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A new family of singular integral operators whose $L^2$-boundedness implies rectifiability (1601.07319v2)

Published 27 Jan 2016 in math.CA

Abstract: Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}1(E)<\infty$. David and L\'eger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|2$ and $\textrm{Im}\, z/|z|2$, $z\in \mathbb{C}\setminus{0}$) has the following property $()$: the $L2(\mathcal{H}1\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\textrm{Re}\, z){2n-1}/|z|{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property $()$ is valid for operators associated to the much wider class of kernels $(\textrm{Re}\, z){2N-1}/|z|{2N}+t\cdot(\textrm{Re}\, z){2n-1}/|z|{2n}$, where $n,N$ are positive integer numbers such that $N\ge n$, and $t\in \mathbb{R}\setminus (t_1,t_2)$ with $t_1,t_2$ depending only on $n$ and $N$.

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