Boundedness of Fractional Maximal Operator on Generalized Weighted Morrey Spaces with Variable Exponents

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.

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Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

Abstract and Applied Analysis ◽

10.1155/2010/291345 ◽

2010 ◽

Vol 2010 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Emin V. Guliyev ◽

Yagub Y. Mammadov

Keyword(s):

Maximal Operator ◽

Morrey Space ◽

Shift Operator ◽

Morrey Spaces ◽

Dunkl Operator ◽

Generalized Shift Operator ◽

Fractional Maximal Operator ◽

Generalized Shift

We consider the generalized shift operator, associated with the Dunkl operatorΛα(f)(x)=(d/dx)f(x)+((2α+1)/x)((f(x)-f(-x))/2),α>-1/2. We study some embeddings into the Morrey space (D-Morrey space)Lp,λ,α,0≤λ<2α+2and modified Morrey space (modifiedD-Morrey space)L̃p,λ,αassociated with the Dunkl operator onℝ. As applications we get boundedness of the fractional maximal operatorMβ,0≤β<2α+2, associated with the Dunkl operator (fractionalD-maximal operator) from the spacesLp,λ,αtoL∞(ℝ)forp=(2α+2-λ)/βand from the spacesL̃p,λ,α(ℝ)toL∞(ℝ)for(2α+2-λ)/β≤p≤(2α+2)/β.

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Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

Journal of Function Spaces and Applications ◽

10.1155/2012/543475 ◽

2012 ◽

Vol 2012 ◽

pp. 1-20

Author(s):

Vagif S. Guliyev ◽

Kamala R. Rahimova

Keyword(s):

Maximal Operator ◽

Morrey Space ◽

Morrey Spaces ◽

Reverse Hölder Class ◽

Fractional Maximal Operator ◽

Hölder Class ◽

Holder Class ◽

Homogeneous Dimension ◽

Reverse Hölder ◽

Limiting Case

We prove that the parabolic fractional maximal operatorMαP,0≤α<γ, is bounded from the modified parabolic Morrey spaceM̃1,λ,P(ℝn)to the weak modified parabolic Morrey spaceWM̃q,λ,P(ℝn)if and only ifα/γ≤1-1/q≤α/(γ-λ)and fromM̃p,λ,P(ℝn)toM̃q,λ,P(ℝn)if and only ifα/γ≤1/p-1/q≤α/(γ-λ). Hereγ=trPis the homogeneous dimension onℝn. In the limiting case(γ-λ)/α≤p≤γ/αwe prove that the operatorMαPis bounded fromM̃p,λ,P(ℝn)toL∞(ℝn). As an application, we prove the boundedness ofMαPfrom the parabolic Besov-modified Morrey spacesBM̃pθ,λs(ℝn)toBM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

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On the boundedness of the fractional maximal operator on global Orlicz-Morrey spaces

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS ◽

10.31489/2021m1/17-24 ◽

2021 ◽

Vol 101 (1) ◽

pp. 17-24

Author(s):

N.А. Bokayev ◽

◽

А.А. Khairkulova ◽

Keyword(s):

Characteristic Function ◽

Maximal Operator ◽

Sufficient Conditions ◽

Morrey Spaces ◽

Boundedness Condition ◽

Fractional Maximal Operator

The article deals with the global Orlia-Morrey spaces GMΦ,ϕ,θ(Rn). We find sufficient conditions on pairs of functions (ϕ, η) and (Φ, Ψ), which ensure the boundedness of the fractional maximal operator Mα from GMΦ,ϕ,θ(Rn) in GMΨ,η,θ(Rn). It is proved that under some additional conditions on the function ϕ, the conditions obtained are also necessary. In the proof, the boundedness condition is essentially used, the maximal Hardy-Littlewood functions and the estimate of the norm of the characteristic function in global Orlicz-Morrey spaces are used.

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