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 limit lim ε→0 y∈E:|x−y|>ε x −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. © 2007 Elsevier Inc. All rights reserved.