Some Endpoint Estimates for the Multiplier Operator

In this paper,the author proves that Multiplier operator is bounded on BMO（Rⁿ）,LMO（Rⁿ） and CBMO^ρ,λ（Rⁿ）respectively if some concellation conditions are satisfied.     

关于乘子的几个端点估计

In this paper,the author proves that Multiplier operator is bounded on BMO（Rⁿ）,LMO（Rⁿ） and CBMO^ρ,λ（Rⁿ）respectively if some concellation conditions are satisfied.     

多线性分数次积分的Hardy-Littlewood-Sobolev定理

该文讨论了Lebesgue空间上多线性分数次积分算子的有界性,通过将多线性分数次积分转化为相对应的分数次积分来考虑,得到算子TΩ,α,A1,A2和MΩ,α,A1,A2的Hardy-Littlewood- Sobolev定理的弱型结果,并得到一种简明的方法．     

A Correction Theorem for Functions with Integral Smoothness

A theorem similar to the correction theorem of Oskolkov is proved. Namely, for a function with a given kth continuity modulus calculated in a symmetric space X and for every , there exists a set of measure at least and such that we can give a sharp quantitative estimate of the uniform kth continuity modulus of the considered function. It is shown that this estimate depends only on and on the fundamental function of the symmetric space. Bibliography: 9 titles.     

广义分数次积分多线性交换子的端点估计

设函数b=（b1,b2,…,bm）和广义分数次积分L-a／2（0〈α〈n）,它们生成多线性算子定义如下 Lb -a/2 f = [bm …, [b2[b1, L-a/2]],…, ]f,其中m ∈ Z＋ , bi ∈ Lipβi （0 〈βi 〈 1）,其中（1≤i≤m）．将讨论Lb -1a/2。从Mp^q（Rn）到Lip（α＋β-n/ q） （ Rn ）和q^q （ Rn ）到BMO（Rn）的有界性．     

Endpoint Estimates for Generalized Multilinear Fractional Integrals on the Non-homogeneous Metric Spaces

In this paper, some endpoint estimates for the generalized multilinear fractional integrals I_(α,m) on the non-homogeneous metric spaces are established.     

Hormander's Hp Multiplier Theorem for the Heisenberg Group

Let Hⁿ denote the (2n+1)-dimensional Heisenberg group. Givenan operator-valued function M, define the operator T_M with ‘^’ denoting theFourier transform. Hörmander-type sufficient conditionsare determined on M for the H^p-boundedness, p 1, of the operatorT_M on Hⁿ.     

广义Sublaplace算子的谱乘子定理

本文通过建立与广义Ｓｕｂｌａｐｌａｃｅ算子Ｌ相关的Ｌｉｔｌｅｗｏｏｄ－Ｐａｌｅｙ理论，得到Ｌ的谱乘子定理，作为该结果的应用，并给出一类Ｌａｇｕｅｒｅ函数展开的乘子定理     

On decomposability of Multilinear sets

We consider the Multilinear set \({\mathcal {S}}\) defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form \(y_I = \prod _{i \in I} x_i\), \(I \in {\mathcal {I}}\), where \({\mathcal {I}}\) denotes a family of subsets of \(\{1,\ldots , n\}\) of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when \({\mathcal {S}}\) is decomposable into simpler Multilinear sets \({\mathcal {S}}_j\), \(j \in J\); namely, the convex hull of \({\mathcal {S}}\) can be obtained by convexifying each \({\mathcal {S}}_j\), separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which \({\mathcal {S}}\) is decomposable into \({\mathcal {S}}_j\), \(j \in J\), based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.     

多线性Calder\'on-Zygmund算子的加权弱型端点估计及其应用

A weighted weak type endpoint estimate is established for the m-linear operator with Calderón-Zygmund kernel, which was introduced by Coifman and Meyer. As applications, the mapping properties on weighted L^p1（Rn）×···×L^pm（Rn） with weight MBw for certain maximal operator MB and general weight w, and a two-weight weighted norm estimate for this operator, are obtained.     

New Sufficient Conditions for the Denseness of Norm Attaining Multilinear Forms

This paper gives new sufficient conditions for the density ofthe set of norm attaining multilinear forms in the space ofall continuous multilinear forms on a Banach space. The symmetriccase is also discussed.     

鞍点定理在Lagrange乘数法上的应用

将鞍点的概念运用在Lagrange乘数法上,给出了多元函数的条件极值问题存在的一个充要条件.     

On the k—smoothness and k—strong smoothness

5l.IntroductionFrompal,er[11,L2],to[3j,tI1efol1owingresultsareobtained:1)LetXbeaBanachspace,thenxes(X)={xeX;llxIl=1}isG-differetiableifandonIyif,foreachyeX,letXI=span(x,y).2)LetXbeaBanachspace,thenxeS(X)isF-differetiableifandonlyifIn[4j,NanchaoxunandWangjianhuaintroducedk-smoothnessandk-strongsmooth-ness.lnthisPaPer,wegive:Theorem1LetXbeaBanachspace,thcnthefollowingstatementsareequivalent.(i)xo6S(X)isk-smooth.(ii)IfX*cyXisk-dimensionalsubspaceandx,eX*,thenwhereU(x,,r)={xeX;l1x-xo…     

On Bivariate Smoothness Spaces Associated with Nonlinear Approximation

In recent years there have been various attempts at the representations of {\mbox multivariate} signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves of singularities like the Mumford–Shah functional and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further simplified remains open.     

On Components of the Partial Schur Multiplier

In this paper we study properties of the partial Schur multiplier. As an application a characterization of components of the partial Schur multiplier for some groups is given. We also describe the classical Schur multiplier of elementary abelian 2-groups by using partial factor sets.     

On Equivalence of Moduli of Smoothness

It is known that iffW^k_p, thenω_m(f, t)_ptω_m−1(f′, t)_p…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S, t)_ptω_m−1(S′, t)_pt²ω_m−2(S″, t)_p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.     

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

We prove Szegös strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary N-functions both satisfying the ⁰ ₂-condition and by weight sequences satisfying some regularity and compatibility conditions.