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  • 1.
    Abramovich, Shoshana
    et al.
    University of Haifa.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On some new developments of Hardy-type inequalities2012In: 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012 / [ed] Seenith Sivasundaram, Melville, NY: American Institute of Physics (AIP), 2012, p. 739-746Conference paper (Refereed)
    Abstract [en]

    In this paper we present and discuss some new developments of Hardy-type inequalities, namely to derive (a) Hardy-type inequalities via a convexity approach, (b) refined scales of Hardy-type inequalities with other “breaking points” than p = 1 via superquadratic and superterzatic functions, (c) scales of conditions to characterize modern forms of weighted Hardy-type inequalities.

  • 2.
    Abramovich, Shoshana
    et al.
    Department of Mathematics, University of Haifa.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On γ-quasiconvexity, superquadracity and two-sided reversed Jensen type inequalities2015In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 18, no 2, p. 615-627Article in journal (Refereed)
    Abstract [en]

    In this paper we deal with γ -quasiconvex functions when −1γ 0, to derive sometwo-sided Jensen type inequalities. We also discuss some Jensen-Steffensen type inequalitiesfor 1-quasiconvex functions. We compare Jensen type inequalities for 1-quasiconvex functionswith Jensen type inequalities for superquadratic functions and we extend the result obtained forγ -quasiconvex functions to more general classes of functions.

  • 3.
    Abramovich, Shoshana
    et al.
    Department of Mathematics, University of Haifa.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Some new scales of refined Jensen and Hardy type inequalities2014In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 17, no 3, p. 1105-1114Article in journal (Refereed)
    Abstract [en]

    Some scales of refined Jensen and Hardy type inequalities are derived and discussed. The key object in our technique is ? -quasiconvex functions K(x) defined by K(x)x-? =? (x) , where Φ is convex on [0,b) , 0 < b > ∞ and γ > 0.

  • 4.
    Barza, Sorina
    et al.
    Karlstads Universitet.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Some new sharp limit Hardy-type inequalities via convexity2014In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2014, article id 6Article in journal (Refereed)
    Abstract [en]

    Some new limit cases of Hardy-type inequalities are proved, discussed and compared. In particular, some new refinements of Bennett's inequalities are proved. Each of these refined inequalities contain two constants, and both constants are in fact sharp. The technique in the proofs is new and based on some convexity arguments of independent interest.

  • 5.
    Burtseva, Evgeniya
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Lundberg, Staffan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. UiT, The Arctic University of Norwa.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Multi–dimensional Hardy type inequalities in Hölder spaces2018In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 12, no 3, p. 719-729Article in journal (Refereed)
    Abstract [en]

    Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in Hölder spaces.

  • 6. Burtseva, Evgeniya
    et al.
    Lundberg, Staffan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Potential type operators in PDEs and their applications2017In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1798, article id 020178Article in journal (Refereed)
    Abstract [en]

     We prove the boundedness of Potential operator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation in r3 with a free term in such a space. We also give a short overview of some typical situations when Potential type operators arise when solving PDEs.

  • 7.
    Burtseva, Evgeniya
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On weighted generalized fractional and Hardy-type operators acting between Morrey-type spaces2017In: Fractional Calculus and Applied Analysis, ISSN 1311-0454, E-ISSN 1314-2224, Vol. 20, no 6, p. 1545-1566Article in journal (Refereed)
    Abstract [en]

    We study weighted generalized Hardy and fractional operators acting from generalized Morrey spaces Lp,φ,(n) into Orlicz-Morrey spaces LΦ,φ,(n). We deal with radial quasi-monotone weights and assumptions imposed on weights are given in terms of Zigmund-type integral conditions. We find conditions on φ,Φ, the weight w and the kernel of the fractional operator, which insures such a boundedness. We prove some pointwise estimates for weighted generalized fractional operators via generalized Hardy operators, which allow to obtain the weighted boundedness for fractional operators from those for Hardy operators. We provide also some easy to check numerical inequalities to verify the obtained conditions.

  • 8.
    Burtseva, Evgeniya
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Weighted Adams type theorem for the Riesz fractional integral in generalized Morrey Space2016In: Fractional Calculus and Applied Analysis, ISSN 1311-0454, E-ISSN 1314-2224, Vol. 19, no 4, p. 954-972Article in journal (Refereed)
    Abstract [en]

    We prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space L-p,L-phi to a certain Orlicz-Morrey space L-Phi,L-phi which covers the Adams result for Morrey spaces. We also give a generalization to the case of weighted Riesz fractional integration operators for some class of weights.

