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- W3098133670 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 $textrm{Im}, z/|z|^2$, $zin mathbb{C}setminus{0}$) has the following property $(*)$: the $L^2(mathcal{H}^1lfloor E)$-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 $(textrm{Re}, z)^{2n-1}/|z|^{2n}$, $nin mathbb{N}$. In this paper, we prove that the property $(*)$ is valid for operators associated to the much wider class of kernels $(textrm{Re}, z)^{2N-1}/|z|^{2N}+tcdot(textrm{Re}, z)^{2n-1}/|z|^{2n}$, where $n,N$ are positive integer numbers such that $Nge n$, and $tin mathbb{R}setminus (t_1,t_2)$ with $t_1,t_2$ depending only on $n$ and $N$." @default.
- W3098133670 created "2020-11-23" @default.
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- W3098133670 date "2017-02-18" @default.
- W3098133670 modified "2023-09-23" @default.
- W3098133670 title "A New Family of Singular Integral Operators Whose $$L^2$$ L 2 -Boundedness Implies Rectifiability" @default.
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- W3098133670 doi "https://doi.org/10.1007/s12220-017-9780-9" @default.
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