M(o)bius不变的Qp空间:结果、技巧和问题

本文是关于Mobius不变的Qp函数空间方面的综述文章，给出了近10年来有关Qp函数空间的若干结果、技巧和问题。     

ON THE EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR SINGULAR p-LAPLACE EQUATIONS

In this paper N-dimensional singular, p-Laplace equations of the following form are considered, where p ≥N, β > 0, and f : [0,∞)×(0,∞)×[0, ∞)→[0,∞) is a continuous function. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of |x|(p-N)/(p-1) for p > N or log|x| for N = p as |x|→∞.     

The kernel theorem for Cauchy ultradistribution

Kernel theorems for spaces of Cauchy ultradistribution, supported by an n-dimensional tube and cone of the product type, are investigated.     

On harmonic analysis of causal operators

We follow the group representation theory approach to define causal operators on Banach modules and present some of their spectral properties.     

白噪声分析中的梯度算子和散度算子

在白噪声分析的框架中，我们给出了广义Weiner泛函空间上的梯度算子和散度算子的定义与公式，并利用梯度和散度算子以及适应投影建立了广义泛函的表示公式．也证明了积分核算子可用梯度与散度算子表出．     

Smooth and path connected Banach submanifold Σ r of B(E,F) and a dimension formula in B(ℝ n ,ℝ m )

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ _r the set of all operators of finite rank r in B(E,F), and Σ _r ^# the number of path connected components of Σ _r . It is known that Σ _r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ _r . In this paper,the equality Σ _r ^# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ _r is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T _A Σ _r = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ _r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of Σ _r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ ⁿ and F = ℝ ^m , then Σ _r is a smooth and path connected submanifold of B(ℝ ⁿ , ℝ ^m ) and its dimension is dimΣ _r = (m+n)r−r ² for each r, 0 <- r < min {n,m}. Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).     

LEGENDRE SERIES SOLUTIONS FOR TIME-VARIATION DYNAMICS

In this topic, a new approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solution of time-variation dynamics is therefore established, which is beneficial to further research of time-variation science. Supported by Shanghai Key Project Research Foundation.     

加权Bergman空间Aα^2（Ω）上的Schatten-p类加权复合算子

刻画加权Bergman空间Aα^2（Ω）上的加权复合算子Cφ,Ф的Schatten-p类．     

不定Sturm—Liouville算子的特征函数的振荡问题

研究了定义在有限区间（0，l）上的具有一般分离型边条件的不定Sturm—Liouville算子的特征函数的振荡问题．利用Prufer变换，给出了上述Sturm-Liouville算子特征值的符号指标的具体形式；得到了特征值的符号指标与Weyl函数以及Prufer角在该特征值处的罗朗展式（泰勒展式）的首项系数的符号之间的关系；最后，在上述两个结果的基础上给出了上述Sturm—Liouville算子的第n个正特征值所对应的特征函数在[0，l]内的零点个数的计算公式．     

强伪压缩算子带误差的Ishikawa迭代的强稳定性

设X是任意实Banach空间，T：X→X是一Lipschitz强伪压缩算子，本给出T带误差的Ishikawa迭代过程的强稳定性，并给出一个涉及Lipschitz强增生算子T的非线性方程Tx=f迭代解的强稳定性。