$\Omega$-symmetric measures and related singular integrals

Abstract

Let $\mathbb{S}\subset \mathbb{C}$ be the circle in the plane, and let $\Omega :\mathbb{S}\to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+{\delta }_{\Omega }$, where ${\delta }_{\Omega }\ge 0$ is small. Assume also that $\Omega$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss, we prove the following: If a Radon measure $\mu$ has positive lower density and finite upper density almost everywhere, and the limit

exists $\mu$-almost everywhere, then $\mu$ is $1$-rectifiable. To achieve this, we prove first that if an Ahlfors–David 1-regular measure $\mu$ is symmetric with respect to $\Omega$, that is, if

then $\mu$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $\mu =c{\mathcal{H}}^1{|}_L$.