Non existence of principal values of signed Riesz transforms of non   integer dimension
  In this paper we prove that, given s> 0, if E is a subset of R^m with positive and bounded s-dimensional Hausdorff measure H^s and the principal values of the s-dimensional signed Riesz transform of H^s|E exist H^s-almost everywhere in E, then s is integer. Other more general variants of this result are also proven. 
Aleix Ruiz de Villa and Xavier Tolsa