齐型空间上的T(1)定理和交换子

本文在齐型空间上讨论弱核形式的　Ｔ（１）　定理，　并得到弱核形式的奇异积分算子与　ＢＭＯ　函数的交换子的Ｌｐ有界性及端点估计　（１＜ｐ＜∞　）．     

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderón–Zygmund operators and multilinear Calder′on–Zygmund operators with Dini type kernels.     

Classes of singular integral operators along variable lines

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main L^p estimates are proved by interpolating L² bounds with suitable bounds in Hardy spaces on product domains. The L² bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.     

Some Estimates for Convolution Operators with Kernels of Type （l, r） on Homogeneous Groups

Some estimates for the convolution operators with kernels of type （l, r） on Lebesgue spaces with power weights and Herz-type spaces in the setting of homogeneous groups are established.     

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

A Sum Operator with Applications to Self-Improving Properties of Poincaré Inequalities in Metric Spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.     

Interpolation,Projection and Hierarchical Bases in Discontinuous Galerkin Methods

The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities.     

齐型空间上的Herz空间及其应用

定义了一类齐型空间上的Herz空间及弱Herz空间,研究了定义在这些空间中的一类次线性算子的性质．     

Weyl numbers and eigenvalues of abstract summing operators

We estimate Weyl numbers and eigenvalues of operators via studying their abstract summing norms. In particular we prove estimates of these summing norms for abstract interpolation Lorentz spaces. For this we combine factorization theorems with estimates of concavity constants. Finally we apply our general eigenvalue results to integral operators with kernels of weakly singular type. We obtain asymptotically optimal estimates which extend the well-known classical results.     

Volterra Type and Weighted Composition Operators on Weighted Fock Spaces

Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan ${\mathbb{C}}$ . All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.     

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.     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

一类次线性算子在非齐型空间上的弱Herz空间中的弱型估计

作者引入了非齐型空间上的弱Herz空间,并建立了一类次线性算子在这些空间中的弱型估计. 作为应用, 证明了由Calder\'on-Zygmund算子和$\os$函数生成的交换子在弱Herz空间中的弱型估计,其中$r\ge1$. 并且Orlicz空间$\os$当$r=1$时即为$\rb$空间;当$r>1$时为$\rb$的子空间.     

Calderón-Zygmund型算子及其交换子的

该文建立了Calderón-Zygmund型算子及其交换子的sharp极大函数估计. 作为应用, 可以得到这些算子在Lebesgue空间和Morrey型空间上的有界性.     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.     

Extrapolation results in grand Lebesgue spaces defined on product sets

Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces.     

多线性强奇异Calderón-Zygmund算子的多线性迭代交换子的Sharp极大和加权估计

本文主要建立了由多线性强奇异Calderón-Zygmund算子和BMO函数生成的多线性迭代交换子的Sharp极大估计.作为应用,也分别得到了该类多线性迭代交换子在乘积加权Lebesgue空间和乘积变指数Lebesgue空间上的有界性.     

单边算子有界性和KdV型色散方程的适定性研究

数学和物理中许多重要问题均可归结为算子在某些函数空间中的有界性质.奇异积分算子有界性质的研究是调和分析理论的核心课题之一,由此发展起来的各种方法和技巧已广泛应用于偏微分方程的研究.借助奇异积分算子在Lebesgue空间或Morrey型空间中建立的时空估计和半群理论,可以得到非线性色散方程在低阶Sobolev空间中Cauchy问题的适定性.本文首次定义一类单边振荡奇异积分算子并研究该类算子的经典加权有界性质.受经典交换子刻画理论的启发,本文首次引入Morrey空间的交换子刻画理论.利用不同于常规极大函数的方法得到两类象征函数在Morrey空间中的交换子刻画.以上结果为偏微分方程的研究提供了新的工具.最后,结合能量方法和数论知识,本文解决几类KdV型色散方程的适定性问题.     

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces.     

Two-weight Inequalities for Commutators of Potential Operators on Spaces of Homogeneous Type

We derive the two-weight inequalities for the commutators of potential integral operators in the context of spaces of homogeneous type. The conditions required of the weights involve the Orlicz bumps. As a consequence, we obtain the norm estimates for these commutators for arbitrary weights. Supported by the National Natural Science Foundation of China (Nos. 10771049, 60773174) and the Natural Science Foundation of Hebei Province (08M001).