On the boundedness of integral operators in general spaces of the Morrey type.

The known results on the boundedness of the classical Riesz potential in general local spaces of the Morrey type are generalized to the case of integral operators of the Riesz type, in which the power function is replaced by a general function satisfying certain conditions. For all acceptable values of numerical parameters characterizing common local Morrey-type spaces, sufficient conditions have been obtained for the functional parameters characterizing these spaces, ensuring the boundedness of such operators from one common local Morrey-type space to another, and for some range of values of numerical parameters, these conditions are also necessary. For the Riemann-Liouville operator and the generalized Riemann-Liouville operator, accurate estimates of the norms of these operators are obtained, as operators acting from one local Morrey space on a parallelepiped to another, depending on the dimensions of the parallelepiped. Similar exact estimates are also obtained for anisotropic local Morrey spaces.