Principal values for Riesz transforms and rectifiability

Let E⊂RdE⊂Rd with Hn(E)<∞Hn(E)<∞, where HnHn stands for the n-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limitlimε→0∫y∈E:|x−y|>εx−y|x−y|n+1dHn(y) exists HnHn-almost everywhere in E  . To prove this result we obtain precise estimates from above and from below for the L2L2 norm of the n-dimensional Riesz transforms on Lipschitz graphs.