Ukrainian Mathematical Bulletin
Volume 2, Issue 3, 2005 pp. 403-424.
Finite Mean Oscillation in the Mapping TheoryAuthors: Andrei A. Ignatyev and Vladimir I. Ryazanov
Author institution: Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 74 Rosa Luxemburg Str., 83114, Donetsk, Ukraine
Summary: We say that a function $Q(x)$ has a finite mean oscillation at a point if its mean deviation from the mean value is bounded for all balls with sufficiently small radii centered at this point (or, in other words, if the variance is bounded for all small balls centered at the indicated point). It is shown that isolated singularities are removable for $Q$-homeomorphisms provided that $Q(x)$ has a finite mean oscillation at the point. We also prove an analog of the well-known Painlev\'{e} theorem for analytic functions under the condition that the mean oscillation of $Q(x)$ is finite on a singular set of length zero. The results can be applied to various classes of mappings with finite distortion.
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