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	1. Commutators of Riesz transforms
	Gong Ru-Ming
	Mathematics 08 December 2015
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	Abstract:Let $A$ be a Laplacian operator associated with a quadratic formon $Omega$ where $Omega$ is the Euclidean space $mathbb{R}^n$ ora domain of $mathbb{R}^n$. In this paper, we show that when afunction $bin BMO(Omega)$, the commutators $[b,igtriangledownA^{-1/2}]$ are bounded on $L^p(Omega)$ for all $1<p<2$, where theoperators $igtriangledown A^{-1/2}$ are Riesz transformsassociated with $A$.
	TO cite this article:Gong Ru-Ming. Commutators of Riesz transforms[OL].[ 8 December 2015] http://en.paper.edu.cn/en_releasepaper/content/4669969


	2. Bilinear Operator and the Decomposition of $H^{1} imes BMO$
	LI Peng-Tao
	Mathematics 01 December 2015
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	Abstract:In this paper, we prove that, for every $bin BMO(R^{n})$ and $finH^{1}(R^{n})$, by use of a kind of compensated quantities, we canget a decomposition of the product space $BMO(R^{n}) imesH^{1}(R^{n})$. Precisely, we obtain, for $fin H^{1}(R^{n})$, $binBMO(R^{n})$, the point-wise product $bcdot f$ as a Schwartzdistribution, denoted by $b imes fin S'(R^{n})$, can be decomposedinto two parts associated with the bilinear operators, that is$b imes f=u+v$, where $uin L^{1}(R^{n})$ and $v$ belongs to theHardy-Orlicz space $H^{mathcal{P}}(R^{n})$.
	TO cite this article:LI Peng-Tao. Bilinear Operator and the Decomposition of $H^{1} imes BMO$[OL].[ 1 December 2015] http://en.paper.edu.cn/en_releasepaper/content/4667259

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	3. Normality and Quasinormality Criteria of Zero-free Meromorphic Functions
	Cheng Chunnuan,Xu Yan
	Mathematics 23 November 2015
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	Abstract:Let $k,K$ be positive integers, $arphi(z)( otequiv0)$ be ananalytic function, and $mathcal{F}$ be a family of zero-free meromorphicfunctions on a domain $D$, all of whose poles are multiple. If for each$finmathcal{F}$, $f^{(k)}(z)-arphi(z)$ has at most $K$ distinctzeros(ignoring multiplicity), then $mathcal{F}$ is quasinormalof order at most $ u$ on $D$, where $ u=[rac{K}{k+2}]$ is equal tothe largest integer not exceeding $rac{K}{k+2}$. In particular, if $K=k+1$, then $cal F$ is normal on $D$.
	TO cite this article:Cheng Chunnuan,Xu Yan. Normality and Quasinormality Criteria of Zero-free Meromorphic Functions[OL].[23 November 2015] http://en.paper.edu.cn/en_releasepaper/content/4665203


	4. Normal functions and shared sets
	Xu Yan, Qiu Huiling
	Mathematics 13 November 2015
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	Abstract:In this paper, we obtain some criteria for normal functions that share sets with their derivatives. The main result is:Let $S_1={a_1, a_2}$ and $ S_2={b_1, b_2}$ be two sets in $Bbb C$ such that $a_1a_2 eq 0$ and $b_1/b_2 otin Bbb{Z}^-cup1/Bbb{Z}^-$. Let $f$ be a meromorphic function in the unit disc $Delta$, and suppose that there exists a positive number $M$ such that $\|f'(z)\|leq M$ whenever $f(z)=0$. If $fin S_1 Leftrightarrow f'in S_2$ in $Delta$, then $f$ is normal. Here $Bbb{Z}^-$ denotes the set of all negative integers, and $1/Bbb{Z}^-$ stands for the set ${1/k; kin Bbb{Z}^-}$.
	TO cite this article:Xu Yan, Qiu Huiling. Normal functions and shared sets[OL].[13 November 2015] http://en.paper.edu.cn/en_releasepaper/content/4661587


	5. 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


	6. 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


	7. Integration of algebroid functions
	Sun Dao-Chun, Huo Ying-Ying, Kong Yin-ying, Chai Fu-Jie
	Mathematics 28 September 2015
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	Abstract:In this paper, we introduce the integration of algebroid functions on Riemann surfaces for the first time. Some properties of integration are obtained.By giving the definition of residues and integral function element, we obtain the condition that the integral is independent of path. At last, we prove that the integral of an irreducible algebroid function is also an irreducible algebroid function if all the residues at critical points are zeros.
	TO cite this article:Sun Dao-Chun, Huo Ying-Ying, Kong Yin-ying, et al. Integration of algebroid functions[OL].[28 September 2015] http://en.paper.edu.cn/en_releasepaper/content/4656329


	8. 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


	9. The boundedness of higher order Riesz transform associated with Sch
	SHEN Jian-Chun,Dong Jianfeng
	Mathematics 04 January 2015
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	Abstract:Let $L=-Delta+V$ be a Schr"{o}dinger operator on $mathbb{R}^n$ ($n geq 3$) , where $V ot equiv 0$ is a nonnegative potential belonging to certain reverse H"{o}lder class $B_s$ for $s geq n$. The Hardy type spaces $H_L^p, rac{n}{n+delta}<pleq 1$ for some $delta >0$, are defined in terms of the maximal function with respect to the semigroup ${e^{-tL} }_{t>0}$. In this article, we investigate the boundedness of some integral operator related to $L$, such as $VL^{-1}$, $Delta L^{-1}$ and $ abla^2 L^{-1}$, on spaces $H_L^p(mathbb{R}^n)$.
	TO cite this article:SHEN Jian-Chun,Dong Jianfeng. The boundedness of higher order Riesz transform associated with Sch[OL].[ 4 January 2015] http://en.paper.edu.cn/en_releasepaper/content/4626619


	10. 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

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