# $Ω$-symmetric measures and related singular integrals

### Michele Villa

University of Jyväskylä, Finland

## Abstract

Let $S⊂C$ be the circle in the plane, and let $Ω:S→S$ be an odd bi-Lipschitz map with constant $1+δ_{Ω}$, where $δ_{Ω}≥0$ is small. Assume also that $Ω$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss, we prove the following: If a Radon measure $μ$ has positive lower density and finite upper density almost everywhere, and the limit

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

then $μ$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $μ=cH_{1}∣_{L}$.

## Cite this article

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

DOI 10.4171/RMI/1245