On planar Sobolev Lᵐₚ-extension domains

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For each m≥ 1 and p>2 we characterize bounded simply connected Sobolev Lᵐₚ-extension domains Ω⊂ 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⊂Ω there exists a similar square Q⊂Ω which touches S and has the property that x and S belong to distinct connected components of Ω Q.