  • 9.
    Kufner, Alois
    et al.
    Mathematical Institute, Czech Academy of Sciences, Department of Mathematics, University of West Bohemia.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Hardy type inequalities with kernels: The current status and some new results2017In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 290, no 1, p. 57-65Article in journal (Refereed)
    Abstract [en]

    We consider the general Hardy type operator inline image where inline image is a positive and measurable kernel. To characterize the weights u and v so that inline image is still an open problem for any parameters p and q. However, for special cases the solution is known for some parameters p and q. In this paper the current status of this problem is described and discussed mainly for the case inline image In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised.

  • 10.
    Kufner, Alois
    et al.
    Mathematical Institute, Czech Academy of Sciences.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Weighted inequalities of Hardy type2017 (ed. 2)Book (Other academic)
    Abstract [en]

    In this second edition, all chapters in the first edition have been updated with new information. Moreover, a new chapter contains new and complementary information concerning: (a) a convexity approach to prove and explain Hardy-type inequalities; (b) sharp constants; (c) scales of inequalities to characterize Hardy-type inequalities; (d) Hardy-type inequalities in other function spaces; and (e) a number of new open questions.

  • 11.
    Lukkassen, Dag
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Meidell, A.
    Narvik University College, 8505 Narvik, Norway.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations2012In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 35, no 11, p. 1300-1311Article in journal (Refereed)
    Abstract [en]

    We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces L p,Ψ(ℝ n,w), defined by an almost increasing function Ψ(r) and radial type weights. We obtain sufficient conditions, in terms of numerical characteristics, that is, index numbers of the weight functions and the function Ψ. In relation with the wide usage of singular integral equations in applications, we show how the solvability of such equations in the generalized Morrey spaces depends on the main characteristics of the space, which allows to better control both the singularities and regularity of solutions

  • 12.
    Lukkassen, Dag
    et al.
    Narvik University College and Norut Narvik.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Hardy Type Operators in Local Vanishing Morrey Spaces on Fractal Sets2015In: Fractional Calculus and Applied Analysis, ISSN 1311-0454, E-ISSN 1314-2224, Vol. 18, no 5, p. 1252-1276Article in journal (Refereed)
    Abstract [en]

    We obtain two-weighted estimates for the Hardy type operators fromlocal generalized Morrey spaces Lp,ϕloc (X,w1) defined on an arbitrary underlyingquasi-metric measure space (X, μ, ) with the growth condition, toLq,ψloc (X,w2), where w1 = w1[(x, x0)], x0 ∈ X is a weight of radial type,while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneouslytreat a similar boundedness V Lp,ϕloc (X,w1) → V Lq,ψloc (X,w2) forvanishing Morrey spaces. We obtain sufficient conditions for such estimatesin terms of some integral inequalities imposed on ϕ, ψ and w1.w2. We alsospecially treat the one weight case where w2(x) is also of radial type. Wedo not impose doubling condition on the measure μ, but base our result onthe growth condition.The obtained results show the explicit dependence of the mapping propertiesof the Hardy type operators on the fractional dimension of the set(X, μ, ). An application to spherical Hardy type operators is also given.

  • 13.
    Lundberg, Staffan
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On some hyperbolic type equations and weighted anisotropic Hardy operators2017In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 40, no 5, p. 1414-1421Article in journal (Refereed)
    Abstract [en]

    We introduce a version of weighted anisotropic Morrey spaces and anisotropic Hardy operators. We find conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in solving some classes of degenerate hyperbolic partial differential equations.

  • 14.
    Oguntuase, James
    et al.
    Department of Mathematics, University of Agriculture, Abeokuta.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Some Hardy type inequalities with "broken" exponent p2014In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 8, no 3, p. 405-416Article in journal (Refereed)
    Abstract [en]

    Some new Hardy-type inequalities, where the parameter p is permitted to take different values in different intervals, are proved and discussed. The parameter can even be negative in one interval and greater than one in another. Moreover, a similar result is derived for a multidimensional case.

