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.

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.

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.

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.

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.

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.

Integrability of the general product Hardy operators on the product Hardy spaces

In this paper,we prove that the general product Hardy operators are bounded from the product Hardy space H1n（Rm1×···×Rmn） to L1（R（∑ni=1mi）.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN PRODUCT G-CONVEX SPACES

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.

K-Spaces Property of Product Spaces

Let K be a class of spaces which are eigher a pseudo-open s-image of a metric space or a k-space having a compact-countable closed k-network. Let K′ be a class of spaces which are either a Fréchet space with a point-countable k-network or a point-G_δ k-space having a compact-countable k-network. In this paper, we obtain some sufficient and necessary conditions that the products of finitely or countably many spaces in the class K or K′ are a k-space. The main results are that Theorem A If X, Y ∈ K. Then X x Y is a k-space if and only if (X, Y) has the Tanaka’s condition. Theorem B The following are equivalent: (a) BF(w2) is false. (b) For each X, Y ∈ K′, X x Y is a k-space if and only if (X, Y) has the Tanaka’s condition.

Riesz Product Spaces and Representation Theory

Let {Ei:i∈I}be a family of Archimedean Riesz spaces.The Riesz product space is denoted by Πi∈I Ei.The main result in this paper is the following conclusion:There exists a completely regular Hausdorff space X such that Πi∈I Ei is Riesz isomorphic to C（X）if and only if for every i ∈ I there exists a completely regular Hausdorff space Xi such that Ei is Riesz isomorphic to C（Xi）.

The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces

Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.

MAXIMAL ELEMENTS FOR G_B-MAJORIZED MAPPINGS IN PRODUCT G_CONVEX SPACES AND APPLICATIONS (Ⅱ)

A new family of set_valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set_valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature.

MAXIMAL ELEMENTS FOR G_B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅰ)

A new family of set_valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set_valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature.