On properties of functions of bounded variation on a~set

T. P. Lukashenko

Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 41-66

Résumé

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.