Lipschitz Estimates for Commutators of N-dimensional Fractional Hardy Operators

In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.

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.

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）.

COMMUTATORS OF GENERALIZED HARDY OPERATORS ON HOMOGENEOUS GROUPS

Let G be a homogeneous group. The author considers the boundedness of commutators generated by the generalized Hardy operators and CMO（G） functions on Herz spaces in the setting of homogeneous group. This article extends some known results.

WEIGHTED BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL HARDY OPERATORS WITH BESOV-LIPSCHITZ FUNCTIONS

In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.

λ-central BMO Estimates for Higher Order Commutators of Hardy Operators

In this paper, the λ-central BMO estimates for higher order commuta-tors of Hardy operators on central Morrey space Lq,λ（Rn） are established. In the meanwhile, the corresponding corollary for central BMO estimates is also obtained.

Lipschitz Estimates for Commutators of p-adic Fractional Hardy Operators

In this paper, the boundedness of commutators generated by the n- dimensional fractional Hardy operators and Lipschitz functions on p-adic function spaces are obtained. The authors show that these commutators are bounded on Herz space and Lebesgue space with suitable indexes. Moreover, the commutator of Hardy- Littlewood-Poly~ operator is also considered.

Sharp estimates for Hardy operators on Heisenberg group

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on R, balls in Rn, or ‘ellipsoids＇ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type （1, 1） inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESàRO TYPE

This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

CBMO Estimates for Multilinear Hardy Operators

The purpose of this paper is to study the multilinear Hardy operators in higher dimensional cases and establish the CBMO estimates for multilinear Hardy operators on some function spaces, such as the Lebesgue spaces, the Herz spaces and the Morrey-Herz spaces.

Commutators of Bilinear Hardy Operators on Two Weighted Herz Spaces with Variable Exponents

In this paper,we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of two weighted Herz spaces.

Sharp weak bounds for n-dimensional fractional Hardy operators

We obtain the operator norms of the n-dimensional fractional Hardy operator Hα（0 〈 α 〈 n） from weighted Lebesgue spaces Lp｜x｜p（R^n） to weighted weak Lebesgue spacesLq,∞｜x｜β（R^n）.

Function characterizations via commutators of Hardy operator

This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years.More precisely,the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.

Compactness of the Commutators of Fractional Hardy Operator with Rough Kernel

The more explicit decomposition of the operator and the kernel are utilized to investigate a characterization of the central BMO(R^(n))-closure of C^(∞)(R^(n))space via the compactness of the commutators of fractional Hardy operator with rough kernel.

Boundedness for Hardy Type Operators on Herz Spaces with Variable Exponents

In this paper, we will prove the boundedness of Hardy type operators Hβ（x） and Hβ^＊（x） of variable order β（x） on Herz spaces Kp（·）^α（·）q and Kp（·）^α（·）q′,where α（·） and p（·）are both variable.

Hausdorf 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.

(L^p, L^q)-Boundedness of Hausdorff Operators with Power Weight on Euclidean Spaces

In this paper, we prove the （L^p, L^q）-boundedness of （fractional） Hausdorff operators with power weight on Euclidean spaces. As special cases, we can obtain some well known results about Hardy operators.

一类多线性Hardy型算子在变指数Herz-Morrey空间的有界性（英文）

In this paper, we will obtain that the boundedness of multilinear n-dimensional fractional Hardy operators of variable order β(x) on variable exponent Herz-Morrey spaces.

Sharp Estimates of p-Adic Hardy and Hardy-Littlewood-Pólya Operators

In this paper we get the sharp estimates of the p-adic Hardy and Hard^Littlewood-Pdlya operators on L^q （｜x｜apdx）. Also, we prove that the commutators generated by the p-adic Hardy operators （Hardy-Littlewood-Pdlya operators） and the central BMO functions are bounded on L^q （｜x｜apdx）, more generally, on Herz spaces.

Multilinear Fractional Hausdorff Operators

In this paper,by introducing the space with weak mixed norms,weak type estimates of two kinds of multilinear fractional Hausdorff operators RΦ,β and SΦ,β on Lebesgue spaces are shown.By virtue of Marcinkiewicz interpolation,strong type estimates of these two operators on Lebesgue spaces are also obtained.Our methods shed some new light on dealing with the case of non-radial function Φ.