Principal values for the signed Riesz kernels of non-integer dimension

For positive measures μ in R n and 0 < α < 1, we study the μ-almost everywhere existence of the principal values of the α-Riesz transform of μ, lime→0∫Ιy-xΙ>ε y-x/Ιy-xΙ1+αdμ(y). We show that the L 2 (μ)-boundedness of the a-Riesz transform implies the existence of the above principal value for μ-almost all x &isin; R n . We also prove that if μ has positive and finite upper density μ-almost everywhere, then the set where the principal value does not exist has positive μ-measure. © 2011 Rocky Mountain Mathematics Consortium.