On the boundedness of the fractional maximal operator, riesz potential and their commutators in generalized morrey spaces

Abstract

In the paper the authors find conditions on the pair (? 1 ,? 2 ) which ensure the Spanne type boundedness of the fractional maximal operator M ? and the Riesz potential operator I ? from one generalized Morrey spaces Mp,?1 to another M q,?2 ,1<p<q<?,1/p-1/q=?/n, and from M 1,?1 to the weak space W M q,?2 ,1<p<q<?,1-1/q=?/n, We also find conditions on ? which ensure the Adams type boundedness of the M ? and I ? from M p,?1/p to M q,?1/q for 1<p<q<? and from M 1,? to WM q,?1/p for 1<q<?. As applications of those results, the boundeness of the commutators of operators I ? and I ? on generalized Morrey spaces is also obtained. In the case b?BMO(R)n and 1<p<q<?, we find the sufficient conditions on the pair (?1,?2) which ensures the boundedness of the operators Mb,?and[b,I?]fromMp,?1toMq,?2with1/p-1/q=?/n. We also find the sufficient conditions on ? which ensures the boundedness of the operators Mb,?and[b,I ? ]from Mp,?1ptoMq,?1pfor1<p<q<?. In all cases conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (? 1 ,? 2 )and ?, which do not assume any assumption on monotonicity of ? 1 ,? 2 and ? in r, As applications, we get some estimates for Marcinkiewicz operator and fractional powers of the some analytic semigroups on generalized Morrey spaces. © 2013 Springer Basel.