scholarly journals Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Hongbin Wang ◽  
Dunyan Yan
Keyword(s):  
Variable Exponent ◽  
Homogeneous Function ◽  
Higher Order ◽  
Marcinkiewicz Integral ◽  
Degree Zero ◽  
Herz Spaces ◽  
Marcinkiewicz Integrals ◽  
Bmo Functions ◽  
Parametric Marcinkiewicz Integral ◽  
Marcinkiewicz Integral Operator

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.

scholarly journals ESTIMATES FOR MARCINKIEWICZ INTEGRALS IN BMO AND CAMPANATO SPACES

2007 ◽  
Vol 49 (2) ◽  
pp. 167-187 ◽  
Author(s):  
GUOEN HU ◽  
YAN MENG ◽  
DACHUN YANG
Keyword(s):  
Regularity Condition ◽  
Mean Value ◽  
Marcinkiewicz Integral ◽  
Degree Zero ◽  
Marcinkiewicz Integrals ◽  
Almost Everywhere ◽  
Higher Dimensional ◽  
Lusin Area Integral ◽  
Image Position ◽  
Marcinkiewicz Integral Operator

AbstractIn this paper, the authors consider the behavior on BMO($\mathbb R^n$) and Campanato spaces for the higher-dimensional Marcinkiewicz integral operator which is defined by where Ω is homogeneous of degree zero, has mean value zero and is integrable on the unit sphere. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO($\mathbb R^n$) or to a certain Campanato space, then [μΩ(f)]2 is either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness is also obtained. The corresponding Lusin area integral is also considered.

scholarly journals Higher Order Commutators of Fractional Integral Operator on the Homogeneous Herz Spaces with Variable Exponent

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Liwei Wang ◽  
Meng Qu ◽  
Lisheng Shu
Keyword(s):  
Integral Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Higher Order ◽  
Lipschitz Functions ◽  
Fractional Integral Operator ◽  
Herz Spaces ◽  
Bmo Functions

By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMO functions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent. These estimates extend some known results in the literatures.

scholarly journals Rough Marcinkiewicz integral operators

2001 ◽  
Vol 27 (8) ◽  
pp. 495-503 ◽  
Author(s):  
Hussain Al-Qassem ◽  
Ahmad Al-Salman
Keyword(s):  
Integral Operator ◽  
Unit Sphere ◽  
Homogeneous Function ◽  
Integral Operators ◽  
Mean Value ◽  
Marcinkiewicz Integral ◽  
Degree Zero ◽  
Polynomial Mapping ◽  
Marcinkiewicz Integral Operator ◽  
Boundedness Result

We study the Marcinkiewicz integral operatorM𝒫f(x)=(∫−∞∞|∫|y|≤2tf(x−𝒫(y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where𝒫is a polynomial mapping fromℝnintoℝdandΩis a homogeneous function of degree zero onℝnwith mean value zero over the unit sphereSn−1. We prove anLpboundedness result ofM𝒫for roughΩ.

On the boundedness of Marcinkiewicz integrals on continual variable exponent Herz spaces

2019 ◽  
Vol 26 (1) ◽  
pp. 105-116 ◽  
Author(s):  
Alexander Meskhi ◽  
Humberto Rafeiro ◽  
Muhammad Asad Zaighum
Keyword(s):  
Variable Exponent ◽  
Marcinkiewicz Integral ◽  
Herz Spaces ◽  
Marcinkiewicz Integrals

AbstractIn this paper, we obtain the boundedness of the Marcinkiewicz integral on continual Herz spaces with variable exponent, where all parameters defining the space are variable.

scholarly journals Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

2020 ◽  
Vol 53 (1) ◽  
pp. 44-57
Author(s):  
Mohammed Ali ◽  
Qutaibeh Katatbeh
Keyword(s):  
Integral Operators ◽  
Marcinkiewicz Integral ◽  
Rough Kernels ◽  
Marcinkiewicz Integrals ◽  
Natural Extensions ◽  
Parametric Marcinkiewicz Integral ◽  
Weaker Conditions

AbstractIn this article, we study the generalized parabolic parametric Marcinkiewicz integral operators { {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)} related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to Lp spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.

BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES

2015 ◽  
Vol 99 (3) ◽  
pp. 380-398 ◽  
Author(s):  
FENG LIU ◽  
SUZHEN MAO
Keyword(s):  
Radial Function ◽  
Integral Operators ◽  
Marcinkiewicz Integral ◽  
Marcinkiewicz Integrals ◽  
Parametric Marcinkiewicz Integral ◽  
Marcinkiewicz Operators ◽  
L Log L

In an extrapolation argument, we prove certain $L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$-functions are given.

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