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Multilinear Singular Integrals with rough kernel

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Acta Mathematica Sinica,English Series 2003年第1期19卷 51-62页

作者： ShanZhenLU HuoXiongWU PuZHANG DepartmentofMathermatics BeijngNormalUniversityBeijing100875 DepartmentofMathermatics ZhejiangUniversity(atXixiCampus)Hangzhou310028

For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.

关键词： Multilinear operator Singular integral Lipschitz spaces Triebel Lizorkin spaces rough kernel

Parabolic Littlewood-Paley g-Function with rough kernel

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Acta Mathematica Sinica,English Series 2008年第12期24卷 2049-2060页

作者： Qing Ying XUE Yong DING Kozo YABUTA School of Mathematical Sciences Laboratory of Mathematics and Complex Systems( BNU) Ministry of Education China Beijing Normal University Beijing 100875 P. R. China School of Science and Technology Kwansei Gakuin University Gakuen 2-1 Sanda 669-1337 Japan

In this note, we give the L^p （1 〈 p 〈∞） boundedness of the parabolic Littlewood Paley g-function with rough kernel.

关键词： parabolic Littlewood-Paley g-function mixed homogeneity spaces rough kernel

BOUNDEDNESS OF MULTILINEAR OPERATORS WITH rough kernel ON HERZ SPACES

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Applied Mathematics(A Journal of Chinese Universities) 2003年第4期18卷 467-476页

作者： Chen Dongxiang Chen JiechengDept. of Math., Zhejiang Univ.,Hangzhou 310028,China. Dept. of Math. Zhejiang Univ. Hangzhou China

In this paper,the boundedness of the multilinear operators with rough kernel on the Herz-type spaces is *** is proved that M A,Ω,T A,Ω are bounded on α,p q(Rn) while Mλ A,Ω,TL A,Ω are bounded from... 详细信息

关键词： multilinear operator rough kernel Herz space maximal operator

Commutators of Marcinkiewicz Integral with rough kernels on Sobolev Spaces

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Acta Mathematica Sinica,English Series 2011年第7期27卷 1345-1366页

作者： Yan Ping CHEN Yong DING Xin Xia WANG Department of Mathematics and Mechanics School of Applied ScienceUniversity of Science and Technology of Beijing Beijing 100083 P. R. China School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems (BNU) Ministry of Education Beijing 100875 P. R. China The College of Mathematics and Systems Science Xinjiang University U~'umqi 830046 P. R. China

In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP（R^n） to the Lebesgue space L^p（R^n） for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz integral with rough hypersingular kernel defined by μΩ,γf（x）=（∫0^∞｜∫｜x-y｜≤tΩ（x-y）/｜x-y｜^n-1f（y）dy｜^2dt/t^3＋2γ）^1/2,with Ω∈L^1（S^n-1）for 0〈γ〈min（n/2,n/p）or Ω∈L（log＋L）^β（S^n-1）for｜1-2/p｜〈β〈1（0〈γ〈n/2）,respectively.

关键词： Marcinkiewicz integral commutator rough kernel Sobolev space Bony paraproduct

A Note on the Generalized Marcinkiewicz Integral Operators with rough kernels

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Acta Mathematica Sinica,English Series 2012年第12期28卷 2395-2406页

作者： Yu Rong WU Huo Xiong WU School of Mathematical Sciences Xiamen University School of Mathematical Sciences Fudan University

In this paper, we study the generalized Marcinkiewicz integral operators MΩ,r on the product space Rn ×Rm. Under the condition that Ω is a function in certain block spaces, which is optimal in some senses, the boun... 详细信息

关键词： Marcinkiewicz integral Triebel-Lizorkin space vector-valued multiplier theorem rough kernel product space

λ-Central BMO Estimates for Commutators of Singular Integral Operators with rough kernels

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Acta Mathematica Sinica,English Series 2008年第3期24卷 373-386页

作者： Zun Wei FU Yan LIN Shan Zhen LU School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China Department of Mathematics Linyi Normal Universtiy Linyi 276005 P. R. China

The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed. As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, resDectivelv.

关键词： λ-central BMO space central Morrey space Calderon-Zygmund operator fractionalintegral operator rough kernel

L^p Bounds for Singular Integrals with rough kernels on Product Domains

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Acta Mathematica Sinica,English Series 2012年第1期28卷 133-144页

作者： Li MA Da Shan FAN Huo Xiong WU School of Mathematics and Computer Sciences Gannan Normal UniversityGanzhou 341000P.R.China Department of Mathematics University of Wisconsin-MilwaukeeMilwaukeeWI 53201USA School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface. Some rather weaker size conditions, which imply the L^P-boundedness ... 详细信息

关键词： Singular integral rough kernel product domain Littlewood-Paley theory Fourier transform estimate

L^p Boundedness for Parabolic Littlewood-Paley Operators with rough kernels Belonging to Block Spaces

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Acta Mathematica Sinica,English Series 2010年第2期26卷 277-286页

作者： Dong Xiang CHEN Shan Zhen LU School of Mathematics Beijing Normal University Beijing 100875 P. R. China and Department of Mathematics Jiangxi Normal University Nanchang 330022 P. R. China School of Mathematics Beijing Normal University Beijing 100875 P. R. China

This paper is primarily concerned with the proof of the Lp boundedness of parabolic Littlewood-Paley operators with kernels belonging to certain block spaces. Some known results are essentially extended and improved.

关键词： parabolic Littlewood-Paley operator Fourier transform rough kernel block space

Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces

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Applied Mathematics(A Journal of Chinese Universities) 2013年第1期28卷 90-100页

作者： ZHANG Chun-jie ZHANG Yan-dan School of Science Hangzhou Dianzi University Department of Mathematics Zhejiang University of Science & Technology

In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel （x） satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.

关键词： Oscillatory singular integral Triebel-Lizorkin space rough kernel phase function.