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arXiv · 0708.0109

Principal values for Riesz transforms and rectifiability

Xavier Tolsa
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Abstract

Let $E\subset R^d$ with $H^n(E)<\infty$, where H^n stands for the $n$-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit $$\lim_{\ve\to0}\int_{y\in E:|x-y|>\ve} \frac{x-y}{|x-y|^{n+1}} dH^n(y)$$ exists H^n-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the $L^2$ norm of the n-dimensional Riesz transforms on Lipschitz graphs.

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Xavier Tolsa. 2007-08-01. Principal values for Riesz transforms and rectifiability. https://arxiv.org/abs/0708.0109

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