A note on vanishing Morrey → VMO result for fractional integrals of variable order

2020 ◽  
Vol 23 (1) ◽  
pp. 298-302 ◽  
Author(s):  
Humberto Rafeiro ◽  
Stefan Samko
Keyword(s):  
Morrey Space ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operator ◽  
Limiting Case

AbstractIn the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.

scholarly journals Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces

2020 ◽  
Vol 27 (1) ◽  
pp. 157-164
Author(s):  
Stefan Samko
Keyword(s):  
Growth Condition ◽  
Lebesgue Space ◽  
Measure Space ◽  
Variable Exponent ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operator ◽  
Open Set ◽  
Measure Spaces ◽  
Limiting Case

AbstractWe show that the fractional operator {I^{\alpha(\,\cdot\,)}}, of variable order on a bounded open set in Ω, in a quasimetric measure space {(X,d,\mu)} in the case {\alpha(x)p(x)\equiv n} (where n comes from the growth condition on the measure μ), is bounded from the variable exponent Lebesgue space {L^{p(\,\cdot\,)}(\Omega)} into {\mathrm{BMO}(\Omega)} under certain assumptions on {p(x)} and {\alpha(x)}.

scholarly journals Riesz potential on the Heisenberg group and modified Morrey spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 189-212
Author(s):  
Vagif S. Guliyev ◽  
Yagub Y. Mammadov
Keyword(s):  
Heisenberg Group ◽  
Fractional Integral ◽  
Morrey Space ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Group Setting ◽  
Riesz Potential Operator ◽  
Homogeneous Dimension ◽  
Limiting Case

Abstract In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

scholarly journals A note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Stefan Samko
Keyword(s):  
Fractional Integration ◽  
Fractional Integrals ◽  
Variable Order ◽  
Integration Operator ◽  
Hölder Continuous ◽  
Open Set ◽  
Fractional Integration Operator ◽  
Limiting Case

AbstractWe show that the Riesz fractional integration operator I α(·) of variable order on a bounded open set in Ω ⊂ ℝn in the limiting Sobolev case is bounded from L p(·)(Ω) into BMO(Ω), if p(x) satisfies the standard logcondition and α(x) is Hölder continuous of an arbitrarily small order.

scholarly journals Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

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.

Maximal and Potential Operators in Variable Exponent Morrey Spaces

2008 ◽  
Vol 15 (2) ◽  
pp. 195-208 ◽  
Author(s):  
Alexandre Almeida ◽  
Javanshir Hasanov ◽  
Stefan Samko
Keyword(s):  
Variable Exponent ◽  
Morrey Spaces ◽  
Bmo Space ◽  
Sobolev Type ◽  
Potential Operator ◽  
Variable Order ◽  
Potential Operators ◽  
Open Set ◽  
Limiting Case ◽  
Littlewood Maximal Operator

Abstract We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿𝑝(·), λ(·)(Ω) over a bounded open set Ω ⊂ ℝ𝑛 and a Sobolev type 𝐿𝑝(·), λ(·) → 𝐿𝑞(·), λ(·)-theorem for potential operators 𝐼 α(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

scholarly journals Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

2013 ◽  
Vol 21 (2) ◽  
pp. 111-130
Author(s):  
Malik S. Dzhabrailov ◽  
Sevinc Z. Khaligova
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Riesz Potential ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Potential Operator ◽  
Fractional Maximal Operator ◽  
Riesz Potential Operator ◽  
Necessary And Sufficient ◽  
Limiting Case

Abstract We prove that the anisotropic fractional maximal operator Mα,σ and the anisotropic Riesz potential operator Iα,σ, 0 < α < ∣σ∣ are bounded from the anisotropic modified Morrey space L̃1,b,σ(Rn) to the weak anisotropic modified Morrey space WL̃q,b,σ(Rn) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃p,b,σ(Rn) to L̃q,b,σ(Rn) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case we prove that the operator Mα,σ is bounded from L̃p,b,σ(Rn) to L∞ (Rn) and the modified anisotropic Riesz potential operator Ĩα,σ is bounded from L̃p,b,σ(Rn) to BMOσ(Rn).

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.

scholarly journals Weighted Hardy operators in local generalized Orlicz-Morrey spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 522-533
Author(s):  
C. Aykol ◽  
Z.O. Azizova ◽  
J.J. Hasanov
Keyword(s):  
Morrey Space ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Strong Type ◽  
Young Functions ◽  
Weighted Hardy Operators ◽  
Hardy Operators

In this paper, we find sufficient conditions on general Young functions $(\Phi, \Psi)$ and the functions $(\varphi_1,\varphi_2)$ ensuring that the weighted Hardy operators $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ are of strong type from a local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ into another local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$. We also obtain the boundedness of the commutators of $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ from $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ to $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$.

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.

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