  • 15. Oguntuase, James
    et al.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Sonubi, A.
    Federal University of Agriculture, Abeokuta, Ogun State.
    On the equivalence between some multidimensional Hardy-type inequalities2014In: Banach Journal of Mathematical Analysis, ISSN 1735-8787, Vol. 8, no 1Article in journal (Refereed)
    Abstract [en]

    We prove and discuss some power weighted Hardy-type inequalities on finite and infinite sets. In particular, it is proved that these inequalities are equivalent because they can all be reduced to an elementary inequality, which can be proved by Jensen inequality. Moreover, the corresponding limit (Pólya-Knopp type) inequalities and equivalence theorem are proved. All constants in these inequalities are sharp.

  • 16.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Ragusa, Alessandra
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On some results in weighted Morrey spaces with applications to PDE2012Report (Other academic)
  • 17.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Ragusa, Maria Alessandra
    University of Catania.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Commutators of Hardy operators in vanishing Morrey spaces2012In: 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012 / [ed] Seenith Sivasundaram, Melville, NY: American Institute of Physics (AIP), 2012, p. 859-866Conference paper (Refereed)
    Abstract [en]

    In this paper we study boundedness of commutators of the multi-dimensional Hardy type operators with BMO coefficients, in weighted global and/or local generalized Morrey spaces LΠp,φ(Rn,w) and vanishing local Morrey spaces VLlocp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(∣x∣). This study is made in the perspective of posterior applications of the weighted results to some problems in the theory of PDE. We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, and also in terms of the Matuszewska-Orlicz indices of φ and w, for such a boundedness.

  • 18.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Inequalities and Convexity2014In: Operator Theory, Operator Algebras and Applications, Basel: Encyclopedia of Global Archaeology/Springer Verlag, 2014, p. 279-306Chapter in book (Refereed)
    Abstract [en]

    It is a close connection between various kinds of inequalities and the concept of convexity. The main aim of this paper is to illustrate this fact in a unified way as an introduction of this area. In particular, a number of variants of classical inequalities, but also some new ones, are derived and discussed in this general frame.

  • 19.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Session-workshop on analysis, inequalities and homogenization theory and applications2010In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1281, p. 489-Article in journal (Other academic)
    Abstract [en]

    Analysis, Inequalities and Homogenization Theory are increasingly important areas for various kinds of applications both to other fields of Mathematics and to other sciences, e.g. physics, material science, numerical analysis and geophysics.The main aim of the session is to bring together researchers with different backgrounds and interests in all aspects of these areas of mathematics and plan for future cooperation and new directions of joint research. As background the participants will present the newest developments and present “status of the art” of their research fields. Special meetings with informal discussions will be organized, where in particular various kinds of applications will be highlighted. The topics of interest include (but are not limited to): General Inequalities, Hardy type inequalities, Real and complex analysis, Functional analysis, q-analysis, Interpolation theory, Function Spaces and Homogenization Theory

  • 20.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Singular integral equations in generalized weighted Morrey spaces2010In: Numerical analysis and applied mathematics: Conference on Numerical Analysis and Applied Mathematics 2010, Rhodes, Greece, 19 - 25 September 2010 / [ed] Theodore E. Simos, American Institute of Physics (AIP), 2010, p. 498-501Conference paper (Refereed)
    Abstract [en]

    We consider singular integral equations with discontinuous coefficients in generalized weighted Morrey spaces. We prove a result on Fredholmness of such equations. Moreover, we give explicit formulas showing direct dependence of the number of solutions on the parameters defining the space. Finally we apply our result to derive concrete solutions, in this space, of Sönghen equation which is of great interest in aerodynamics.

  • 21.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Some remarks and new developments concerning Hardy-type inequalities2010In: Studies in the History of Modern Mathematics: Supplemento di Rendiconti del Circolo Matematico di Palermo, no 82, p. 93-122Article in journal (Refereed)
  • 22.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Weighted Hardy and potential operators in the generalized Morrey spaces2011In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 377, no 2, p. 792-806Article in journal (Refereed)
    Abstract [en]

    We study the weighted p→q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces Lp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p→q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p→q-boundedness of the Riesz potential operator

  • 23.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Calderón–Zygmund Type Singular Operators in Weighted Generalized Morrey Spaces2016In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, no 2, p. 413-426Article in journal (Refereed)
    Abstract [en]

    We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces L p,φ (R n ,w), defined by a function φ(x,r) and radial type weight w(|x−x 0 |),x 0 ∈R n . These conditions are given in terms of inclusion into L p,φ (R n ,w), of a certain integral constructions defined by φ and w. In the case of φ=φ(r) we also provide easy to check sufficient conditions for that in terms of indices of φ and w.

