$\Omega$-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 ${\delta }_{\Omega }>0$ 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 $$\underset{\epsilon \downarrow 0}{\lim }{\int }_{\mathbb{C}\setminus B(x,\epsilon )}\frac{\Omega \left((x-y)/|x-y|\right)}{|x-y|}d\mu (y)$$ 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 $$ \int_{B(x,r)} |x-y|Ω\left(\frac{x-y}{|x-y|}\right) \, dμ(y) = 0 \mbox{ for all } x \in \mbox{spt}(μ) \mbox{ and } r>0, $$ then $\mu$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $\mu =c{\mathcal{H}}^1{|}_L$.

44 pages. To appear in Revista Matemática Iberoamericana