Boundedness of the potential operators and their commutators in the local “complementary” generalized variable exponent Morrey spaces on unbounded sets

2020 ◽  
Vol 11 (2) ◽  
pp. 423-438
Author(s):  
Canay Aykol ◽  
Xayyam A. Badalov ◽  
Javanshir J. Hasanov
Keyword(s):  
Variable Exponent ◽  
Morrey Spaces ◽  
Potential Operators

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 Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

2010 ◽  
Vol 107 (2) ◽  
pp. 285 ◽  
Author(s):  
Vagif S. Guliyev ◽  
Javanshir J. Hasanov ◽  
Stefan G. Samko
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Morrey Spaces ◽  
Variable Order ◽  
General Function ◽  
Singular Operators ◽  
Potential Operators ◽  
Type Integral ◽  
Bounded Sets ◽  
Littlewood Maximal Operator

We consider generalized Morrey spaces ${\mathcal M}^{p(\cdot),\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded sets $\Omega \subset {\mathsf R}^n$ we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type ${\mathcal M}^{p(\cdot),\omega} (\Omega)\rightarrow {\mathcal M}^{q(\cdot),\omega} (\Omega)$-theorem for the potential operators $I^{\alpha(\cdot)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\omega(x,r)$, which do not assume any assumption on monotonicity of $\omega(x,r)$ in $r$.

Norm inequalities on variable exponent vanishing Morrey type spaces for the rough singular type integral operators

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ferit Gürbüz ◽  
Shenghu Ding ◽  
Huili Han ◽  
Pinhong Long
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Variable Exponent ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Rough Kernel ◽  
Type Integral ◽  
Type Spaces ◽  
Bounded Sets ◽  
Littlewood Maximal Operator

AbstractIn this paper, applying the properties of variable exponent analysis and rough kernel, we study the mapping properties of rough singular integral operators. Then, we show the boundedness of rough Calderón–Zygmund type singular integral operator, rough Hardy–Littlewood maximal operator, as well as the corresponding commutators in variable exponent vanishing generalized Morrey spaces on bounded sets. In fact, the results above are generalizations of some known results on an operator basis.

scholarly journals Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Xukui Shao ◽  
Shuangping Tao
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Variable Kernel ◽  
Fractional Integral Operators ◽  
Weak Type Estimates ◽  
Weak Morrey Spaces

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent function p(·) equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.

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