# $\Omega$-symmetric measures and related singular integrals

### Michele Villa

University of Jyväskylä, Finland

## Abstract

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega\colon \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega\geq 0$ 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 $\mu= c\mathcal{H}^{1}|_{L}$.

## Cite this article

Michele Villa, $\Omega$-symmetric measures and related singular integrals. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1669–1715

DOI 10.4171/RMI/1245