  We obtain a characterization of the weighted inequalities for the Riesz
transforms on weighted local Morrey spaces. The condition is sufficient for the
boundedness on the same spaces of all Calder\'on-Zygmund operators suitably
defined on the functions of the space. In the case of the fractional maximal
operator and the fractional integral we obtain a characterization valid for
exponents satisfying the Sobolev relation. For power weights we get sharp
results in a larger range of exponents for the usual versions of weighted
Morrey spaces.
