SHARP BOUNDS FOR HARDY OPERATORS ON PRODUCT SPACES

In this article, we obtain the sharp bounds from LP（Gn） to the space wLP（Gn） for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from LP（Gn） to the space LPI （Gn） are obtained.

SOME ROUGH OPERATORS ON PRODUCT SPACES

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.

Nested Commutators of Calderon-Zygmund Operators on Product Spaces

Commutators of Calderon-Zygmund operators on product spaces to be study in the paper is the commutators of Sarah H Ferguson and Michael T Lacey. The L^P boundedness of the nested commutators is proved on product spaces, where 1＜p＜∞.

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective and projective tensor product spaces E F and E F. Special attention is paid to l2 l2. In addition, two counter-examples are given.

TOEPLITZ-HANKEL TYPE OPERATORS ON PRODUCT SPACES

Via a series of orihogonal two-dimensional wavelets, an orthogonal decomposition of the space of square integral functions on Ux U (U is the upper half-plane) with the meaaure y_1^(_1) y_2~(_2 dx_1 dx_2 dy_1 dy_2 is given. Four kinds of Toeplitz-Hankel type operators between the decomposition components are defined and boundedness. S_p properties of them are established.

Singular Integrals on Product Spaces with Mixed Norms

In this paper,we prove the boundedness of the singular integral operators on product spaces with mixed norms and obtain the endpoint weak-type estimates.

Some Results on Reverses of Cauchy-Schwarz Inequality in Inner Product Spaces

Some new reverses of the Cauchy-Schwarz inequality in inner product spaces are presented in this paper. As an application of the main result, a formula for error estimate concerning Cauchy-Schwarz’s inequality is provided. The results obtained in the paper complement and improve some recent work about this topic.

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan （2006）. Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integrals. Links of these newly introduced risk measures to multi-period comonotonic risk measures are represented. Finally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.

ON APPROXIMATELY(p,q)-WRIGHT AFFINE FUNCTIONS AND INNER PRODUCT SPACES

We prove, using the fixed point approach, some results on hyperstability （in normed spaces） of the equation that defines the generalization of p-Wright affine functions and show that they yield a simple characterization of the complex inner product spaces.

Angles of Norm ed Linear Spaces and Characterizations of Inner Product Spaces

In [1],a family of angles are defined in normed linear spaces. In this paper,it is shown that if anyone of the angles satisfies certain euclidean triangle congruence properties,the space must be an inner product space.

Intuitionistic Menger inner product spaces and applications to integral equations

In this paper, first introduce and define an intuitionistic Menger inner product space, and then, obtain a new fixed point theorem in a complete intuitionistic Menger inner product space. As an application, the results are used to study the existence and uniqueness of the solution to a linear Volterra integral equation.

Evolving patterns of agricultural production space in China:A network-based approach

The agricultural production space,as where and how much each agricultural product grows,plays a vital role in meeting the increasing and diverse food demands.Previous studies on agricultural production patterns have predominantly centered on individual or specific crop types,using methods such as remote sensing or statistical metrological analysis.In this study,we characterize the agricultural production space(APS)by bipartite network connecting agricultural products and provinces,to reveal the relatedness between diverse agricultural products and the spatiotemporal characteristic of provincial production capabilities in China.The results show that core products are cereal,pork,melon,and pome fruit;meanwhile the milk,grape,and fiber crop show an upward trend in centrality,which is in line with diet structure changes in China over the past decades.The little changes in community components and structures of agricultural products and provinces reveal that agricultural production patterns in China are relatively stable.Additionally,identified provincial communities closely resemble China's agricultural natural zones.Furthermore,the observed growth in production capabilities in North and Northeast China implies their potential focus areas for future agricultural production.Despite the superior production capa-bilities of southern provinces,recent years have witnessed a notable decline,warranting special attentions.The findings provide a comprehensive perspective for understanding the complex relationship of agricultural prod-ucts'relatedness,production capabilities and production patterns,which serve as a reference for the agricultural spatial optimization and agricultural sustainable development.

Best constants for Hausdorf operators on n-dimensional product spaces

In this paper, we study certain Hausdorff operators in the high-dimensional product spaces. We obtain their power weighted boundedness from Lp to Lq and characterize the necessary and sufficient conditions for their boundedness on the power weighted Lp spaces. Moreover, in the case p = q, we obtain the sharp bound constants.

CHARACTERIZATIONS OF JORDAN (?)-SKEW MULTIPLICATIVE MAPS ON OPERATOR ALGEBRAS OF INDEFINITE INNER PRODUCT SPACES

Let H and K be indefinite inner product spaces. This paper shows that a bijective map φ：B（H） →B（K） satisfies φ（AB^＋ ＋ B^＋A） = φ（A）φ（B）^＋ ＋ φ（B）^＋φ（A） for every pair A, B ∈ B（H） if and only if either φ（A） = cUAU^＋ for all A or φ（A） = cUA^＋U^＋ for all A; φ satisfies φ（AB^＋A） = φ;（A）φ;（B）^＋φ;（A） for every pair A, B ∈ B（H） if and only if either φ（A） = UAV for all A or φ（A） = UA^＋V for all A, where At denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U^tU = c^-1I and V^＋V = cI for some nonzero real number c.

The maximal super-singular integral operators on product spaces

We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ∈ S（R^n ×R^m）, where 0 ≤ α,β〈∞, b1 b2 ∈ L∞（R＋1 ）,Ω satisfies certain cancellation conditions and Ω∈L1（S^n-1×S^m-1）in the case α,β〉0;Ω∈L（log＋L）（S^n-1×S^m-1）in the case αβ=0 and α＋β 〉0. It is proved that, for 1〈p〈∞.T＊Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p（R^n×R^m）to the Lebesgue space L^p（R^n×R^m）.

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.

Rough Marcinkiewicz Integrals with Mixed Homogeneity on Product Spaces

In this paper, the parabolic Marcinkiewicz integral operators on the product spaces R^m × R^n（m, n ≥ 2） are studied. The LP-boundedness for such operators are established under rather weak size conditions of the kernels, which essentially improve or extend certain previous results.

On Linearly Topological Structure and Orthogonality of Menger Probabilistic Inner Product Spaces

In this paper the linearly topological structure of Menger Probabilistic inner product space is discussed. In virtue of these, some more general convergence theorems, Pythagorean theorem, and the orthogonal projective theorem are established.

Tensor Product of 2-Frames in 2-Hilbert Spaces

2-frames in 2-Hilbert spaces are studied and some results on it are presented. The tensor product of 2-frames in 2-Hilbert spaces is introduced. It is shown that the tensor product of two 2-frames is a 2-frame for the tensor product of Hilbert spaces. Some results on tensor product of 2-frames are established.

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B0,1 1 （Rn ×Rm） and on the local product Hardy space h1 （Rn ×Rm）. We establish some boundedness criteria for Hausdorff operators on these function spaces.