Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

2021 ◽  
Vol 24 (6) ◽  
pp. 1643-1669
Author(s):  
Natasha Samko
Keyword(s):  
Upper Bound ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Target Space ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Hardy Type ◽  
Measure Spaces ◽  
Hardy Operators ◽  
Hardy Type Operators

Abstract We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.

GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES

2016 ◽  
Vol 103 (2) ◽  
pp. 268-278 ◽  
Author(s):  
GUANGHUI LU ◽  
SHUANGPING TAO
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Measure Spaces ◽  
Definition Of ◽  
Marcinkiewicz Integral Operator

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

scholarly journals Weighted Hardy Operators in Complementary Morrey Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Dag Lukkassen ◽  
Lars-Erik Persson ◽  
Stefan Samko
Keyword(s):  
Morrey Spaces ◽  
Bounded Domains ◽  
Weighted Lebesgue Space ◽  
Power Functions ◽  
Logarithmic Factor ◽  
Hardy Type ◽  
Weighted Hardy Operators ◽  
Increasing Function ◽  
Hardy Operators ◽  
Hardy Type Operators

We study the weightedp→q-boundedness of the multidimensional weighted Hardy-type operatorsHwαandℋwαwith radial type weightw=w(|x|), in the generalized complementary Morrey spacesℒ∁{0}p,ψ(ℝn)defined by an almost increasing functionψ=ψ(r). We prove a theorem which provides conditions, in terms of some integral inequalities imposed onψandw, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the functionψand the weightware power functions. We also prove that the spacesℒ∁{0}p,ψ(Ω)over bounded domains Ω are embedded between weighted Lebesgue spaceLpwith the weightψand such a space with the weightψ, perturbed by a logarithmic factor. Both the embeddings are sharp.

On weighted generalized fractional and Hardy-type operators acting between Morrey-type spaces

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Evgeniya Burtseva ◽  
Natasha Samko
Keyword(s):  
Morrey Spaces ◽  
Fractional Operators ◽  
Generalized Morrey Spaces ◽  
Hardy Type ◽  
Type Spaces ◽  
Hardy Type Operators

AbstractWe study weighted generalized Hardy and fractional operators acting from generalized Morrey spaces

Generalized fractional integral operators on generalized Orlicz–Morrey spaces of the second kind over non-doubling metric measure spaces

2018 ◽  
Vol 25 (2) ◽  
pp. 303-311
Author(s):  
Yoshihiro Sawano ◽  
Tetsu Shimomura
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Integral Transforms ◽  
Morrey Spaces ◽  
Metric Measure Spaces ◽  
Fractional Integral Operators ◽  
Generalized Morrey Spaces ◽  
Measure Spaces ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

Abstract In this paper, we aim to deal with the boundedness and the weak-type boundedness for the generalized fractional integral operators on generalized Orlicz–Morrey spaces of the second kind over non-doubling metric measure spaces, as an extension of [Y. Sawano and T. Shimomura, Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces, Z. Anal. Anwend. 36 2017, 2, 159–190], [Y. Sawano and T. Shimomura, Generalized fractional integral operators over non-doubling metric measure spaces, Integral Transforms Spec. Funct. 28 2017, 7, 534–546] and [I. Sihwaningrum, H. Gunawan and E. Nakai, Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces, Math. Nachr., to appear].

scholarly journals Embeddings of weighted generalized Morrey spaces into Lebesgue spaces on fractal sets

2019 ◽  
Vol 22 (5) ◽  
pp. 1203-1224
Author(s):  
Natasha Samko
Keyword(s):  
Morrey Space ◽  
General Setting ◽  
Morrey Spaces ◽  
General Nature ◽  
Sufficient Condition ◽  
Lebesgue Spaces ◽  
Generalized Morrey Spaces ◽  
Fractal Sets ◽  
Measure Spaces ◽  
Logarithmic Type

Abstract We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls(X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

Hardy Type Operators In Local Vanishing Morrey Spaces On Fractal Sets

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Dag Lukkassen ◽  
Lars-Erik Persson ◽  
Natasha Samko
Keyword(s):  
Morrey Spaces ◽  
Fractional Dimension ◽  
Explicit Dependence ◽  
Generalized Morrey Spaces ◽  
Weighted Estimates ◽  
Fractal Sets ◽  
Hardy Type ◽  
Hardy Type Operators

AbstractWe obtain two-weighted estimates for the Hardy type operators from local generalized Morrey spaces LThe obtained results show the explicit dependence of the mapping properties of the Hardy type operators on the fractional dimension of the set (X, μ, ϱ). An application to spherical Hardy type operators is also given.

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

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