scholarly journals Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces

2006 ◽  
Vol 181 ◽  
pp. 103-148 ◽  
Author(s):  
Yong Ding ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Single Point ◽  
Morrey Spaces ◽  
Rough Kernel ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lipschitz Spaces ◽  
Area Function ◽  
Bmo Function ◽  
Lusin Area Function ◽  
Lipschitz Kernel

AbstractLet g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

scholarly journals A problem on rough parametric Marcinkiewicz functions

2002 ◽  
Vol 72 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu ◽  
Kôzô Yabuta
Keyword(s):  
Kernel Function ◽  
Radial Function ◽  
Marcinkiewicz Integral ◽  
Area Function ◽  
Lusin Area Function ◽  
Parametric Marcinkiewicz Integral

AbstractIn this note the authors give the L2(n) boundedness of a class of parametric Marcinkiewicz integral with kernel function Ω in L log+L(Sn−1) and radial function h(|x|) ∈ l ∞ l(Lq)(+) for 1 < q ≦.As its corollary, the Lp (n)(2 < p < ∞) boundedness of and and with Ω in L log+L (Sn-1) and h(|x|) ∈ l∞ (Lq)(+) are also obtained. Here and are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g*λ-function and the Lusin area function S, respectively.

Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces

2012 ◽  
Vol 55 (3) ◽  
pp. 646-662 ◽  
Author(s):  
Jiang Zhou ◽  
Bolin Ma
Keyword(s):  
Hardy Spaces ◽  
Lipschitz Function ◽  
Homogeneous Spaces ◽  
Lipschitz Functions ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Nondoubling Measure

AbstractUnder the assumption that μ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

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.

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 Sublinear operators with rough kernel on generalized Morrey spaces

1998 ◽  
Vol 27 (1) ◽  
pp. 219-232 ◽  
Author(s):  
Shanzhen LU ◽  
Dachun YANG ◽  
Zusheng ZHOU
Keyword(s):  
Morrey Spaces ◽  
Rough Kernel ◽  
Generalized Morrey Spaces ◽  
Sublinear Operators

scholarly journals Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Wei Wang ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Bmo Function

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

scholarly journals Applications of Littlewood-Paley Theory forB˙σ-Morrey Spaces to the Boundedness of Integral Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Yasuo Komori-Furuya ◽  
Katsuo Matsuoka ◽  
Eiichi Nakai ◽  
Yoshihiro Sawano
Keyword(s):  
Riesz Potential ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Lipschitz Spaces ◽  
Potential Operators

The boundedness of the various operators onB˙σ-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization ofB˙σ-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization ofB˙σ-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators onB˙σ-Lipschitz spaces is discussed.

The existence and boundedness of square functions on generalized Orlicz–Campanato spaces

2010 ◽  
Vol 17 (2) ◽  
pp. 405-421
Author(s):  
Songyan Zhang
Keyword(s):  
Single Point ◽  
Positive Function ◽  
Square Functions ◽  
Doubling Property ◽  
Area Function ◽  
Almost Everywhere

Abstract Let T(ƒ) denote the Littlewood–Paley square operators, including the g-function g(ƒ), Luzin area function S(ƒ) and Stein's function , on the generalized Orlicz–Campanato spaces , where Φ is a N-function satisfying the Δ2 condition and φ a positive function satisfying the doubling property. It is proved that if T(ƒ)(x 0) < ∞ for a single point , then T(ƒ)(x) exists almost everywhere in , and T(ƒ) is bounded on the spaces .

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