广义Greiner算子的几类Hardy型不等式

对构成广义Greiner算子的向量场$X_j = \frac{\partial }{\partial x_j} + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial }{\partialt}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j \vert z\vert^{2k - 2}\frac{\partial }{\partial t}$, j = 1,... ,n, x,y∈ Rⁿ, $z = x + \sqrt { - 1} \,y$, t ∈ R, k ≥1, 得到了拟球域内和拟球域外的Hardy型不等式;建立了广义Picone型恒等式,并由此导出比文献[3]更一般的全空间上的Hardy型不等式;并在$p = 2$时建立了具最佳常数的Hardy型不等式.     

The Generalized Riemann^Haseman Problem

In this paper we study the generalized Riemann—Haseman problem which was given by Vekua, I. N. Problem (R-H). Find a sectionally generalized holomorphic function w(z) = {w^+(z), w^-(z)} such that $\frac {\partial w}{\partial \bar z}+B(z)\bar w=0,z\in E$ Here $B(z)\in C_\alpha ^n-1(D^++L),B(z)\in C_\alpha ^n-1(D^- +L),L\in C_\alpha ^n-1,0<\alpha \leq 1,|B(z)|\leq \frac {K}{|z|^1+s}(z\rightarrow \infty),K>0,\varepsilon >0;w(z)$ Satisfies the boundary condition $\sum\limits_{k=0}^n {a_k(t)\frac {\partial ^k w^+}{\partial t^k}|b_k \frac{\bar \partial ^kw^+}{\partial t^k}}_{t=\alpha(z)}-\sum\limits_{k=0}^n{c_k(t)\frac{\partial ^k w^-}{\partial t^k}+d_k(t)\frac{\bar \partial ^kw^-}{\partial t^k}}=f(t),t\in L$, Where a_k(t)、 b_k (t) 、 c_k (t) 、d_k(t)、f(t)\in H;\alpha(t) is a mapping of L into itself, $\alpha'(t)\ne 0$and \alpha [\alpha(t)] \equal t. We study the conditions of the solubility and the number of linearly independent solutions.     

On the Growth Properties of Solutions for a Generalized Bi-Axially Symmetric Schr\"{o}dinger Equation

In this paper, we have considered the generalized bi-axially symmetric Schr\"{o}dinger equation $$\frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2} + \frac{2\nu} {x}\frac{\partial \varphi} {\partial x} + \frac{2\mu} {y}\frac{\partial \varphi} {\partial y} + \{K^2-V(r)\} \varphi=0,$$ where $\mu,\nu\ge 0$, and $rV(r)$ is an entire function of $r=+(x^2+y^2)^{1/2}$ corresponding to a scattering potential $V(r)$. Growth parameters of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

On a Class of Mixed Partial Differential Equations of higher Order

就作者所知，高阶（阶数超过2）的混合型偏微分方程还是一个未曾讨论过的领域。本文的目的在于讨论一类高阶的混合型方程。 设P(\frac{\partial}{\partial t},\frac{\partial}{\partial y_1},\cdots,\frac{\partial}{\partial y_n})是齐m阶的实常系数的偏微分算子关于t=0是双曲的。定义R_n上的微分算子L(x,\frac{\partial}{\partial x},a)使得 $P(\frac{\partial}{\partial t},\frac{\partial}{\partial y})(t^\a+1u(\frac{y_1}{t},\cdots,\frac{y_n}{t}))=t^a+1-m[L(x,\partial/\partial x,a)u(x_1,\cdots,x_n)]_{x_i\y_i/t}$ 这样定义起来的算子L(x,\partial/=partial x,a)是依赖于一个参数a的m阶混合型算子。在一个有界闭区域之外，L为双曲型的。记 \bar \Omega为一有界闭区域，其边界\partial \Omega为充分光滑，落在双区域之中，又L关于\partial \Omega 为双曲的。 我们考虑了两类的边值问题，它们的提法和参数\alpha的数值有关。主要结果是： 我们考虑了区域\Omega上算子L(x,\partial/\partial x,a)的两类边值问题，证明了这两类边值问题的适应性，且得到了古典解，同时也讨论了C^\infty解得存在性和唯一性。 本文是[7,14]在高阶混合型方程情形时的推广。     

A Note on the Approximate Solution of the Cauchy Problem by Number-Theoretic Nets

рябенький, в. с.曾提出用数论网络构造的常微分方程组的解来构造偏微分方程 $\frac{\partial u}{\partial t}=Q(\frac{\partial}{\partial x_1},\cdots,\frac{\partial}{\partial x_s})u,0 \leq t \leq T,-\infty相似文献   

某类$q$差分方程亚纯解的性质

对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.     

