Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

The problem of the boundedness of the Riesz potential I (alpha) , 0 < alpha < n, in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted L (p) -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, these sufficient conditions coincide with the necessary ones.