Principal values for Riesz transforms and rectifiability

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.