Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

Guliyev, Vagif S.

doi:10.1155/2009/503948

Cited by 131 publications

(105 citation statements)

References 17 publications

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“…We refer for instance, to [11] and also the survey [22] where one can find a detailed historical account on such spaces.…”

Section: Introductionmentioning

confidence: 99%

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

Samko

2017

Mediterr. J. Math.

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Abstract. We prove theorems on the boundedness of commutators [a, H The main impacts of these theorems are 1. the use of CMOs-class of coefficients a for the commutators; 2. the general setting when the function ϕ defining the Morrey space and the weight w are independent of one another and the weight w is not assumed to be in Ap; 3. recovering the Sobolev-Adams exponent q instead of SobolevSpanne type exponent in the case of classical Morrey spaces 4. boundedness from local to global Morrey spaces; 5. the obtained estimates contain the parameter s > 1 which may be arbitrarily chosen. Its choice regulates in fact an equilibrium between assumptions on the coefficient a and the characteristics of the space. The obtained results are new also in non-weighted case.Mathematics Subject Classification. 46E30, 42B35, 42B25, 47B38.

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“…We refer for instance, to [11] and also the survey [22] where one can find a detailed historical account on such spaces.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Samko

2017

Mediterr. J. Math.

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“…The classical Morrey spaces were originally introduced by Morrey in [36] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [17,18,23,36,41]. Mizuhara [35] introduced generalized Morrey spaces.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Mizuhara [35] introduced generalized Morrey spaces. Later, in [23] Guliyev defined the generalized Morrey spaces M p;' with normalized norm. Recently, Komori and Shirai [34] considered the weighted Morrey spaces L p;Ä .w/ and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Akbulut

Hasanov

2016

Open Mathematics

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Abstract:In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛, which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces M p;' .w/. We find the sufficient conditions on the pair .' 1 ; ' 2 / with w 2 A p;q which ensures the boundedness of the operators T ;˛a re given in terms of Zygmund-type integral inequalities on .' 1 ; ' 2 / and w, which do not assume any assumption on monotonicity of ' 1 .x; r/; ' 2 .x; r/ in r.

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“…There are many papers discussed the conditions on ϕ to obtain the boundedness of integral operators on the generalized Morrey spaces (see Guliyev, 1994Guliyev, , 1999Guliyev, & 2009Mizuhara, 1991;Nakai, 1994Nakai, & 2006Softova, 2006).…”

Section: Notationsmentioning

confidence: 99%

Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

Dzhabrailov¹,

Khaligova²

2012

JMR

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In this paper it is proved that anisotropic fractional maximal operator M α,σ , 0 ≤ α < |σ| is bounded on anisotropic generalized Morrey spaces M p,ϕ,σ , where |σ| = n i=1 σ i is the homogeneous dimension of R n . We find the conditions on the pair (ϕ 1 , ϕ 2 ) which ensure the Spanne-Guliyev type boundedness of the operator M α,σ from anisotropic generalized Morrey space M p,ϕ 1 ,σ to M q,ϕ 2 ,σ , 1 < p ≤ q < ∞, 1/p − 1/q = α/|σ|, and from the space M 1,ϕ 1 ,σ to the weak space W M q,ϕ 2 ,σ , 1 < q < ∞, 1 − 1/q = α/|σ|. We also find conditions on the ϕ which ensure the AdamsGuliyev type boundedness ofAs applications, we establish the boundedness of some Schödinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

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Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

Cited by 131 publications

References 17 publications

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

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

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

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