THE UNIFORMLY BOUNDEDNESS OF THE RIESZ TRANSFORMS ON THE CAYLEY HEISENBERG GROUPS

In this article,the authors estimate some functions by using the explicit expression of the heat kernels for the Cayley Heisenberg groups,and then prove the uniform boundedness of the Riesz transforms on these nilpotent Lie groups.

A NOTE ON H-w^p-BOUNDEDNESS OF RIESZ TRANSFORMS AND θ-CALDERóN-ZYGMUND OPERATORS THROUGH MOLECULAR CHARACTERIZATION

Let 0 〈 p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and yang[11] established that the Riesz transforms R j, j = 1,2,..., n, are bounded on Hwp （Rn）. In this note we extend this to the general case of weight w in the Muckenhoupt class A.. through molecular characterization. One difficulty, which has not been taken care in [11] consists in passing from atoms to all functions in HwP（Rn）. Furthermore, the HwP-boundedness of θ- Calderon-Zygmund operators are also given through molecular characterization and atomic decomposition.

OSCILLATION AND VARIATION OF THE LAGUERRE HEAT AND POISSON SEMIGROUPS AND RIESZ TRANSFORMS

In this article, we study LP-boundedness properties of the oscillation and vari- ation operators for the heat and Poissson semigroup and Riesz transforms in the Laguerre settings. Also, we characterize Hardy spaces associated to Laguerre operators by using the variation operator of the heat semigroup.

Riesz Transforms Associated with Schrdinger Operators Acting on Weighted Hardy Spaces

Let L = -△＋V be a Schrodinger operator acting on L^2（R^n）, n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp （w） associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform △↓L^-1/2 associated with L is bounded from our new space HP（w） to the classical weighted Hardy sp ace HP （ w ） when n / （n ＋ 1 ） 〈 p 〈 1 and w ∈ A 1 ∩ R H（ 2 / p ）＇.

Boundedness of Commutators Generated by Campanato-type Functions and Riesz Transforms Associated with Schrdinger Operators

Abstract： Let L=-△＋V be a SchrSdinger operator on Rn, n 〉 3, where △ is the Laplacian on Rn and V ≠ 0 is a nonnegative function satisfying the reverse HSlder＇s inequality. Let [b, T] be the commutator generated by the Campanatotype function b ∈∧Lβ and the Riesz transform associated with SchrSdinger operator T =△↓（-△ ＋ V）-1/2. In the paper, we establish the boundedness of [b, T] on Lebesgue spaces and Campanato-type spaces.

Hardy Type Estimates for Riesz Transforms Associated with Schrdinger Operators on the Heisenberg Group

Abstract. Let H^n be the Heisenberg group and Q = 2n＋2 be its homogeneous dimen- sion. In this paper, we consider the Schr6dinger operator -△H^n ＋V, where △H^n is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H61der class Bql for ql _〉 Q/2. We show that the operators T1 = V（-△H^n-In ＋V）-1 and T2 = V1/2（-△H^n-V）-1/2 are both bounded from 1 n HL^1（H^n ） into L1（H^n）. Our results are also valid on the stratified Lie group.

Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Let L =-?+V be a Schr?dinger operator on R^n(n ≥ 3), where the non-negative potential V belongs to reverse H?lder class RH_(q1) for q_1>n/2. Let H_L^p(R^n)be the Hardy space associated with L. In this paper, we consider the commutator[b,T_α], which associated with the Riesz transform T_α= V~α(-?+V)^(-α) with 0 < α ≤ 1,and a locally integrable function b belongs to the new Campanato space Λ_β~θ(ρ). We establish the boundedness of [b,T_α] from L^p(R^n) to L^q(R^n) for 1 < p < q_1/α with 1/q = 1/p-β/n. We also show that [b,T_α] is bounded from H_L^p(R^n) to L^q(R^n) when n/(n+ β) < p ≤ 1,1/q = 1/p-β/n. Moreover, we prove that [b,T_α] maps H_L^(n/n+β)(~Rn)continuously into weak L^1(R^n).

Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

The BMO_L Space and Riesz Transforms Associated with Schrdinger Operators

Let L =△ ＋ V be a SchrSdinger operator in Rd, d ≥ 3, where the nonnegative potential V belongs to the reverse HSlder class Sd. We establish the BMOL-boundedness of Riesz transforms З/ЗxiL-1/2, and give the Fefferman-Stein type decomposition of BMOL functions.

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrodinger Type Operators

Let L2=(-△)^2+ V^2 be the Schrödinger type operator, where V≠0 is a nonnegative potential and belongs to the reverse Holder class RHq1 for q1> n/2, n ≥5. The higher Riesz transform associated with L2 is denoted by R=△^2L2^(-1/2)and its dual is denoted by R^*=L2^(-1/2)△^2. In this paper, we consider the m-order commutators [b^m, R] and [bm, R^*], and establish the(L^p, L^q)-boundedness of these commutators when b belongs to the new Campanato space Λβ^θ(ρ) and 1/q = 1/p-mβ/n.

