Geodesic
Some properties of the normal image of convex functions
Sbornik. Mathematics, Tome 34 (1978) no. 2, pp. 161-171 
N. V. Krylov. Some properties of the normal image of convex functions. Sbornik. Mathematics, Tome 34 (1978) no. 2, pp. 161-171. http://geodesic.mathdoc.fr/item/SM_1978_34_2_a2/
@article{SM_1978_34_2_a2,
     author = {N. V. Krylov},
     title = {Some properties of the normal image of convex functions},
     journal = {Sbornik. Mathematics},
     pages = {161--171},
     year = {1978},
     volume = {34},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1978_34_2_a2/}
}
TY  - JOUR
AU  - N. V. Krylov
TI  - Some properties of the normal image of convex functions
JO  - Sbornik. Mathematics
PY  - 1978
SP  - 161
EP  - 171
VL  - 34
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1978_34_2_a2/
LA  - en
ID  - SM_1978_34_2_a2
ER  - 
%0 Journal Article
%A N. V. Krylov
%T Some properties of the normal image of convex functions
%J Sbornik. Mathematics
%D 1978
%P 161-171
%V 34
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1978_34_2_a2/
%G en
%F SM_1978_34_2_a2

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

Let $z$ be a convex function defined in a convex domain $D$ of a finite-dimensional Euclidean space. Denote by $z^{(n)}$ the convolutions of $z$ with elements of a $\delta$-type sequence of test functions and let $\nu_z$ and $\nu_{z^{(n)}}$ be the measures of normal images corresponding to $z$ and $z^{(n)}$. One of the main results of this work is that $\nu_{z^{(n)}}\to\nu_z$ in variation on a compact $K\subset D$ if and only if $\nu_z$ is absolutely continuous on $K$ with respect to Lebesgue measure. Bibliography: 7 titles.

[1] I. Ya. Bakelman, Geometricheskie metody resheniya ellipticheskikh uravnenii, izd-vo “Nauka”, Moskva, 1964

[2] L. Schwartz, Téorie des distributions 1, Hermann, Paris, 1950 | MR | Zbl

[3] H. Danford, Dzh. T. Shvarts, Lineinye operatory, obschaya teoriya, t. 1, IL, Moskva, 1962

[4] V. I. Smirnov, Kurs vysshei matematiki, t. 5, Fizmatgiz, Moskva, 1959 | MR

[5] A. D. Aleksandrov, “Suschestvovanie pochti vezde vtorogo differentsiala vypukloi funktsii i nekotorye svyazannye s nim svoistva vypuklykh poverkhnostei”, Uchenye zapiski LGU, seriya matem., 37:6 (1939), 3–35 | Zbl

[6] N. V. Krylov, “Posledovatelnosti vypuklykh funktsii i otsenki maksimuma resheniya parabolicheskogo uravneniya”, Sib. matem. zh., 17:2 (1976), 290–303 | MR | Zbl

[7] R. Sh. Liptser, A. N. Shiryaev, Statistika sluchainykh protsessov, izd-vo “Nauka”, Moskva, 1974 | MR