Ahlfors regular domains are Lyapunov (C^{1,ω}) precisely when the Riesz transforms on constants have the same modulus of continuity as the outward unit normal, via a generalized Plemelj-Privalov theorem.
Casey,Quantitative control on the Carlesonε-function determines regularity, arXiv:2410.18422v2 [math.CA] 29 Nov
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Characterizations of Lyapunov domains in terms of Riesz transforms and the Plemelj-Privalov theorem
Ahlfors regular domains are Lyapunov (C^{1,ω}) precisely when the Riesz transforms on constants have the same modulus of continuity as the outward unit normal, via a generalized Plemelj-Privalov theorem.