Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space

We consider generalized Morrey spaces L( X) on quasi-metric measure spaces X, d, μ, in general unbounded, with variable exponent p(x) and a general function φ( x, r) defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function φ( x, r) , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions φ. Our conditions do not suppose any assumption on monotonicity of φ( x, r) in r.