Geodesic
On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 762-775 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is proved that if a convex density-like differential basis $B$ is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to $B$ can boundedly diverge only on a set of measure zero (this generalizes a theorem of Guzmán and Menarguez); it is established that both translation and homothety invariances are necessary. 
@article{MZM_2004_76_5_a11,
     author = {G. G. Oniani},
     title = {On {Possible} {Values} of {Upper} and {Lower} {Derivatives} with {Respect} to {Convex} {Differential} {Bases}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {762--775},
     year = {2004},
     volume = {76},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a11/}
}
TY  - JOUR
AU  - G. G. Oniani
TI  - On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases
JO  - Matematičeskie zametki
PY  - 2004
SP  - 762
EP  - 775
VL  - 76
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a11/
LA  - ru
ID  - MZM_2004_76_5_a11
ER  - 
%0 Journal Article
%A G. G. Oniani
%T On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases
%J Matematičeskie zametki
%D 2004
%P 762-775
%V 76
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a11/
%G ru
%F MZM_2004_76_5_a11
G. G. Oniani. On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 762-775. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a11/

[1] Gusman M., Differentsirovanie integralov v $\mathbb R^n$, Mir, M., 1978

[2] Saks S., “Remarks on the differentiability of the Lebesgue indefinite integral”, Fund. Math., 22 (1934), 257–261 | Zbl

[3] Bezikovich A. S., “On differentiation of Lebesgue double integrals”, Fund. Math., 25 (1935), 209–216 | Zbl

[4] Ward A., “On the derivation of additive functions of intervals in $m$-dimensional space”, Fund. Math., 28 (1937), 265–279

[5] Saks S., Teoriya integrala, IL, M., 1949

[6] de Guzmán M., Real Variable Methods in Fourier Analysis, Math. Stud., 46, North-Holland, Amsterdam, 1981 | Zbl

[7] Korenovskyy A., Lerner A., Stokolos A., On multidimensional F. Riesz's “rising sun” lemma, , 2003, 22 August E-print math.CA/0308211v1

[8] Zerekidze T., “Differentiation of integrals by bases of type $\Pi$”, Proc. A. Razmadze Math. Inst., 133 (2003), 119–130 | MR | Zbl

[9] Zerekidze T., “On the equivalence and nonequivalence of some differential bases”, Proc. A. Razmadze Math. Inst., 133 (2003), 166–169 | MR | Zbl

[10] Oniani G. G., “O vozmozhnykh znacheniyakh verkhnei i nizhnei proizvodnykh”, Matem. zametki, 64:1 (1998), 107–114 | MR | Zbl

[11] Oniani G. G., Differentsirovanie integralov Lebega, Izd-vo Tbilisskogo un-ta, Tbilisi, 1998

[12] Busemann H., Feller W., “Zur Differentiation der Lebesgueschen Integrale”, Fund. Math., 22 (1934), 226–256 | Zbl

[13] Oniani G., “On strong maximal operators corresponding to different frames, II”, Georgian Math. J., 6:2 (1999), 149–168 | DOI | MR | Zbl

[14] Jessen B., Marcinkiewitcz J., Zygmund A., “Note on the differentiability of multiple integrals”, Fund. Math., 25 (1935), 217–234 | Zbl