MATH. SCAND. 107 (2010), 285–304 BOUNDEDNESS OF THE MAXIMAL, POTENTIAL AND SINGULAR OPERATORS IN THE GENERALIZED VARIABLE EXPONENT MORREY SPACES VAGIF S. GULIYEV, JAVANSHIR J. HASANOV and STEFAN G. SAMKO Abstract We consider generalized Morrey spaces M p(·),ω () with variable exponent p(x) and a gen- eral function ω(x,r) defining the Morrey-type norm. In case of bounded sets  ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund sin- gular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type M p(·),ω () → M q(·),ω ()-theorem for the potential operators I α(·) , also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(x,r), which do not assume any assumption on monotonicity of ω(x,r) in r . 1. Introduction In the study of local properties of solutions to partial differential equations, together with weighted Lebesgue spaces, Morrey spaces L p,λ () play an important role, see [14], [25]. Introduced by C. Morrey [27] in 1938, they are defined by the norm ‖f ‖ L p,λ := sup x,r>0 r - λ p ‖f ‖ L p (B(x,r)) , where 0 ≤ λ<n,1 ≤ p< ∞. As is known, last two decades there is an increasing interest to the study of variable exponent spaces and operators with variable parameters in such spaces, we refer for instance to the surveying papers [12], [20], [22], [38], on the progress in this field, including topics of Harmonic Analysis and Operator Theory, see also references therein. Variable exponent Morrey spaces L p(·),λ(·) (), were introduced and stud- ied in [2] and [29] in the Euclidean setting and in [21] in the setting of met- ric measure spaces, in case of bounded sets. In [2] there was proved the boundedness of the maximal operator in variable exponent Morrey spaces L p(·),λ(·) () under the log-condition on p(·) and λ(·) and for potential op- erators, under the same log-condition and the assumptions inf x∈ α(x) > 0, Received 2 July 2009, in revised form 27 September 2009.