GENERALIZED FRACTIONAL MAXIMAL OPERATOR ON GENERALIZED LOCAL MORREY SPACES.

In this paper, we study the boundedness of generalized fractional maximal operator on generalized local Morrey spaces and generalized Morrey spaces, including weak estimates. Firstly, we prove the Spanne type boundedness of generalized fractional maximal operator on generalized local Morrey spaces for 1 < p < q < infinity and, on weak generalized local Morrey spaces for p = 1 and 1 < q < infinity. Secondly, we prove the Adams type boundedness of generalized fractional maximal operator on generalized local Morrey spaces for 1<p<q< \infinity and, on weak generalized local Morrey spaces for p = 1 and 1 < q < \infinity. In all cases the conditions for the boundedness of generalized fractional maximal operators are given in terms of supremal-type integral inequalities on (\varphi_1; \varphi_2; \rho) and (\varphi; \rho), which do not assume any assumption on monotonicity of \varphi_1(x; r), \varphi_2(x; r) and \varphi(x; r) in r. [ABSTRACT FROM AUTHOR]