The Cauchy Integral, Analytic Capacity, and Uniform Rectifiability

Variational inequalities for singular integral operators

“…The "if" implication in the question above was proved by P. Mattila, M. Melnikov and J. Verdera in [21] for the case of the Cauchy transform, that is n = 1 and d = 2.…”

A symmetry theorem revisited

“…From (3.14) and Theorem 1 in [MMV96] we see that ∂D is contained in a rectifiable curve. The argument in [LV92] can now be used essentially verbatim to get that D = ball.…”

“…In this respect our argument is in the spirit of [MMV96]. In fact Theorem 1 for an Ahlfors regular domain in R 2 can be deduced from [MMV96] as we point out in a remark following Theorem 1.…”

“…In this respect our argument is in the spirit of [MMV96]. In fact Theorem 1 for an Ahlfors regular domain in R 2 can be deduced from [MMV96] as we point out in a remark following Theorem 1. Our proof of Theorem 1 does not rely on the apriori knowledge of nontangential limits in cones and seems simpler to us than the proof in [LV92].…”

“…The argument in [LV92] can now be used essentially verbatim to get that D = ball. Finally we note that since it is unknown whether Theorem 1 in [MMV96] has an analogue in R n we were forced to come up with the simpler and more general proof of Theorem 1 above.…”