ON DIRICHLET PROBLEMS FOR SECOND ORDER QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

The purpose of this paper is to study the existence of the classical solutions of some Dirichlet problems for quasilinear elliptic equations $$\[{a_{11}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2{a_{12}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {a_{22}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {y^2}}} + f(x,y,u,\frac{{\partial u}}{{\partial x}},\frac{{\partial u}}{{\partial y}}) = 0\]$$ Where $\[{a_{ij}}(x,y,u)(i,j = 1,2)\]$ satisfy $$\[\lambda (x,y,u){\left| \xi \right|^2} \le \sum\limits_{i,j = 1}^2 {{a_{ij}}(x,y,u)} {\xi _i}{\xi _j} \le \Lambda (x,y,u){\left| \xi \right|^2}\]$$ for all $\[\xi \in {R^2}\]$ and $\[(x,y,u) \in \bar \Omega \times [0, + \infty ),i.e.\lambda (x,y,u),\Lambda (x,y,u)\]$ denote the minimum and maximum eigenvalues of the matrix $\[[{a_{ij}}(x,y,u)]\]$ respectively, moreover $$\[\lambda (x,y,0) = 0,\Lambda (x,u,0) = 0;\Lambda (x,y,u) \ge \lambda (x,y,u) > 0,(u > 0).\]$$ Some existence theorems under tire “ natural conditions imposed on $\[f(x,y,u,p,q)\]$ are obtained.     

含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题其中Ω是RN中具有C1边界的有界区域,0∈■Ω,N≥5.2*(s)=2(N-s)/N-2(0≤s≤2)是临界Sobolev-Hardy指标, 10.利用变分方法和对偶喷泉定理,证明了这个方程无穷多解的存在性.     

