On planar Sobolev Lpm-extension domains

For each m≥1m≥1 and p>2p>2 we characterize bounded simply connected Sobolev Lpm-extension domains Ω⊂R2Ω⊂R2. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in Ω. 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 Ω.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⊂ΩS⊂Ω there exists a similar square Q⊂ΩQ⊂Ω which touches S and has the property that x and S   belong to distinct connected components of Ω∖QΩ∖Q.