Non existence of principal values of signed Riesz transforms of non integer dimension Aleix Ruiz de VillaXavier Tolsa 28A7531B1042B25Riesz transformsHausdorff measures In this paper we prove that, given $s \geq 0$, and a Borel non zero measure $\mu$ in $\mathbb{R}^{m}$, if for $\mu$-almost every $x \in \mathbb{R}^{m}$ the limit \[ \lim_{\epsilon\to 0} \int_{|x-y| > \epsilon} \frac{x-y}{|x-y|^{s+1}} \mathrm{d}\mu(y) \] exists and $0 < \limsup_{r\to 0} \mu(B(x,r))/r^{s} < \infty$, then $s$ in an integer. In particular, if $E \subset \mathbb{R}^{m}$ is a set with positive and bounded $s$-dimensional Hausdorff measure $H^{s}$ and for $H^{s}$-almost every $x \in E$ the limit \[ \lim_{\epsilon\to 0} \int_{|x-y| > \epsilon} \frac{x-y}{|x-y|^{s+1}} \mathrm{d}H^{s}_{|E}(y) \] exists, then $s$ is an integer. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.3884 10.1512/iumj.2010.59.3884 en Indiana Univ. Math. J. 59 (2010) 115 - 130 state-of-the-art mathematics http://iumj.org/access/