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	1. Boundedness and compactness of multilinear singular integrals on Morrey spaces
	MEI Ting,LI Ao-Bo
	Mathematics 20 April 2023
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	Abstract:In this paper, we consider the boundedness and compactness of the multilinear singular integral operator on Morrey spaces, which is defined by\begin{align}T_Af(x)={\rm{p.v.}}\int_{\mathbb{R}^n} \frac{\Omega(x-y)}{\|x-y\|^{n+1}} R(A;x,y)f(y)dy,\end{align}where $R(A;x,y)=A(x)-A(y)-\nabla A(y)\cdot(x-y)$ with $D^\beta A\in BMO(\mathbb{R}^n)$ for all $\|\beta\|=1$.We prove that $T_A$ is bounded and compact on Morrey spaces $L^{p,\lambda}(\mathbb{R}^n)$ for all $1<p<\infty$ with $\Omega$ and $A$ satisfying some conditions. Moreover, the boundedness and compactness of the maximal multilinear singular integral operator $T_{A,*}$ on Morrey spaces are also given in this paper.
	TO cite this article:MEI Ting,LI Ao-Bo. Boundedness and compactness of multilinear singular integrals on Morrey spaces[OL].[20 April 2023] http://en.paper.edu.cn/en_releasepaper/content/4760244


	2. Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
	HE ZIYI, YANG DACHUN, YUAN WEN
	Mathematics 02 October 2015
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	Abstract:In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({mathbb R}^n)$,with $pin [2,infty)$ and $ninmathbb N$ or $pin (1,2)$ and $nin{1,2,3}$, via the Lusin areafunction and the Littlewood-Paley $g_lambda^st$-function in terms of ball means.
	TO cite this article:HE ZIYI, YANG DACHUN, YUAN WEN. Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls[OL].[ 2 October 2015] http://en.paper.edu.cn/en_releasepaper/content/4656719

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	3. Weighted Endpoint Estimates for Commutators ofCalder'on-Zygmund Operators
	LIANG YIYU, KY LUONG DANG, YANG DACHUN
	Mathematics 02 October 2015
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	Abstract:Let $deltain(0,1]$ and $T$ be a $delta$-Calder'on-Zygmund operator.Let $w$ be in the Muckenhoupt class $A_{1+delta/n}({mathbb R}^n)$ satisfying$int_{{mathbb R}^n}rac {w(x)}{1+\|x\|^n},dx<infty$.When $bin{ m BMO}(mathbb R^n)$,it is well known that the commutator $[b, T]$ is not bounded from $H^1(mathbb R^n)$to $L^1(mathbb R^n)$ if $b$ is not a constant function.In this article, the authors find out a proper subspace${mathopmathcal{BMO}_w({mathbb R}^n)}$of $mathopmathrm{BMO}(mathbb R^n)$ such that,if $bin {mathopmathcal{BMO}_w({mathbb R}^n)}$, then $[b,T]$ is bounded from theweighted Hardy space $H_w^1(mathbb R^n)$ to the weighted Lebesguespace $L_w^1(mathbb R^n)$.Conversely, if $bin{ m BMO}({mathbb R}^n)$ and the commutators of theclassical Riesz transforms ${[b,R_j]}_{j=1}^n$are bounded from $H^1_w({mathbb R}^n)$ into $L^1_w(R^n)$,then $bin {mathopmathcal{BMO}_w({mathbb R}^n)}$.
	TO cite this article:LIANG YIYU, KY LUONG DANG, YANG DACHUN. Weighted Endpoint Estimates for Commutators ofCalder'on-Zygmund Operators[OL].[ 2 October 2015] http://en.paper.edu.cn/en_releasepaper/content/4656709


	4. A note on high order commutators of some bilinear operators
	ZHANG　JUAN, HE QIANJUN, LIU ZONGGUANG
	Mathematics 19 August 2015
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	Abstract:Denote by $T$ and $I_{lpha}$ the bilinear Calder'{o}n-Zygmund singular integral operator and bilinear fractional integral operator. In this paper, we give some characterizations of $mathrm{BMO}$ space via the high order commutators of the bilinear singular integral operator $T_{b}^{m}$ and the bilinear fractional integral operator $I_{lpha,b}^{m}$, respectively. More precisely, we prove that the corresponding commutators $T_{b}^{m}$ and $I_{lpha,b}^{m}$ are all bounded operators from $L^{p_{1}} imes L^{p_{2}}$ into $L^{p}$ if $binmathrm{BMO}$ for some suitable indexes $p_{1}$, $p_{2}$ and $p$. Conversely, $binmathrm{BMO}$ if the commutators $T_{b}^{m}$ and $I_{lpha,b}^{m}$ map $L^{p_{1}} imes L^{p_{2}}$ into $L^{p}$ for some suitable indexes $p_{1}$, $p_{2}$, $p$ and $m$ is an even integer.
	TO cite this article:ZHANG　JUAN, HE QIANJUN, LIU ZONGGUANG. A note on high order commutators of some bilinear operators[OL].[19 August 2015] http://en.paper.edu.cn/en_releasepaper/content/4652329


