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On properties of functions of bounded variation on a set
T. P. Lukashenko
Abstract:
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.
Received: 24.06.1982
Citation:
T. P. Lukashenko, “On properties of functions of bounded variation on a set”, Mat. Sb. (N.S.), 122(164):1(9) (1983), 41–63; Math. USSR-Sb., 50:1 (1985), 41–66
Linking options:
https://www.mathnet.ru/eng/sm2273https://doi.org/10.1070/SM1985v050n01ABEH002732 https://www.mathnet.ru/eng/sm/v164/i1/p41
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Abstract page: | 684 | Russian version PDF: | 153 | English version PDF: | 14 | References: | 72 | First page: | 3 |
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