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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. 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


	5. 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


	6. A restriction theorem for the quaternion Heisenberg group
	Liu Heping,Wang Yingzhan
	Mathematics 28 January 2011
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	Abstract:We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from Lp to Lp' if 1<=p=<3/4 . This is different from the Heisenberg group, on which the restriction operator is not bounded from Lp to Lp' unless p=1.
	TO cite this article:Liu Heping,Wang Yingzhan. A restriction theorem for the quaternion Heisenberg group[OL].[28 January 2011] http://en.paper.edu.cn/en_releasepaper/content/4409526


	7. Inversion Formulas for the Spherical Radon-Dunkl Transform
	Zhongkai Li,Song Futao
	Mathematics 14 January 2009
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	Abstract:The spherical Radon-Dunkl transform R_{kappa}, associated to weight functions invariant under a finite reflection group, is introduced,and some elementary properties are obtained in terms of $h$-harmonics. Several inversion formulas of $R_{kappa}$ are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.
	TO cite this article:Zhongkai Li,Song Futao. Inversion Formulas for the Spherical Radon-Dunkl Transform[OL].[14 January 2009] http://en.paper.edu.cn/en_releasepaper/content/27760


	8. The Decomposition of Product Space $H^{1}_{L}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\times BMO_{L}$
	Li Pengtao ,Peng Lizhong
	Mathematics 27 May 2008
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	Abstract:In analogy with classical results([BIJZ]), we prove that functions in the product of the Hardy space $H^{1}_{L}$ associated with Schr“{o}dinger operators $L=-triangle+V$ and its dual space $BMO_{L}$ admit a suitable decomposition. We obtain that for $fin H^{1}_{L}$ and $bin BMO_{L}$, the point-wise product $b cdot f$ as a Schwartz distribution, denoted by $b times f in S＇(R^{n})$, can be decomposed in two parts; precisely, $b times f=u+v$ where $u in L^{1}(R^{n})$ while $v$ lies in Hardy-Orlicz space associated with Schr”{o}dinger operators $H^{{mathcal{P}}}_{L}(R^{n},d mu)$.
	TO cite this article:Li Pengtao ,Peng Lizhong . The Decomposition of Product Space $H^{1}_{L}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\times BMO_{L}$[OL].[27 May 2008] http://en.paper.edu.cn/en_releasepaper/content/21797


	9. L2--oundedness of Hilbert transforms along variable curves
	Chen Jiecheng,Zhu Xiangrong
	Mathematics 01 December 2006
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	Abstract:For $\phi \in C^1(R^1)$, $\gamma \in C^2(R^1)$, odd or even, $\gamma (0)=\gamma ^{\prime }(0)=0$, convex on $(0,\infty )$, define a Hilbert transform along variable curves by $$H_{\phi ,\gamma }(f)(x_1,x_2)=p.v.\int_{-\infty }^{+\infty }f(x_1-t,x_2-\phi (x_1)\gamma (t))\frac{dt}t. $$In this paper, we shall first give a counter-example to show that under the condition of Nagel-Vance-Wainger-Weinberg on $\gamma $, the $L^2-$boundedness of $H_{\phi ,\gamma }$ may fail even if $\phi \in C^\infty (R^1)$. Then, we relax Bennett
	TO cite this article:Chen Jiecheng,Zhu Xiangrong. L2--oundedness of Hilbert transforms along variable curves[OL].[ 1 December 2006] http://en.paper.edu.cn/en_releasepaper/content/10103


	10. Stability of G-frames
	Sun Wenchang
	Mathematics 16 February 2006
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	Abstract: G-frames are natural generalizations of frames which cover many other recent generalizations of frames, e.g., bounded quasi-projectors, frames of subspaces, outer frames, oblique frames, pseudo-frames and a class of time-frequency localization operators. Moreover, it is known that g-frames are equivalent to stable space splittings. In this paper, we study the stability of g-frames. We first present some properties for g-Bessel sequences. Then we prove that g-frames are stable under small perturbations. We also study the stability of dual g-frames.
	TO cite this article:Sun Wenchang. Stability of G-frames[OL].[16 February 2006] http://en.paper.edu.cn/en_releasepaper/content/5252

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