For chord-arc curves whose conformal derivative is an A2 weight, the Douglas and Littlewood-Paley fractional Sobolev spaces coincide for every order 0<s<1, and the Cauchy integral operator is bounded on them.
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Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves
For chord-arc curves whose conformal derivative is an A2 weight, the Douglas and Littlewood-Paley fractional Sobolev spaces coincide for every order 0<s<1, and the Cauchy integral operator is bounded on them.