Diagonalization modulo a class of Orlicz ideals

We consider a class of norm ideals CG which are non-commutative analogues of Orlicz spaces and which satisfy a previously introduced condition called (QK). We give a spectral condition which is necessary and sufficient for a commuting tuple of self-adjoint operators A=(A₁,…,An) to be simultaneously diagonalizable modulo CG.     

A condition for diagonalization modulo arbitrary norm ideals

We give a spectral condition which is sufficient for the simultaneous diagonalization of a commuting tuple of self-adjoint operators modulo a given norm ideal. For diagonalization modulo certain norm ideals this condition is also necessary, while for other norm ideals this condition seems to be close to being necessary. Moreover, this condition is easy to verify in applications.     

Normal operators and inequalities in norm ideals

In this work we characterize normal invertible operators via inequalities with unitarily invariant norm of elementary operators.     

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type

Weighted norm inequalities with general weights are established for the maximal singular integral operators on spaces of homogeneous type, when the kernel satisfies a Hörmander regularity condition on one variable and a Hölder regularity condition on the other variable.     

Weighted norm inequalities with multiple-weight for singular integral operators with non-smooth kernels

By sharp maximal function, we establish a weighted estimate with multiple-weight for the multilinear singular integral operators with non-smooth kernels.     

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.     

Compact inverses of first-order normal differential operators

In this work, firstly we describe all normal extensions of a minimal operator generated by linear differential-operator expression of first order in the Hilbert space of vector functions in finite interval. Later on, we investigate discreteness of spectrum and asymptotical behavior of s-numbers of the inverses of these normal extensions.     

On the norm of an integral operator and applications

In this paper, the norm of an integral operator (r>1) is obtained. As applications, a new bilinear integral operator inequality with the norm and the equivalent forms are given, and some new Hilbert's type inequalities with the best constant factors as the particular cases are established.     

Singular integral operators on Triebel-Lizorkin spaces of para-accretive type

In this article, we study the boundedness of singular integral operators acting on Triebel-Lizorkin spaces of para-accretive type and show a Tb theorem on these spaces, which extends previous results in David, Journé and Semmes (1985) [3], Han (1994) [9], Han, Lee and Lin (2004) [10], Han and Sawyer (1990) [12], Lin and Wang (2009) [15], Wang (1999) [22].     

一类带齐次核的奇异重积分算子的范数及其应用

设核函数K(u,v)具有对称性和齐次性,对如下定义的奇异重积分算子T:(Tf)(y)=∫R_+~n K(‖x‖α,‖y‖α)f(x)dx,y∈R_+~n,其中‖x‖α=(x_1~α+…+x_n~α)~1/α(α>0),研究了T的范数及其应用.     

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight (p,q) inequalities for commutators of potential type integral operators.     

带非卷积核的分数次振荡积分算子及其交换子的加权有界性质——献给陆善镇教授75华诞

振荡积分算子的有界性质是调和分析研究的中心内容之一.本文建立一类由Ricci和Stein定义的带非卷积核的分数次振荡积分算子在加权Lebesgue空间中的有界性质.特别地,结合复分析和数学归纳等方法得到该类算子和有界平均振幅（BMO）函数生成交换子的加权有界性质.     

A class of integral operators on mixed norm spaces in the unit ball

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball. The author is supported in part by the NNSF China (No. 10671115), grants from Specialized Research Fund for the doctoral program of Higher Education (No. 20060560002) and NSF of Guangdong Province (No. 06105468).     

Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380-391] and V? and Gorenflo [Kim Tuan V?, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174-185] it is known that the jth singular value of the fractional integral operator of order α>0 is approximately (πj)^−α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is (πj)^−α[1+O(j⁻¹)].     

Sharp norm estimates for a class of integral operators

ABSTRACT

Norm comparison inequalities for two integral operators with radial kernels are established. Sharp norm estimates for operators with monotone and convex/concave kernels are obtained. Integral analogues of Bennett's estimates for summability matrices are given. The exact operator norms with power weights are also obtained for a class of integral operators with radial quasimonotone kernels.     

Boundedness and compactness of Volterra type integral operators

We introduce some nested classes of Volterra type integral operators. For the operators of these classes we establish criteria for boundedness and compactness in Lebesgue spaces.     

Weighted norm inequalities for the maximal commutators of singular integral operators

The maximal operator associated with the commutator of Calderón-Zygmund operator is considered. It is shown that the maximal commutator enjoys some two-weight norm estimates which are similar to those of the commutator of Calderón-Zygmund operator.     

Rank one perturbations and singular integral operators

We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tε are uniformly (in ε) bounded operators from L²(μ) to L²(μα), where μ and μα are the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aα to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L² potentials are considered.     

Area integral functions for sectorial operators on L spaces

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^infin functional calculus of sectorial operators on L^p-spaces hold true when the square functions are replaced by the area integral functions.