Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

Endpoint estimates for commutators of multilinear fractional integral operators

In this paper, the weak type LlogL estimate for the multilinear fractional commutator is obtained by introducing a new kind of maximal operator of the multilinear fractional order associated with the mean Luxumburg norm and using the technique of sharp function.     

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

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

Weighted endpoint estimates for commutators of fractional integrals

Given α, 0 < α < n, and b ∈ BMO, we give sufficient conditions on weights for the commutator of the fractional integral operator, [b, I _α ], to satisfy weighted endpoint inequalities on ℝⁿ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on ℝⁿ.     

Weighted endpoint estimates for commutators of multilinear fractional integral operators

Let m be a positive integer, 0 < α < mn, $\overrightarrow b $ = (b ₁, …, b _m ) ε BMO ^m . We give sufficient conditions on weights for the commutators of multilinear fractional integral operators ${\rm I}_a^{\overrightarrow b }$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362, (2010), 355–373.     

Weighted inequalities for multilinear commutators of some sublinear operators

We establish sharp estimates for some multilinear commutators related to the Littlewood-Paley and Marcinkiewicz operators. As an application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear commutators.      

Sharp estimates of multilinear operators in polydisk and unit ball

Sharp estimates for some multifunctional analytic expressions and multilinear operators are given, generalizing the corresponding estimates for a single function or for a linear operator.     

Weighted estimates for multilinear pseudodifferential operators

In this paper,we study the weighted estimates for multilinear pseudodifferential operators.We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions.Our result generalizes similar ones from the classical Ap weights to multiple weights.     

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights is defined as

     

Boundedness of multilinear commutators of generalized fractional integrals

Let L be the infinitesimal generator of an analytic semigroup on L²(?ⁿ ) with Gaussian kernel bound, and let L^{–α /2} be the fractional integral of L for 0 < α < n. Suppose that b = (b₁, b₂, …, b_m ) is a finite family of locally integral functions, then the multilinear commutator generated by b and L^{–α /2} is defined by L^{–α /2} _b f = [b_m , …, [b₂, [b₁, L^{–α /2}]], …, ] f, where m ∈ ?⁺. When b₁, b₂, …, b_m ∈ BMO or b_j ∈ Λ (0 < β_j < 1) for 1 ≤ j ≤ m, the authors study the boundedness of L^{–α /2} _b . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Triebel-lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel-Lizorkin space. The operators include Littlewood-Paley operator and Marcinkiewicz operator.     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

向量值极大积分交换子的加权估计

考虑了极大向量值交换子的加权有界性.分别得到了强型和弱型的加权模不等式,其中权函数是非负局部可积函数.     

Multiple weighted estimates for commutators of multilinear fractional integral operators

Multilinear commutators and iterated commutators of multilinear fractional integral operators with BMO functions are studied. Both strong type and weak type endpoint weighted estimates involving the multiple weights for such operators are established and the weak type endpoint results are sharp in some senses. In particular, we extend the results given by Cruz-Uribe and Fiorenza in 2003 and 2007 to the multilinear setting. Moreover, we modify the weak type of endpoint weighted estimates and improve the strong type of weighted norm inequalities on the multilinear commutators given by Chen and Xue in 2010 and 2011.     

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let(X, d, μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderón-Zygmund operator and ■:=(b1,..., bm) be a finite family of ■(μ) functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator ■Tgenerated by T and ■ are obtained.     

多线性分数次积分算子一般框架下的交换子的有界性——献给陆善镇教授75华诞

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）=∫（Rn）^m ∏（i,j）∈S（bi（x）-bi（yj））/（｜x-y1｜＋…＋｜x-ym｜）^mn-α∏（j=1→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（L^p1（ω1）×···×L^pm（ωm）,L^q（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.     

Sharp maximal function estimates for multilinear singular integrals with non-smooth kernels

In this article we obtain weighted norm estimates for multilinear singular integrals with non-smooth kernels and the boundedness of certain multilinear commutators by making use of a sharp maximal function.     

Weak estimates for commutators of fractional integral operators

By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type LlogL estimates for the commutators of the fractional integral operator and the related maximal operator are established.     

Weighted endpoint estimates for maximal commutators associated with the sections

We prove endpoint estimates for maximal commutators for a class of singular integral operators related to the real analysis of the Monge Ampere equation.     

Weighted norm inequalities for the commutators of multilinear singular integral operators

In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p₁ ∈ （1,∞）, p₂,…,p_m ∈(1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L^P1（Rⁿ,M_l（logL）^σw）× p2（Rn,M^w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p₁ and the order of commutator