### On boundedness of maximal functions in Sobolev spaces.

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We consider the simplest form of a second order, linear, degenerate, elliptic equation with divergence structure in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.

We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Hölder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p > 1 but not necessarily for p = 1.

We establish continuity, openness and discreteness, and the condition (N) for mappings of finite distortion under minimal integrability assumptions on the distortion.

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