Obobschennye funktsii i differentsialnye uravneniya v lineinykh prostranstvakh. I: Differentsiruemye mery”, Trudy Mosk. matem. ob-va, XXIV (1971), 133–174

[2] A. N. Kolmogorov, V. M. Tikhomirov, “$\varepsilon$-entropiya i $\varepsilon$-emkost mnozhestv v funktsionalnykh prostranstvakh”, Uspekhi matem. nauk, XXIV:2 (146) (1959), 3–86 | MR