Advances in Mathematics
For each m > = 1 and p > 2 we characterize bounded simply connected Sobolev L-p(m)-extension domains Omega subset of R-2. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in Omega. Its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in Omega. An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point x and a square S subset of Omega there exists a similar square Q subset of Omega which touches S and has the property that x and S belong to distinct connected components of Omega \ Q. (C) 2015 Elsevier Inc. All rights reserved.
Shvartsman, P., & Zobin, N. (2016). On planar Sobolev Lpm-extension domains. Advances in Mathematics, 287, 237-346.