  • 24.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Commutators with Coefficients in CMO of Weighted Hardy Operators in Generalized Local Morrey Spaces2017In: Mediterranean Journal of Mathematics, ISSN 1660-5446, E-ISSN 1660-5454, Vol. 14, no 2, article id 64Article in journal (Refereed)
    Abstract [en]

    We prove theorems on the boundedness of commutators [a,Hwα] of the weighted multidimensional Hardy operator Hwα:=wHα1w from a generalized local Morrey space Lp , φ ; 0(Rn) to local or global space Lq , ψ(Rn). The main impacts of these theorems are1.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 Sobolev–Spanne type exponent in the case of classical Morrey spaces4.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

  • 25.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Maximal, potential and singular operators in vanishing generalized Morrey spaces2013In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916, Vol. 57, no 4, p. 1385-1399Article in journal (Refereed)
    Abstract [en]

    We introduce vanishing generalized Morrey spaces {Mathematical expression} with a general function {Mathematical expression} defining the Morrey-type norm. Here {Mathematical expression} is an arbitrary subset in Ω including the extremal cases {Mathematical expression} and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces {Mathematical expression} we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type {Mathematical expression} -theorem for the potential operator I α. The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on {Mathematical expression}. No monotonicity type condition is imposed on {Mathematical expression}. In case {Mathematical expression} has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function {Mathematical expression}. The proofs are based on pointwise estimates of the modulars defining the vanishing spaces

  • 26.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On a Muckenhoupt-type condition for Morrey spaces2013In: Mediterranean Journal of Mathematics, ISSN 1660-5446, E-ISSN 1660-5454, Vol. 10, no 2, p. 941-951Article in journal (Refereed)
    Abstract [en]

    As is known, the class of weights for Morrey type spaces {Mathematical expression} for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class A p of such weights for the Lebesgue spaces L p(Ω). For instance, in the case of power weights {Mathematical expression}, {Mathematical expression}, the singular operator (Hilbert transform) is bounded in {Mathematical expression}, if and only if -1 < ν < p - 1, while it is bounded in the Morrey space {Mathematical expression}, if and only if the exponent α runs the shifted interval λ - 1 < ν < λ + p - 1. A description of all the admissible weights similar to the Muckenhoupt class A p is an open problem. In this paper, for the one-dimensional case, we introduce the class A p,λ of weights, which turns into the Muckenhoupt class A p when λ = 0 and show that the belongness of a weight to A p,λ is necessary for the boundedness, in Morrey spaces, of the Hilbert transform in the one-dimensional case. In the case n > 1 we also provide some λ-dependent á priori assumptions on weights and give some estimates of weighted norms {Mathematical expression} of the characteristic functions of balls

  • 27.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On two-weight estimates for the maximal operator in local Morrey spaces2014In: International Journal of Mathematics, ISSN 0129-167X, Vol. 25, no 11, article id 1450099Article in journal (Refereed)
    Abstract [en]

    For two weighted local Morrey spaces and we obtain general type sufficient conditions and necessary conditions imposed on the functions φ and ψ and the weights u and v for the boundedness of the maximal operator from to , with some "logarithmic gap" between the sufficient and necessary conditions. Both the conditions formally coincide if we omit a certain logarithmic factor in these conditions.

  • 28.
    Samko, Natasha
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Weighted Hardy operators in the local generalized vanishing Morrey spaces2013In: Positivity (Dordrecht), ISSN 1385-1292, E-ISSN 1572-9281, Vol. 17, no 3, p. 683-706Article in journal (Refereed)
    Abstract [en]

    In this paper we study {Mathematical expression}-boundedness of the multi-dimensional Hardy type operators in the vanishing local generalized Morrey spaces {Mathematical expression} defined by an almost increasing function {Mathematical expression} and radial type weight {Mathematical expression}. We obtain sufficient conditions, in terms of some integral inequalities imposed on {Mathematical expression} and {Mathematical expression}, for such a boundedness. In the case where the function {Mathematical expression} and the weight are power functions, these conditions are also necessary

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