Planar Sobolev extension domains and a Square Separation Theorem

For each positive integer $m$ and each $p>2$ we characterize bounded simply connected
Sobolev $W^m_p$-extension domains $\Omega$ in $R^2$. 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
$S\subset\Omega$ there exists a similar square $Q\subset\Omega$ which touches $S$ and has
the property that $x$ and $S$ belong to distinct connected components of $\Omega\setminus Q$.

 This is a joint work with Nahum Zobin.