Best constants for Hausdorf operators on n-dimensional product spaces

In this paper,we study certain Hausdorf operators in the high-dimensional product spaces.We obtain their power weighted boundedness from Lpto Lqand characterize the necessary and sufcient conditions for their boundedness on the power weighted Lpspaces.Moreover,in the case p=q,we obtain the sharp bound constants.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Hausdorff operators on Euclidean spaces

Hausdorff operator is an important operator raised from the dilation on Euclidean space and rooted in the classical summability of number series and Fourier series. It is also connected to many well known operators in real and complex analysis. This article is a survey of some recent developments and extensions on the Hausdorff operator. Particularly, various boundedness properties of the Hausdorff operators, studied recently by our research group, are addressed.     

Defining Hausdorff operators on Euclidean spaces

After 2000, an interest in the Hausdorff operators grew, first of all in the sense of a diversity of spaces on which these operators were considered. We try to introduce a ‘correct’ definition of the Hausdorff operator on Euclidean spaces. This is supplemented by a variety of known and possible applications.     

Hausdorff算子在Triebel-Lizorkin型空间上的有界性

证明了Hausdorff算子HΦ及HΦ,A在Triebel-Lizorkin型空间Fα,τp,q,上的有界性,并且求得HΦ相应的算子范数.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

Multilinear Hausdorff operators and their best constants

In this paper, we study two different extensions of the Hausdorff operator to the multilinear case. Boundedness on Lebesgue spaces and Herz spaces is obtained. The bound on the Lebesgue space is optimal. Our results are substantial extensions of some known results on Multilinear high dimensional Hardy operator.     

Convolution operators on Lebesgue spaces of an n-dimensional cube

This paper is concerned with the problem of diagonally scaling a given nonnegative matrix a to one with prescribed row and column sums. The approach is to represent one of the two scaling matrices as the solution of a variational problem. This leads in a natural way to necessary and sufficient conditions on the zero pattern of a so that such a scaling exists. In addition the convergence of the successive prescribed row and column sum normalizations is established. Certain invariants under diagonal scaling are used to actually compute the desired scaled matrix, and examples are provided. Finally, at the end of the paper, a discussion of infinite systems is presented.     

Hausdorff type operators on the power weighted Hardy spaces

In this paper, we study the high‐dimensional Hausdorff type operators and establish their boundedness on the power weighted Hardy spaces for . As a consequence, we obtain that the Hausdorff type operator is bounded on if Φ is the Gauss function, or the Poisson function.     

Toeplitz-Hankel type operators on product spaces

Via a series of orthogonal two-dimensional wavelets, an orthogonal decomposition of the space of square integral functions on U×U (U is the upper half-plane) with the measurey ^a ₁ y ^a ₁ dx ₁ dx ₂ dx ₁ dx 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. Research was supported by the National Natural Science Foundation of China.     

Sharp bounds for Hardy operators on product spaces

In this article, we obtain the sharp bounds from L^P$\left({\mathbb{G}}^n\right)$ to the space wL^P$\left({\mathbb{G}}^n\right)$ for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L^P$\left({\mathbb{G}}^n\right)$ to the space L^P_I$\left({\mathbb{G}}^n\right)$ are obtained.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

Best constants for tensor products of Bernstein type operators

For the tensor product of k copies of the same one-dimensional Bernstein-type operator L, we consider the problem of finding the best constant in preservation of the usual modulus of continuity for the lp-norm on Rk. Two main results are obtained: the first one gives both necessary and sufficient conditions in order that 1+k1−1/p is the best uniform constant for a single operator; the second one gives sufficient conditions in order that 1+k1−1/p is the best uniform constant for a family of operators. The general results are applied to several classical families of operators usually considered in approximation theory. Throughout the paper, probabilistic concepts and methods play an important role.     

Singular integral operators on product Triebel-Lizorkin spaces

In this paper, we consider the rough singular integral operators on product Triebel-Lizorkin spaces and prove certain boundedness properties on the Triebel-Lizorkin spaces. We also use the same method to study the fractional integral operator and the Littlewood-Paley functions. The results extend some known results.     

Sharp bounds for Hardy type operators on higher-dimensional product spaces

We investigate a class of fractional Hardy type operators \({\mathscr{H}_{{\beta _1},{\beta _2}, \ldots ,{\beta _m}}}\) defined on higher-dimensional product spaces \({\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times \cdots \times {\mathbb{R}^{{n_m}}}\) and use novel methods to obtain their sharp bounds. In particular, we optimize the result due to S. M. Wang, S. Z. Lu, and D. Y. Yan [Sci. China Math., 2012, 55(12): 2469–2480].     

The multiplier operators on the weighted product spaces

In this paper, we proved the boundedness of multiplier operators on the weighted product spaces.

     

Calderón-Zygmund operators on product Hardy spaces

Let T be a product Calderón-Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of Hp(Rn×Rm) and Journé's geometric covering lemma, R. Fefferman proved the remarkable Hp(Rn×Rm)−Lp(Rn×Rm) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood-Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journé's covering lemma to show that T is bounded on product Hp(Rn×Rm) for if and only if , where ε is the regularity exponent of the kernel of T.