Abstract. The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.For x ∈ R n and r > 0, let B(x, r) denote the open ball centred at x of radius r. Definition 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞). We denote by LM pθ,w and GM pθ,w the local and global Morrey-type spaces respectively, defined to be the spaces of all functions f ∈ L loc p (R n ) with finite quasinormsrespectively.Lemma 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞).
The problem of the boundedness of the fractional maximal operator M , 0 < < n, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.
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