Smooth measures and the law of the iterated logarithm

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.
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