Geodesic
Smooth measures and the law of the iterated logarithm
Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 455-463.

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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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N. G. Makarov. Smooth measures and the law of the iterated logarithm. Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 455-463. http://geodesic.mathdoc.fr/item/IM2_1990_34_2_a12/

[1] Zigmund L., Trigonometricheskie ryady, Mir, M., 1965

[2] Karleson L., Izbrannye problemy teorii isklyuchitelnykh mnozhestv, Mir, M., 1971 | MR | Zbl

[3] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[4] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, Mir, M., 1984

[5] Duren P., Shapiro H., Shields A., “Singular measures and domains not of Smirnov type”, Duke Math. J., 33 (1966), 247–254 | DOI | MR | Zbl

[6] Hamilton D., Conformal distortion of boundary sets, Preprint, 1986 | MR

[7] Kahane J.-P., “Trois notes sur les ensembles parfaits linéaires”, Enseignement Math., 15 (1969), 185–192 | MR

[8] Lohwater A., “The boundary behaviour of derivatives of univalent functions”, Math. Z., 113 (1971), 115–120 | DOI | MR

[9] Makarov N. G., “On the distortion of boundary sets under conformal mappings”, Proc. London Math. Soc. (3), 51 (1985), 369–384 | DOI | MR | Zbl

[10] Makarov N. G., On the harmonic measure of the snowflake, Preprint No E-4-86, LOMI, L., 1986

[11] Piranian G., “Two monotonic, singular, uniformly almost smooth functions”, Duke Math. J., 33 (1966), 255–262 | DOI | MR | Zbl

[12] Pommerenke Ch. (ed.), “Problems in complex function theory”, Bull. London Math. Soc., 4 (1972), 354–366 | DOI | MR | Zbl

[13] Pommerenke Ch., Univalent Functions, Vandenhoeck Ruprecht, Göttingen, 1975 | MR | Zbl

[14] Pommerenke Ch., “The growth of the derivative of a univalent function”, Math. Surveys and Monographs, 21, 1986, 143–152 | MR