Necessary and sufficient conditions for the boundedness of the Riesz potential in modified morrey spaces

Abstract

We prove that the fractional maximal operator M? and the Riesz potential operator I?, 0 < ? < n are bounded from the modified Morrey space L˜1,? (R{double-struck}n) to the weak modified Morrey space WL˜q,? (R{double-struck}n) if and only if, ?/n ? 1-1/q ? ?/(n-?) and from L˜p,? (R{double-struck}n) to L˜q,? (R{double-struck}n) if and only if, ?/n ? 1/p-1/q ? ?/(n-?). As applications, we establish the boundedness of some Schödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on modified Morrey spaces which are estimated by Riesz potentials. © ELEMENT, Zagreb.