Sobolev Inequalities, Riesz Transforms, and the Ricci Flow

A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented.Various techniques are employed,including Bessel potentials and Riesz transforms.Combining these results with theW1,2 Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmoρ(ω) and establish the duality between h_ρ~1(ω) and bmo _ρ(ω),where ω∈A_1^(ρ, ∞)(R^n) and ω's locally behave as Muckenhoupt's weights but actually include them. We also give the Fefferman-Stein type decomposition of bmoρ(ω) with respect to Riesz transforms associated to Schrdinger operator L, where L =-? + V is a Schrdinger operator on R^n(n≥3) and V is a non-negative function satisfying the reverse Hlder inequality.

Inequalities for Higher Order Riesz-Laguerre Transforms

The weak-type (1, 1) boundedness of the higher order Riesz-Laguerre transforms associated with the Laguerre polynomials and the boundedness for the Riesz-Laguerre transforms of order 2 are considered. We discuss a polynomial weight w that makes the Riesz-Laguerre transforms of order greater than or equal to 2 continuous from L1 (wdμα) into L1,∞ (dμα), under specific value α, where μα is the Laguerre measure.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

基于Riesz变换的结构相似度图像质量评价方法

不同的图像处理过程,会对图像引入各种各样的失真,如何对图像的质量进行评价成为一个热点问题。针对传统的基于像素差值统计的峰值信噪比方法及结构相似度方法与人眼主观评价不够符合的情况,本文提出了一种基于Riesz变换的结构相似度图像质量评价方法。该方法先将参考图像和失真图像进行一阶Riesz变换和二阶Riesz变换,并利用得到的5组对应特征图计算出5幅相似度图和5幅权重图,利用平均法进行融合得到最终的相似度图和权重图,然后加入原参考图像和失真图像的亮度比较项,得到最终的图像质量评价指标。在LIVE图像数据库上的实验表明,本文方法对于5种失真的质量预测准确性和一致性都很高,在交叉失真实验中,本文方法也优于结构相似度方法,PLCC和SROCC值达到了0.9482和0.9532。与几种公认较好的方法相比,本文方法能够更好地预测图像质量,更加符合人眼的主观感知。

基于Riesz变换的图像边缘检测

针对传统边缘检测算法易受光线变化影响的问题,提出一种基于Riesz变换空间下的相位一致性边缘检测方法。用Riesz变换替代Hilbert变换对图像进行处理,通过Riesz变换的不同尺度因子构造计算相位一致性所需要的变换空间,根据相位一致性计算方法在新的Riesz变换空间获得特征图像;使用非极大值抑制检测出图像边缘信息;与传统相位一致特征提取方法、2DLog-Gabor变换以及Canny边缘提取算法对比实验。仿真实验结果说明,该边缘检测方法在不均匀关照条件下可用于图像的边缘特征提取,相比在Hilbert变换空间下的相位一致特征提取方法运算速度快,算法可行有效。

齐次Morrey-Herz空间上广义Riesz变换

主要研究与二阶散度型椭圆算子L相伴的Riesz变换▽L-1/2"及其与BMO(Rn)函数生成的交换子,采用对函数进行环形分解的技术和对算子转化为相应的截断算子的方法,得出它们从MKp1,qα,λ(Rn)到MKp2,qα,λ(Rn)是有界的,从而推广了以前学者的结论.

由Campanato型函数和与薛定谔算子相关的Riesz变换生成的Toeplitz算子的有界性（英文）

本文研究了由Campanato型函数及与Schrdinger算子相关的Riesz变换生成的Toeplitz算子的有界性.利用Sharp极大函数估计得到了Toeplitz算子θ~b在Lebesgue空间的有界性,拓广了已有交换子的结果.

Riesz空间分数阶对流扩散方程的一种计算有效求解方法

Riesz空间分数阶对流扩散方程是从混沌动力系统导出的.继续Ilic,Liu等的工作,我们提出在有界区域内求解Riesz空间分数阶对流-扩散方程的一种新的计算有效方法.即基于这两个Riesz空间分数阶导数的矩阵表示.这个方法的创新在于这个算子的标准离散得到包含具有相同分数次幂的矩阵的一个常微分方程组,并利用计算有效的分数阶行方法求解.同时借助于分数阶导数的谱表示和拉普拉斯变换,导出这个Riesz空间分数阶对流扩散方程的解析解.最后给出了数值例子来证实数值方法的有效性.

广义弱亚正规算子的Riesz幂等元

研究了广义弱亚正规算子T的Riesz幂等元Eλ和T的广义Aluthge变换Tt的Riesz幂等元Eλ的性质,其中λ∈isoσ(T)。证明了EλH=EλH,Eλ是自伴算子,和EλH=ker(T-λ)ker(T-λ)*。