Fractional maximal operator and its higher order commutators on generalized weighted Morrey spaces

In this paper, we study the boundedness of the fractional maximal operator M and its kth order commutators [b;M]k on generalized weighted Morrey spacesMp;'(w). We find the sufficient conditions on the pair ('1; '2) with b 2 BMO(Rn) and w 2 Apq which ensures the boundedness of the operators M and [b;M]k from Mp;'1 (w) to Mp;'2 (w) for 1 < p < 1. In all cases the conditions for the boundedness of the operators M, [b;M]k are given in terms of supremal-type inequalities on ('1; '2) and w, which do not assume any assumption on monotonicity of '1(x; r); '2(x; r) in r. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.