Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Abstract

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, ∞).

Commutators with Coefficients in CMO of Weighted Hardy Operators in Generalized Local Morrey Spaces

“…Various versions of Morrey spaces were widely investigated during the past decades, including the study of classical operators of harmonic analysismaximal, singular and potential operators in Morrey spaces and their generalizations and modifications, see for instance [3][4][5]11,15,20,21] and references therein.…”

Sobolev's theorem and duality for Herz-Morrey spaces of variable exponent

“…Further, Sobolev's inequality was also studied on generalized Morrey spaces (see [31]). This result was extended to local and global Morrey type spaces by Burenkov, Gogatishvili, Guliyev and Mustafayev [8] (see also [7,9,10]). The local Morrey type spaces are also called Herz spaces introduced by Herz [23].…”

Weighted multilinear Hardy operators and commutators

“…Later on in [9], the conclusions in Theorems A and B were further generalized to the central Morrey spacesḂ p,λ (R n ) and the central BMO space CṀ O q (R n ). Here the space CṀ O q (R n ) was first introduced by Lu and Yang in [19], and the spaceḂ p,λ (R n ) is a generalization of CṀ O q (R n ) introduced by Alvarez, Guzman-Partida and Lakey in [1]; see also [2]. Definition 1.1.…”