$\Omega$-symmetric measures and related singular integrals

TL;DR

This paper proves that measures with certain symmetric properties related to a bi-Lipschitz map on the circle are rectifiable, addressing a question by Mattila and Preiss, and establishing conditions for measure flatness.

Contribution

It introduces a new symmetry condition involving $mbda$-symmetric measures and proves rectifiability under these conditions, extending previous understanding of singular integrals.

Findings

Measures with $mbda$-symmetry are flat (line-supported)

Existence of certain singular integrals implies measure rectifiability

Characterization of symmetric measures in the plane

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 $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss in [MP95], we prove the following: if a Radon measure $\mu$ has positive lower density and finte upper density almost everywhere, and the limit $$ \lim_{\epsilon \downarrow 0} \int_{\mathbb{C} \setminus B(x,\epsilon)} \frac{\Omega\left((x-y)/|x-y|\right)}{|x-y|} \, d\mu(y) $$ 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 $$ \int_{B(x,r)} |x-y|\Omega\left(\frac{x-y}{|x-y|}\right) \, d\mu(y) = 0 \mbox{ for all } x \in \mbox{spt}(\mu) \mbox{ and } r>0, $$ then $\mu$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $\mu= c \mathcal{H}^{1}|_{L}$.

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