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 Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 2, Pages 439–446 (Mi izv1250)

Smooth measures and the law of the iterated logarithm

N. G. Makarov

Abstract: A measure $\mu$ defined on the unit circle $\partial\mathbf D$ is called smooth if $|\mu(I')-\mu(I")|\leqslant C|I'|$ for any two adjacent intervals, $I',I"\subset\partial\mathbf D$ of equal length. It is shown that smooth measures are absolutely continuous with respect to Hausdorff measure with weight function $t(\log\frac1t\log\log\log\frac1t)^{1/2}$, and that this result is sharp. The results are applied to the well-known problem of the angular derivative of a univalent function.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:2, 455–463

Bibliographic databases:

UDC: 517.5
MSC: Primary 26A30; Secondary 60F15, 30C35

Citation: N. G. Makarov, “Smooth measures and the law of the iterated logarithm”, Izv. Akad. Nauk SSSR Ser. Mat., 53:2 (1989), 439–446; Math. USSR-Izv., 34:2 (1990), 455–463

Citation in format AMSBIB
\Bibitem{Mak89} \by N.~G.~Makarov \paper Smooth measures and the law of the iterated logarithm \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1989 \vol 53 \issue 2 \pages 439--446 \mathnet{http://mi.mathnet.ru/izv1250} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=998306} \zmath{https://zbmath.org/?q=an:0691.30026|0685.30025} \transl \jour Math. USSR-Izv. \yr 1990 \vol 34 \issue 2 \pages 455--463 \crossref{https://doi.org/10.1070/IM1990v034n02ABEH000664}