scholarly journals Some Estimates of Rough Bilinear Fractional Integral

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Yun Fan ◽  
Guilian Gao
Keyword(s):  
Fractional Integral ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Morrey Spaces ◽  
Central Morrey Space ◽  
Sufficient And Necessary Conditions

We study the boundedness of rough bilinear fractional integralBΩ,αon Morrey spacesLp,λ(ℝn)and modified Morrey spacesL~p,λ(ℝn)and obtain some sufficient and necessary conditions on the parameters. Furthermore, we consider the boundedness ofBΩ,αon generalized central Morrey spaceB˙p,φ(ℝn). These extend some known results.

scholarly journals Third Version of Weak Orlicz–Morrey Spaces and Its In-clusion Properties

2019 ◽  
Vol 4 (2) ◽  
pp. 257-262
Author(s):  
Al Azhary Masta ◽  
Siti Fatimah ◽  
Muhammad Taqiyuddin
Keyword(s):  
Morrey Space ◽  
Necessary Conditions ◽  
Orlicz Spaces ◽  
Morrey Spaces ◽  
The Third ◽  
Inclusion Relations ◽  
Sufficient And Necessary Conditions ◽  
Weak Morrey Spaces

Orlicz–Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are  three  versions  of  Orlicz–Morrey  spaces.  In  this  article,  we discussed  the  third  version  of  weak  Orlicz–Morrey  space, which is an enlargement of third version of (strong) Orlicz– Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as  a  generalization  of  weak  Orlicz  spaces,  weak  Morrey spaces,  and  generalized  weak  Morrey  spaces.  This  study investigated  some  properties  of the third  version of weak Orlicz–Morrey spaces, especially the sufficient and necessary conditions for inclusion relations between two these spaces. One of the keys to get our result is to estimate the quasi- norm of characteristics function of open balls in ℝ.

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 Estimates of Some Operators on One-Sided Weighted Morrey Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaoguang Shi ◽  
Zunwei Fu
Keyword(s):  
Harmonic Analysis ◽  
Fractional Integral ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Weighted Morrey Space ◽  
Weighted Morrey Spaces

A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.

scholarly journals Necessary and sufficient conditions on weighted multilinear fractional integral inequality

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhou Yongliang ◽  
Deng Yangkendi ◽  
Wu Di ◽  
Yan Dunyan
Keyword(s):  
Integral Inequality ◽  
Sobolev Inequality ◽  
Fractional Integral ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integral Inequalities ◽  
Multilinear Fractional Integral ◽  
Sufficient And Necessary Conditions ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.</p>

scholarly journals Weighted multilinear Hardy operators and commutators

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Zun Wei Fu ◽  
Shu Li Gong ◽  
Shan Zhen Lu ◽  
Wen Yuan
Keyword(s):  
Necessary Conditions ◽  
Morrey Spaces ◽  
Bmo Space ◽  
Weight Functions ◽  
Lebesgue Spaces ◽  
Sharp Bounds ◽  
Sharp Estimates ◽  
Sufficient And Necessary Conditions ◽  
Hardy Operators

AbstractIn this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann–Liouville and Weyl.

On two-weight estimates for the maximal operator in local Morrey spaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450099 ◽  
Author(s):  
Natasha Samko
Keyword(s):  
Maximal Operator ◽  
General Type ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Logarithmic Factor ◽  
Sufficient And Necessary Conditions ◽  
Local Morrey Spaces

For two weighted local Morrey spaces [Formula: see text] and [Formula: see text] 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 [Formula: see text] to [Formula: see text], 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.

scholarly journals On the boundedness of a generalized fractional integral on generalized Morrey spaces

2002 ◽  
Vol 33 (4) ◽  
pp. 335-340
Author(s):  
Eridani Eridani
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Fractional Integral Operator ◽  
Campanato Space ◽  
Generalized Morrey Space ◽  
Generalized Morrey Spaces ◽  
Generalized Fractional Integral

In this paper we extend Nakai's result on the boundedness of a generalized fractional integral operator from a generalized Morrey space to another generalized Morrey or Campanato space.

scholarly journals Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

2021 ◽  
Vol 7 (1) ◽  
pp. 1000-1014
Author(s):  
Suixin He ◽  
◽  
Shuangping Tao
Keyword(s):  
Bounded Domain ◽  
Maximal Operator ◽  
Fractional Integral ◽  
Morrey Space ◽  
Homogeneous Spaces ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Generalized Morrey Space ◽  
Fractional Integral Operators ◽  
Littlewood Maximal Operator

<abstract><p>The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ over non-homogeneous spaces, where $ G\subset $ $ \mathbb{R}^{n} $ is a bounded domain. Under assumption that functions $ \varphi $ and $ \phi $ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $ \theta $-type Calderón-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $. Moreover, the boundedness of commutator $ [b, T^{G}_{\theta}] $ which is generated by $ \theta $-type Calderón-Zygmund operator $ T_{\theta} $ and $ b\in\mathrm{RBMO}(\mu) $ on spaces $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ is also established.</p></abstract>

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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