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Let be the circle in the plane, and let be an odd bi-Lipschitz map with constant , where is small. Assume also that is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss, we prove the following: If a Radon measure has positive lower density and finite upper density almost everywhere, and the limit
exists -almost everywhere, then is -rectifiable. To achieve this, we prove first that if an Ahlfors–David 1-regular measure is symmetric with respect to , that is, if
then is flat, or, in other words, there exists a constant and a line so that .
Cite this article
Michele Villa, -symmetric measures and related singular integrals. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1669–1715DOI 10.4171/RMI/1245