Geodesic
On properties of functions of bounded variation on a set
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 41-66 
T. P. Lukashenko. On properties of functions of bounded variation on a set. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 41-66. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Kolmogorov inequality for conjugate functions is generalized in § 1. Theorem 2 is the main result; it shows, for example, that if a function $F$ is $2\pi$-periodic to within linearity and of bounded variation in the narrow sense on a set $E\subset[0,2\pi)$, then for any $\lambda>0$ $$ \bigg|\bigg\{x\in E:\sup_{0\leqslant r>1}|\overline{F'}(r,x)|>\lambda\bigg\}\bigg|^*\leqslant\frac C\lambda{\operatornamewithlimits{Var}_E}^*F. $$ In § 2 a well-known theorem of F. and M. Riesz is generalized. In particular, the following is proved. Theorem 5. {\it Suppose that a $2\pi$-periodic integrable function $\Phi$ and its conjugate $\overline\Phi$ are defined everywhere$,$ bounded$,$ and of bounded variation in the narrow sense on a set $E\subset[0,2\pi),$ and that $\Phi(x)=\lim_{E\ni t\to x}\Phi(t)$ and $\overline\Phi(x)=\lim_{E\ni t\to x}\overline\Phi(t)$ if $\lim_{E\ni t\to x}\Phi(t)$ and $\lim_{E\ni t\to x}\overline\Phi(t)$ exist at a point $x$. Then $\Phi$ and $\overline\Phi$ are absolutely continuous in the narrow sense on $E$.} Bibliography: 14 titles.

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[11] Lukashenko T. P., “O funktsiyakh s ogranichennym izmeneniem na mnozhestve”, Matematika. Tezisy dokl. sedmoi kazakhst. mezhvuz. nauch. konf. po matem. i mekh. (15-18 sentyabrya 1981 g.), Izd-vo Karag. gos. un-ta, Karaganda, 1981, 28–29

[12] Lukashenko T. P., “O funktsiyakh obobschennoi ogranichennoi variatsii”, Izvestiya AN SSSR. Ser. matem., 46 (1982), 276–313 | MR

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