Principal values for Riesz transforms and rectifiability

2007/08/01 by Xavier Tolsa, Tolsa, Xavier
Mathematics · #28A75 #42B20 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.CA #math.FA #msc:28A75 #msc:42B20

paper · pdf · doi:10.48550/arxiv.0708.0109

47 pages

arxiv created 2007/08/01 · openalex publication_date 2007/08/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E⊂ R^d with Hⁿ(E)<∞, where Hⁿ 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→0∫_{y∈ E:|x-y|>\ve} \fracx-y|x-y|ⁿ⁺¹ dHⁿ(y) exists Hⁿ-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the L² norm of the n-dimensional Riesz transforms on Lipschitz graphs.

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