Generalized fractional maximal operator on generalized local morrey spaces

In this paper, we study the boundedness of generalized fractional maximal operator M-rho on generalized local Morrey spaces LMP,phi{x0} and generalized Morrey spaces M-p,M-phi, including weak estimates. Firstly, we prove the Spanne type boundedness of M-rho from the space LMp,phi 1{x0} to another LMq,phi 2{x0} 1 < p < q < infinity and from LM1,phi 1{x0} to the weak space WLMq,phi 2{x0} for p = 1 and 1 < q < infinity. Secondly, we prove the Adams type boundedness of M-rho from the space M-p,M-phi 1/p to another M-q,M-phi 1/q for 1 < p < q < infinity and from to the weak M-1,M-phi, space M-q,M-phi 1/q for p = 1 and 1 < q < infinity. In all cases the conditions for the boundedness of M-rho are given in terms of supremal-type integral inequalities on (phi(1), phi(2), rho) and (phi, rho), which do not assume any assumption on monotonicity of phi(1) (x, r), phi(2)(x, r) and phi(x, r) in r