THE RIESZ TRANSFORM, RECTIFIABILITY, AND REMOVABILITY FOR LIPSCHITZ HARMONIC FUNCTIONS

We show that, given a set E subset of Rn+1 with finite n-Hausdorff measure if the H-n dimensional Riesz transform R-Hn (left perpendicularE)f(x) = integral(E) x - y/vertical bar x - y vertical bar(n+1) f(y) H-n (y) is bounded in L-2 (H-n left perpendicularE), then E is n-rectifiable. From this result we deduce that a compact set E subset of Rn+1 with H-n (E) < infinity is removable for Lipschitz harmonic functions if and only if it is purely n-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.