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In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

We introduce a weaker variant of the concept of linear local connectivity, sufficient to guarantee the extendability of a conformal map $f\colon\mathbb D\to\Omega$ to the entire plane as a homeomorphism of locally exponentially integrable distortion. Additionally, we show that a conformal map as above cannot necessarily be extended in this manner if we assume that $\Omega$ is the image of $\mathbb D$ under a self-homeomorphism of the plane that has locally exponentially integrable distortion.

We study the optimal conditions on a homeomorphism f : Ω → Rn which guarantee that the composition u◦f is weakly differentiable and its weak derivative belongs to the some function space. We show that if f has finite distortion and q-distortion Kq = |Df|q /Jf is integrable enough, then the composition operator Tf (u) = u ◦ f maps functions from W1,q loc into space W1,p loc and the well-known chain rule holds. To prove it we characterize when the inverse mapping f−1 maps sets of measure zero onto sets of measure zero (satisfies the Luzin (N−1 ) con- dition). We also fully characterize conditions for Sobolev-Lorentz space WLn,q for arbitrary q and for Sobolev Orlicz space WLq log L for q ≥ n and α > 0 or 1 < q ≤ n and α < 0. We find a necessary condition on f for Sobolev rearrangement invariant function space WX close to WLq , i.e. X has q-scaling property. 1