JournalsrmiVol. 37, No. 5pp. 1669–1715

Ω\Omega-symmetric measures and related singular integrals

  • Michele Villa

    University of Jyväskylä, Finland
$\Omega$-symmetric measures and related singular integrals cover
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Abstract

Let SC\mathbb{S} \subset \mathbb{C} be the circle in the plane, and let Ω ⁣:SS\Omega\colon \mathbb{S} \to \mathbb{S} be an odd bi-Lipschitz map with constant 1+δΩ1+\delta_\Omega, where δΩ0\delta_\Omega\geq 0 is small. Assume also that Ω\Omega is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss, we prove the following: If a Radon measure μ\mu has positive lower density and finite upper density almost everywhere, and the limit

limϵ0CB(x,ϵ)Ω((xy)/xy)xydμ(y)\lim_{\epsilon \downarrow 0} \int_{\mathbb C \setminus B(x,\epsilon)} \frac{\Omega((x-y)/|x-y|)}{|x-y|} \, d\mu(y)

exists μ\mu-almost everywhere, then μ\mu is 11-rectifiable. To achieve this, we prove first that if an Ahlfors–David 1-regular measure μ\mu is symmetric with respect to Ω\Omega, that is, if

B(x,r)xyΩ(xyxy)dμ(y)=0for all xspt(μ) and r>0,\int_{B(x,r)}\lvert x-y|\Omega \Bigl(\frac{x-y}{|x-y|}\Big) \, d\mu(y) = 0\quad \text{for all } x \in \mathrm{spt}(\mu) \text{ and } r > 0,

then μ\mu is flat, or, in other words, there exists a constant c>0c > 0 and a line LL so that μ=cH1L\mu= c\mathcal{H}^{1}|_{L}.

Cite this article

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

DOI 10.4171/RMI/1245