If both sides of an Ahlfors regular boundary have Poisson kernels with logarithms in VMO, then the domain is a vanishing chord-arc domain, and conversely.
Wavelets and singular integrals on curves and surfaces , volume 1465 of Lecture Notes in Mathematics
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Two Phase Free Boundary Problem for Poisson Kernels
If both sides of an Ahlfors regular boundary have Poisson kernels with logarithms in VMO, then the domain is a vanishing chord-arc domain, and conversely.