$\Omega$-symmetric measures and related singular integrals
نویسندگان
چکیده
Let $\mathbb{S} \subset \mathbb{C}$ be the circle in plane, and let $\Omega\colon \mathbb{S} \to \mathbb{S}$ an odd bi-Lipschitz map with constant $1+\delta\_\Omega$, where $\delta\_\Omega\geq 0$ is small. Assume also that $\Omega$ twice continuously differentiable. Motivated by a question raised Mattila Preiss, we prove following: If Radon measure $\mu$ has positive lower density finite upper almost everywhere, limit $$ \lim\_{\epsilon \downarrow 0} \int\_{\mathbb C \setminus B(x,\epsilon)} \frac{\Omega((x-y)/|x-y|)}{|x-y|} , d\mu(y) exists $\mu$-almost then $1$-rectifiable. To achieve this, first if Ahlfors–David 1-regular symmetric respect to $\Omega$, is, \int\_{B(x,r)}\lvert x-y|\Omega \Bigl(\frac{x-y}{|x-y|}\Big) = 0\quad \text{for all } x \in \mathrm{spt}(\mu) \text{ r > 0, flat, or, other words, there $c line $L$ so $\mu= c\mathcal{H}^{1}|\_{L}$.
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ژورنال
عنوان ژورنال: Revista Matematica Iberoamericana
سال: 2021
ISSN: ['2235-0616', '0213-2230']
DOI: https://doi.org/10.4171/rmi/1245