$R^n$空间中的Cauchy积分公式

In this note p(D) = Dm+ b1Dm 1+···+ bmis a polynomial Dirac operator in R~n, where D =nj=1ej xjis a standard Dirac operator in Rn, bjare the complex constant coefficients. In this note we discuss all decompositions of p(D) according to its coefficients bj,and obtain the corresponding explicit Cauchy integral formulae of f which are the solution of p(D)f = 0.     

$\mathbb{C}^n$中单位球上$\rho$次抛物星形映射的一些估计

主要研究了$\mathbb{C}^n$中单位球$B^{n}$上$\rho \ (\rho \in [0,1))$次抛物星形映射的一些几何性质, 给出了该映射类的增长定理和掩盖定理, 及其齐次展开式中二次项的一些估计.     

$\mathbb{C}^{n}$中$\mu$-Bergman空间的刻画和微分复合算子

设$p>0$, $\mu$和$\mu_{1}$是$[0,1)$上的正规函数. 本文首先给出了$\mathbb{C}^{n}$中单位球上$\mu$-Bergman空间$A^{p}(\mu)$的几种等价刻画; 然后 分别刻画了$A^{p}(\mu)$到$A^{p}(\mu_{1})$的 微分复合算子$D_{\varphi}$为有界算子以及紧算子的充要条件, 同时给出了当$p>1$时$D_{\varphi}$为 $A^{p}(\mu)$到$A^{p}(\mu_{1})$上紧算子的一种简捷充分条件和必要条件.     

Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are classified, up to automorphisms of balls.     

Harmonic Analysis Associated with the Heckman-Opdam-Jacobi Operator on $\mathbb{R}^{d+1}$

In this paper we consider the Heckman-Opdam-Jacobi operator $∆_{HJ}$ on $\mathbb{R}^{d+1}.$ We define the Heckman-Opdam-Jacobi intertwining operator $V_{HJ},$ which turns out to be the transmutation operator between $∆_{HJ}$ and the Laplacian $∆_{d+1}.$ Next we construct $^tV_{HJ}$ the dual of this intertwining operator. We exploit these operators to develop a new harmonic analysis corresponding to $∆_{HJ}.$     

On the n-th linear polarization constant of Rn$\mathbb {R}^n$

We prove that given any set of n unit vectors ${\{v_i\}}_{i=1}^n\subset {\mathbb{R}}^n$, the inequality $$\underset{{\parallel x\parallel }_{{\mathbb{R}}^n}=1}{sup}|⟨x,v_1⟩\cdots ⟨x,v_n⟩|\ge n^{-n/2}$$ holds for $n\le 14$. Moreover, the equality is attained if and only if ${\{v_i\}}_{i=1}^n$ is an orthonormal system.     

A note on the Witt group of ${mathbb P}^n$

Using the triangulated Witt theory of Balmer we give a new proof of Arasons theorem: . By the way we prove also that the Witt group of skew-symmetric forms on the projective space is zero. Received November 3, 1999; in final form February 8, 2000 / Published online March 12, 2001     

Rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3}

Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ .     

Stability of Viscoelastic Wave Equation with Structural $\delta$-Evolution in $\mathbb{R}^{n}$

The aim of this paper is to study the Cauchy problem for the viscoelastic wave equation for structural $\delta$-evolution models. By using the energy method in the Fourier spaces, we obtain the decay estimates of the solution to considered problem.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

On Holomorphic Automorphisms of a Class of Non-homogeneous Rigid Hypersurfaces in C^N＋1

The author determines the real-analytic infinitesimal CR automorphisms of a class of non-homogeneous rigid hypersurfaces in C^N＋1 near the origin, and the connected component containing the identity transformation of all locally holomorphic automorphisms of these hypersurfaces near the origin.     

Ball Convergence Theorems for Chebyshev-Halley Method in $\mathbb{B}$-Space

A local convergence analysis of Chebyshev-Halley method having third order of convergence for approximating zero of non-linear operator $f(v)=0$ by using convex majorant function and their condition in $\mathbb{B}$-space (Banach space), is presented in this article. We give the error estimate to show the efficiency of our study. Besides, we established the relation between majorant function and Kantorovich or Smale-type result as special cases of our general theory.     

$\mathbb{R}^n$ 上 $p(x)$-Laplace型椭圆问题的无穷多解

讨论了涉及一般散度型椭圆算子($p(x)$-Laplace算子为其特例) 非线性偏微分方程的弱解存在性和多解性问题, 假定非线性项 $f_1,\, f_2$ 其中之一是超线性的, 且满足 Ambrosetti-Rabinowitz 条件, 另一项是次线性的. 所采用的方法依赖于变指数 Sobolev 空间 $W^{1,p(x)}(\mathbb{R}^n)$ 理论. 主要结果的证明基于喷泉定理和对偶喷泉定理.     

Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.