	5. Second-order Riesz Transforms and Maximal Inequalities Associated toMagnetic Schr"odinger Operators
	YANG DACHUN, YANG SIBEI
	Mathematics 26 August 2014
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	Abstract:Let $A:=-( abla-iec{a})cdot( abla-iec{a})+V$be a magnetic Schr"odinger operator on $mathbb{R}^n$, where$ec{a}:=(a_1,,ldots,, a_n)inL^2_{mathrm{loc}}(mathbb{R}^n,mathbb{R}^n)$ and $0le VinL^1_{mathrm{loc}}(mathbb{R}^n)$ satisfy some reverse H"olderconditions. Let $arphi:\mathbb{R}^n imes[0,infty) o[0,infty)$ be such that$arphi(x,cdot)$ for any given $xinmathbb{R}^n$ is an Orliczfunction, $arphi(cdot,t)in {mathbb A}_{infty}(mathbb{R}^n)$for all $tin (0,infty)$ (the class of uniformly Muckenhouptweights) and its uniformly critical upper type index$I(arphi)in(0,1]$. In this article, the authors prove thatsecond-order Riesz transforms $VA^{-1}$ and$( abla-iec{a})^2A^{-1}$ are bounded from theMusielak-Orlicz-Hardy space $H_{arphi,,A}(mathbb{R}^n)$,associated with $A$, to the Musielak-Orlicz space$L^{arphi}(mathbb{R}^n)$. Moreover, the authors establish theboundedness of $VA^{-1}$ on $H_{arphi,,A}(mathbb{R}^n)$. Asapplications, some maximal inequalities associated to $A$ in thescale of $H_{arphi,,A}(mathbb{R}^n)$ are obtained.
	TO cite this article:YANG DACHUN, YANG SIBEI. Second-order Riesz Transforms and Maximal Inequalities Associated toMagnetic Schr"odinger Operators[OL].[26 August 2014] http://en.paper.edu.cn/en_releasepaper/content/4607531


	6. Strichartz estimates for the wave equation with full Laplacian on the quaternion Heisenberg group
	SONG Naiqi,ZHAO Jiman
	Mathematics 16 January 2014
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	Abstract:In this article, we prove dispersive and Strichartz estimates for the solution of the wave equationrelated to the full Laplacian on the quaternion Heisenberg group, by means of homogeneous Besov spacedefined by a Littlewood-Paley decomposition related to the full Laplacian.
	TO cite this article:SONG Naiqi,ZHAO Jiman. Strichartz estimates for the wave equation with full Laplacian on the quaternion Heisenberg group[OL].[16 January 2014] http://en.paper.edu.cn/en_releasepaper/content/4581622


	7. Hajl asz gradients are upper gradients
	JIANG RENJIN, NAGESWARI SHANMUGALINGAM,YANG DACHUN, YUAN WEN
	Mathematics 10 November 2013
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	Abstract:Let $(X, d, mu)$ be a metric measure space, with $mu$ a Borel regular measure.In this paper, we prove that, if $uin L^1_loc(X)$ and $g$ is aHajl asz gradient of $u$, then there exists $widetilde u$ such that$widetilde u=u$almost everywhere and $4g$ is a $p$-weakupper gradient of $widetilde u$. This result avoids a priori assumptionon the quasi-continuity of $u$ used in [Rev. Mat. Iberoamericana 16 (2000), 243-279].As an application, an embedding of the Morrey-type function spaces based on Hajl asz-gradientsinto the corresponding function spaces based on upper gradients is obtained. We also introducethe notion of local Hajl asz gradient, and investigate the relations between local Hajl asz gradientand upper gradient.
	TO cite this article:JIANG RENJIN, NAGESWARI SHANMUGALINGAM,YANG DACHUN, et al. Hajl asz gradients are upper gradients[OL].[10 November 2013] http://en.paper.edu.cn/en_releasepaper/content/4568999


	8. Interpolation of Morrey Spaces on MetricMeasure Spaces
	Lu Yufeng,Yang Dachun,Yuan Wen
	Mathematics 20 February 2013
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	Abstract:In this article, via the classical complex interpolation methodand some interpolation methods traced to Gagliardo,the authors obtain an interpolation theorem forMorrey spaces on quasi-metric measure spaces, which generalizessome known results on Rn.
	TO cite this article:Lu Yufeng,Yang Dachun,Yuan Wen. Interpolation of Morrey Spaces on MetricMeasure Spaces[OL].[20 February 2013] http://en.paper.edu.cn/en_releasepaper/content/4521984


	9. Weighted Lp boundedness of Carleson type maximal operators
	DING Yong,LIU Honghai
	Mathematics 12 July 2011
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	Abstract:In 2001, Stein and Wainger have proved that Carleson type maximal operators is Lp bounded, where the phase function P(y) is polynomial without linear term and singular kernel K is smooth. In this artical, authors use TT* method to generalize Stein and Wainger's result, that is, Carleson type maximal operators is weighted Lp bounded for 1<p<∞, where phase function P(y) is polynomial without linear terms, K(y)=Ω(y)/\|y\|n, Ω satisfies Lq-Dini condition, 1<q≤∞.
	TO cite this article:DING Yong,LIU Honghai. Weighted Lp boundedness of Carleson type maximal operators[OL].[12 July 2011] http://en.paper.edu.cn/en_releasepaper/content/4435360


	10. Lp boundedness of Carleson type maximal operators with nonsmooth kernels
	DING Yong,LIU Honghai
	Mathematics 12 July 2011
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	Abstract:Stein and Wainger have proved that Carleson type maximal operators is Lp bounded, where the phase function P(y) is polynomial without linear term and singular kernel K is smooth. In this artical, authors consider another kind of Carleson type maximal operators, where the phase function is P(\|y\|), P(t) is a polynomial on R without linear term, K(y)=Ω(y)/\|y\|n, Ω∈H1(Sn-1). They obtain the Lp boundedness for this kind of Carleson type maximal operators by Stein-Wainger's TT* argument and Calderon-Zygmund's rotation method.
	TO cite this article:DING Yong,LIU Honghai. Lp boundedness of Carleson type maximal operators with nonsmooth kernels[OL].[12 July 2011] http://en.paper.edu.cn/en_releasepaper/content/4435357

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