Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces

We consider the singular integral operator T with kernel K(x)=Ω(x)/n|x| and prove its boundedness on the Triebel-Lizorkin spaces provided that Ω satisfies a size condition which contains the case Ω∈Lr(Sn−1), r>1.     

Some properties of the reduced modulus in space

We obtain upper bounds for the conformal modulus of a condenser with uniformly perfect plates and for the reduced modulus of a uniformly perfect set E at . For the reduced moduli of α-uniformly perfect sets we prove the continuity property with respect to the kernel convergence of the complements to these sets in the sense of Carathéodory.     

Boundedness of Commutators of Marcinkiewicz Integral with Rough Variable Kernel

In this paper the authors give the boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, which is an extension of the result in [5] . Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Non-Gaussian upper estimates for heat kernels on spaces of homogeneous type

The authors extend non-Gaussian upper estimates on the positive real axis to a certain sector of including the positive real axis for heat kernels on spaces of homogeneous type, which are known to be holomorphic in that sector.

     

Weyl Transforms,the Heat Kernel and Green Function of a Degenerate Elliptic Operator

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ² related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L²(ℝ²). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e^−tL, t > 0, are given. Communicated by B.-W. Schulze (Potsdam) Mathematics Subject Classifications (2000): 47G30, 47E05.     

Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator TΩ,αf(x)=p.v.∫R^nb(|y|Ω(y′)|y|^-n-αf(x-y)dy, defined on all test functions f, where b is a bounded function, α＞0, Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions. It is proved that, for n/(n α)＜p＜∞,TΩ,α is a bounded operator from the Hardy-Sobolev space H^pα to the Hardy space H^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden‘s paper(1969). The proof is based on a new atomic decomposition of the space H^pα by Han, Paluszynski and Weiss(1995). By using the same proof,the singluar integral operators with variable kernels are also studied.     

The Koppelman-Leray formula on complex Finsler manifolds

By means of the invariant integral kernel (the Berndtsson kernel), the complex Finsler metric and the non-linear connection associated with the Chern-Finsler connection to research into the integral representation theory on complex Finsler manifolds, the Koppelman and Koppelman-Leray formulas are obtained, and the (?)--equations are solved.     

Invariant subspaces for a class of complete Pick kernels

Motivated by the work of McCullough and Trent, we investigate the -invariant subspaces of the Hilbert function spaces associated to the Szego kernels on the open unit disk. In particular, we characterize those kernels for which the the -invariant subspaces are hyperinvariant, and (partially) those for which the so-called BLH subspaces are cyclic, obtaining counterexamples to two questions posed by McCullough and Trent.

     

The Remainder Term for Analytic Functions of Symmetric Gaussian Quadratures

For analytic functions the remainder term of Gaussian quadrature rules can be expressed as a contour integral with kernel . In this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially Gegenbauer weight functions. First a new series representation of the kernel is developed and analyzed. Then the location of the maximum modulus of the kernel on suitable ellipses is determined. Depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis. Finally, a detailed discussion for some special weight functions is given.

     

椭圆型界面问题的破裂再生核方法

本文基于一维椭圆型界面问题提出了一种有效的数值方法.首先,根据模型构建一个崭新的破裂再生核空间.其次,应用破裂再生核方法给出了此类界面问题的近似解,并讨论该方法的收敛性.最后,通过几个有效的数值算例来说明该方法的精确性和稳定性.