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On the bilinear square Fourier multiplier operators and related multilinear square functions

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Science China Mathematics 2017年第8期60卷 1477-1502页

作者： SI ZengYan xue qingying YABUTA Kz School of Mathematics and Information Science Henan Polytechnic University School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems Ministry of Education Research Center for Mathematical Sciences Kwansei Gakuin University

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.

关键词： multilinear square functions Fourier multiplier operator multiple weights commutators

Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces

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Frontiers of Mathematics in China 2014年第6期9卷 1325-1347页

作者： Wen.juan LI qingying xue School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics andComplex Systems Ministry of Education Beijing 100875 China

Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b,Iα] is continuous from an atomic Hardy space Hp b into Lp, where Hp b is a subspace of the Hardy space Hp for n/（n ＋ 1）〈 p ≤ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint （Hp1 b1 ×... × HP2, Lp） boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderdn-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ （Lipβ）m（Rn）.

关键词： Multilinear fractional type operator multilinar Calderdn-Zygmundoperator commutator Hardy type space

Convergence of truncated rough singular integrals supported by subvarieties on Triebel-Lizorkin spaces

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Frontiers of Mathematics in China 2019年第3期14卷 591-604页

作者： Feng LIU qingying xue Kozo YABUTA College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 China Research Center for Mathematical Sciences Kwansei Gakuin University Gakuen 2-1 Sanda 669-1337 Japan

Let Ω be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let Th,Ω,φ be the classical singular Radon transform, and let Th^ε,Ω,φ be its truncated operator with rough kernels associated to polynomial mapping which is defined by Th^ε,Ω,φf(x)=|f|y|>εf(x-φ(y))h(|y|)Ω(y)|y|^-ndy|.In this paper, we show that for any a ∈(-∞,∞) and (p, q) satisfying certain index condition, the operator Th^ε,Ω,φ enjoys the following convergence properties lim ε→0||Th^ε,Ω,φf-Th,Ω,φf||Fα^p,q(R^d)= 0 and limε→0||Th^ε,Ω,φf-Th,Ω,φf||Bα^p,q(R^d)=0, provided that Ω∈L(log+ L)β(S^n-1) for some β∈(0,1], or Ω∈H^1(S^n-1), or Ω∈(U1

关键词： Singular Radon transform truncated singular integral rough kernel convergence

Weighted estimates for the multilinear commutators of the Littlewood-Paley operators

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Science China Mathematics 2009年第9期52卷 1849-1868页

作者： xue qingying DING Yong School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex SystemsMinistry of EducationBeijing 100875China

In this paper, weighted Lp estimates and sharp weighted endpoint estimates for the mul-tilinear commutators of the Littlewood-Paley operators are established.

关键词： Littlewod-Paley operators multilinear commutators sharp function Young function space Oscexp L r endpoint estimates 42B25 47G10

Boundedness and continuity of maximal singular integrals and maximal functions on Triebel-Lizorkin spaces

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Science China Mathematics 2020年第5期63卷 907-936页

作者： Feng Liu qingying xue Kozo Yabuta College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China Laboratory of Mathematics and Complex Systems(Ministry of Education of China) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875China Research Center for Mathematical Science Kwansei Gakuin UniversitySanda 669-1337Japan

In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov *** boundedness and continuity o... 详细信息

关键词： maximal singular integrals maximal functions Fβ(S^n-1) Triebel-Lizorkin spaces Besov spaces

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

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Acta Mathematica Scientia 2021年第2期41卷 596-608页

作者： Yali PAN qingying xue College of Mathematics and Computer Science Zhejiang Normal UniversityJinhua 321004China Department of Mathematics School of InformationHuaibei Normal UniversityHuaibei 235000China School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex SystemsMinistry of EducationBeijing 100875China

Let 0<β<1 andΩbe a proper open and non-empty subset of R^(n).In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by M_(β)((f))(x)=sup_(Q(∈)xQ∈Fβ)Π_(i=1)^m1/|Q|∫_(Q)|f_(i)(y_(i))|dy_(i),where F_(β)={Q(x,l):x∈Ω,l<βd(x,Ω^(c))},Q=Q(x,l)is denoted as a cube with sides parallel to the axes,and x and l denote its center and half its side ***-weight characterizations for the multilinear local maximal operator M_(β)are obtained.A formulation of the Carleson embedding theorem in the multilinear setting is proved.

关键词： Multilinear local maximal operators A_((p,q))^(β)weights two-weight inequalities

Borderline Weighted Estimates for Commutators of Fractional Integrals

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Analysis in Theory and Applications 2021年第3期37卷 404-425页

作者： Zhidan Wang Huoxiong Wu qingying xue School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems Ministry of EducationBeijing 100875China School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005China

Let I_(α,→b)be the multilinear commutators of the fractional integrals Iαwith the symbol→b=(b1,……,bk)We show that the constant of borderline weighted estimates for Iαis I/ξ,and for I_(α,→b)is with each b_(i)... 详细信息

关键词： Commutators fractional integrals borderline weighted estimates Fefferman-Stein inequality.

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带粗糙核的多线性分数次极大算子的加权估计

中国科学：数学 2015年第12期45卷 1939-1952页

作者：蓝森华梅婷薛庆营丽水学院数学系丽水323000 北京师范大学数学科学学院数学与复杂系统教育部重点实验室北京100875

本文给出了带粗糙核的多线性分数次极大算子M_(Ω,α)的加权估计,获得了M_(Ω,α)的一种混合A_(P,q)-A_∞型估计和A_p型估计.作为应用,得到了带粗糙核的多线性分数次积分算子L_(Ω,α)的几乎最优估计.

关键词：多线性分数次极大算子 A(P,q))权 A_Ρ权

多线性分数次积分算子一般框架下的交换子的有界性

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中国科学：数学 2014年第5期44卷 587-600页

作者：薛庆营严井泉徐得三北京师范大学数学科学学院北京100875 北京师范大学数学复杂系统教育部重点实验室北京100875

对任意给定的正整数m,Z+×{1,...,m}的任意一个有限子集S,定义一般化的多线性分数次积分算子的交换子Iα,→b,S(f)(x)=integral from n=(Rn)m to ∞[∏(i,j)∈S(bi(x)-bi(yj)(|x-y1|+···+|x-ym|)mn-α]multiply from j=1 to m[fj(yj)d→y ],其中d→y=dy1···dym.此框架下的交换子包含了以往研究的各类分数次积分算子的交换子,并蕴含了多线性背景下新的交换子形式.在上述非常一般框架下,本文给出带多重A→p,q权的多线性分数次积分算子的交换子Iα,→b,S(→f)的加权强型(Lp1(ω1)×···×Lpm(ωm),Lq(ν→ωq))估计和加权弱型端点估计.本文还得到更一般核条件下的上述结果.

关键词：多线性分数次积分算子交换子 Oscexp Lr空间