Abstract. We consider the generalized weighted Morrey spaces M p(·),ϕ ω (Ω) with variable exponent p(x) and a general function ϕ(x,r) defining the Morrey-type norm. In case of unbounded sets Ω ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular operators with standard kernel, in such spaces. We also prove the boundedness of the commutators of maximal operator and Calderón-Zygmund singular operators in the generalized weighted Morrey spaces with variable exponent Mathematics subject classification (2010): 42B20, 42B25, 42B35.

Let Ω ⊂ R n be an unbounded open set. We consider the generalized weighted Morrey spaces M p(·),ϕ ω (Ω) and the vanishing generalized weighted Morrey spaces V M p(·),ϕ ω (Ω) with variable exponent p(x) and a general function ϕ(x, r) defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces M p(·),ϕ ω (Ω) and V M p(·),ϕ ω (Ω) . This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces. Especially, it gives a unified result for generalized Morrey spaces and variable Morrey spaces which currently gained a lot of attentions from researchers in theory of function spaces.

Abstract. We consider the generalized weighted Morrey spaces M p(·),ϕ ω (Ω) with variable exponent p(x) and a general function ϕ(x,r) defining the Morrey-type norm. In case of unbounded sets Ω ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular operators with standard kernel, in such spaces. We also prove the boundedness of the commutators of maximal operator and Calderón-Zygmund singular operators in the generalized weighted Morrey spaces with variable exponent Mathematics subject classification (2010): 42B20, 42B25, 42B35.

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