Principal values for Riesz transforms and rectifiability

Abstract

Let E ⊂ R d with H n ( E ) < ∞ , where H n 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 d H n ( y ) exists H n -almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the L 2 norm of the n-dimensional Riesz transforms on Lipschitz graphs.