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.

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.

Endpoint Estimates for Hardy Operator's Conjugate Operator with Power Weight on n-Dimensional Space

In this paper, we establish two integral inequalities for Hardy operator＇s conjugate operator at the endpoint on n-dimensional space. The operator Hn is bounded from Lxα1 （Gn） to Lxβq （Gn） with the bound explicitly worked out and the similar result holds for Hn＊.

Sharp Bound for the Generalized m-Linear n-Dimensional Hardy-Littlewood-Polya Operator

In this paper,we calculate the sharp bound for the generalized m-linear n-dimensional Hardy-Littlewood-Polya operator on power weighted central and non-central homogeneous Morrey spaces.As an application,the sharp bound for the Hardy-Littlewood-Polya operator on power weighted central and noncentral homo-geneous Morrey spaces is obtained.Finally,we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces,which generalizes the previous results.

Boundedness of the Multilinear Maximal Operator with the Hausdorff Content

In this paper,we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to theαdimensional Hausdorff content.Our results cover Orobitg and Verdera’s results in[8].

乘积函数的双Hilbert变换和Bedrosian等式

本文研究双Hilbert变换的性质,并对于f∈L^p(R^2), g∈L^q(R^2), p^(-1)+q^(-1)≤1,探讨在双Hilbert变换作用下Bedrosian等式H_2H_1(fg)=fH_2H_1g成立的充要条件,其中H_2H_1是双Hilbert变换.

Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

We will prove that for 1<p<∞and 0<λ<n,the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<γ<+∞.When p=1 and 0<λ<n,it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<λ<+∞.Moreover,the same results are true for the truncated uncentered Hardy-Littlewood maximal operator.Our work extends the previous results of Lebesgue spaces to Morrey spaces.

A restriction theorem for oscillatory integral operator with certain polynomial phase

We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.

Double Hilbert transform on D（R2）

We introduce a space DHH =D（R2） H2H1D（R2）, where D（R2） is the testing function space whose functions are infinitely differentiable and have bounded support, and H2H1D（R2） is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from DHH onto itself.

Sharp L^p decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

In this paper, we obtain the L^p decay of oscillatory integral operators T_λ with certain homogeneous polynomial phase functions of degree d in(n + n)-dimensions; we require that d > 2 n. If d/(d-n) < p < d/n,the decay is sharp and the decay rate is related to the Newton distance. For p = d/n or d/(d-n), we obtain the almost sharp decay, where "almost" means that the decay contains a log(λ) term. For otherwise, the L^p decay of T_λ is also obtained but not sharp. Finally, we provide a counterexample to show that d/(d-n) p d/n is not necessary to guarantee the sharp decay.

双线性分数次积分算子及其交换子在Morrey空间中的加权估计

本文考虑如下形式的双线性分数次积分算子:■,研究它的一般交换子在Morrey空间中的双权估计.本文证明了这些算子的一个极大函数控制定理,即当权属于A∞时,这些算子的加权Morrey范数可以被一种自然的极大算子的加权Morrey范数来控制.作为定理的推论,本文得到双线性形式的Olsen不等式、Fefferman-Stein型对偶不等式以及与双线性Hilbert变换相关的双线性极大函数的新的加权估计.作为重要的应用,本文建立双线性版本的Stein-Weiss分数次积分不等式.