Principal values for Riesz transforms and rectifiability

Xavier Tolsa

doi:https://doi.org/10.1016/j.jfa.2007.07.020

Principal values for Riesz transforms and rectifiability

Xavier Tolsa

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

31 Citations (Scopus)

Abstract

Let E ⊂ Rd 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 limitunder(lim, ε → 0) under(∫, y ∈ E : | x - y | > ε) frac(x - y, | x - y |n + 1) d Hn (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. © 2007 Elsevier Inc. All rights reserved.

Original language	English
Pages (from-to)	1811-1863
Journal	Journal of Functional Analysis
Volume	254
DOIs	https://doi.org/10.1016/j.jfa.2007.07.020
Publication status	Published - 1 Apr 2008

Keywords

Lipschitz graphs
Principal values
Rectifiability
Riesz transforms

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title = "Principal values for Riesz transforms and rectifiability",

abstract = "Let E ⊂ Rd 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 limitunder(lim, ε → 0) under(∫, y ∈ E : | x - y | > ε) frac(x - y, | x - y |n + 1) d Hn (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. {\textcopyright} 2007 Elsevier Inc. All rights reserved.",

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T1 - Principal values for Riesz transforms and rectifiability

AU - Tolsa, Xavier

PY - 2008/4/1

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N2 - Let E ⊂ Rd 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 limitunder(lim, ε → 0) under(∫, y ∈ E : | x - y | > ε) frac(x - y, | x - y |n + 1) d Hn (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. © 2007 Elsevier Inc. All rights reserved.

AB - Let E ⊂ Rd 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 limitunder(lim, ε → 0) under(∫, y ∈ E : | x - y | > ε) frac(x - y, | x - y |n + 1) d Hn (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. © 2007 Elsevier Inc. All rights reserved.

KW - Lipschitz graphs

KW - Principal values

KW - Rectifiability

KW - Riesz transforms

U2 - https://doi.org/10.1016/j.jfa.2007.07.020

DO - https://doi.org/10.1016/j.jfa.2007.07.020

M3 - Article

VL - 254

SP - 1811

EP - 1863

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

ER -