ON THE CHARACTERISTIC FUNCTION OF SEMI-HYPONORMAL OPERATOR

本文继[3]之后，研究拟亚正常算子和半亚正常算子的特征函数.设\[A = U|A{|_r}\]是\[H{\kern 1pt} {\kern 1pt} \] 上拟亚正常算子，\[U\]是酉算子，\[B = |A{|_ + } - |A{|_ - }\],作算子\[A\]的特征函数\[W(\lambda ,A) = I - {B^{\frac{1}{2}}}{(\lambda I - {A_ - })^{ - 1}}U{B^{\frac{1}{2}}}\] 定理1 设\[A = U|A{|_r}\]及\[{A^'} = {U^'}|{A^'}{|_r}\]为\[\varphi - \]拟亚正常算子而且都是简单的.又设 \[U\]与\[{U^'}|\]是酉算子.如果有酉算\[T\]将\[H\]映照成\[{H^'}\]而且\[|{A^'}{|_ \pm } = T|A{|_ \pm }{T^{ - 1}}\]，\[{U^'} = TU{T^{ - 1}}\]那末必有\[{\cal B}(A)\]到\[{\cal B}({A^'}){\kern 1pt} \]上的酉算子\[S{\kern 1pt} {\kern 1pt} \]使当\[\lambda \notin \sigma ({A_ - }) = \sigma (A_ - ^')\]时\[W(\lambda ,{A^'}) = SW(\lambda ,A){S^{ - 1}}\]反之亦真. 下面设\[A\]是半亚正常的.又设\[{\cal D}\]为一辅助的希尔伯特空间，\[K\]为\[{\cal D}\]到\[{\kern 1pt} H\]中的线 性算子使\[Q = |A{|_{\rm{r}}} - |A{|_l} = K{K^*}{\kern 1pt} {\kern 1pt} \]，当\[\lambda \in \rho (A)\]，\[|Z| \ne 1\]时作 \[Y(z,\lambda ) = I - {\kern 1pt} {\kern 1pt} z{K^*}{(I - z{U^*})^{ - 1}}{(A - \lambda I)^{ - 1}}K\] 定理2设\[A = U|A{|_r}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]及\[{A^'} = {U^'}|{A^'}{|_r}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]分别是\[H\]与\[{H^'}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]中的半亚正常算子，\[U\]与 \[{U^'}\]是酉算子而且\[A\]与\[{A^'}\]都是简单的.如果存在\[{\cal D} \to {{\cal D}^'}{\kern 1pt} \]上的酉算子\[S\]使 \[{Y^'}(z,\lambda ) = SY(z,\lambda ){S^{ - 1}}\] 那末必有由\[H\]到\[{H^'}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]上的酉算子\[T\]使(1)成立，反之亦真. 定理3 若\[K\]是希尔伯特-许密特算子则\[Y(z,\lambda )\]的行列式（当\[|Z| \ne 1\]时）存在， 且\[\det (Y(z,\lambda )) = \det ((I - z{U^*})(A - \lambda I){(I - z{U^*})^{ - 1}}{(A - \lambda I)^{ - 1}})\] 下面只考虑奇型积分模型这时\[W(\lambda ,A)\]成为乘法算子，\[(W(\lambda ,A)f)({e^{i\theta }}) = W({e^{i\theta }},\lambda )f({e^{i\theta }})\]其中\[W({e^{i\theta }},\lambda ) = I - \alpha ({e^{i\theta }}){(\lambda {e^{i\theta }}I - \beta ({e^{i\theta }}))^{ - 1}}\alpha ({e^{i\theta }})\] 我们又假设\[A\]是完全非正常的.记\[{Y_ \pm }({e^{i\theta }},\lambda )a = \mathop {\lim }\limits_{r \to 1 \pm 0} Y({e^{i\theta }},\lambda )a\] 定理4设\[\lambda \in \rho (A){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]，\[a \in {\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]为固定的，那末\[{Y_ \pm }({e^{i\theta }},\lambda )a\]为黎曼-希尔伯特问题 \[{Y_ - }({e^{i\theta }},\lambda )a = W({e^{i\theta }},\lambda ){Y_ + }({e^{i\theta }},\lambda )a\] 的解. 设\[{\cal L}({\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} {\kern 1pt} \]为\[{\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]上线性有界算子全体所成的Banach空间，\[H_ \pm ^p({\cal L}{\kern 1pt} ({\cal D}{\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} \]为单位圆 外，内取值于\[{\cal L}({\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} ){\kern 1pt} \]的某些解析函数所成的Hardy空间.设\[f({e^{i\theta }})\]是单位圆周上的函 数，如果有\[{u_ \pm } \in H_ \pm ^p({\cal L}{\kern 1pt} ({\cal D}{\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} (p > 2)\]使\[u_ - ^{ - 1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]存在\[{u_ - }{\kern 1pt} {\kern 1pt} {\kern 1pt} {({e^{i\theta }})^{ - 1}}{u_ + }{\kern 1pt} ({e^{i\theta }}) = f({e^{i\theta }})\]则称\[f\]是可分解的.     

$\mathbb{C}^n$中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

在有界星形圆形域上定义了一个新的星形映射子族, 它包含了$\alpha$阶星形映射族和$\alpha$阶强星形映射族作为两个特殊子类. 给出了此类星形映射子族的增长定理和掩盖定理. 另外, 还证明了Reinhardt域$\Omega_{n,p_{2},\cdots,p_{n}}$上此星形映射子族在Roper-Suffridge算子 \begin{align*} F(z)=\Big(f(z_{1}),\Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{2}}(f'(z_{1}))^{\gamma_{2}}z_{2},\cdots, \Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{n}}(f'(z_{1}))^{\gamma_{n}}z_{n}\Big)' \end{align*} 作用下保持不变, 其中 $\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in {\mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$, $p_{j}\geq1$, $\beta_{j}\in$ $[0, 1]$, $\gamma_{j}\in[0, \frac{1}{p_{j}}]$满足$\beta_{j}+\gamma_{j}\leq1$, 所取的单值解析分支使得 $\big({\frac{f(z_{1})}{z_{1}}}\big)^{\beta_{j}}\big|_{z_{1}=0}=1$, $(f'(z_{1}))^{\gamma_{j}}\mid_{{z_{1}=0}}=1$, $j=2,\cdots,n$. 这些结果不仅包含了许多已有的结果, 而且得到了新的结论.     

一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献   

一类带Hardy项的椭圆方程的无穷多解

假设 $\Omega=B_R:=\{x\in \mathbb{R}^N:|x|0$, $ N \geq 7$, $ 2^*=\frac{2N}{N-2}$, 我们得到了如下半线性问题无穷多解的存在性: $\left\{ \begin{array}{ll} -\Delta u=\frac{\mu}{|x|^2}u+|u|^{2^*-2}u+\la u, &; x\in\Omega, \\ u=0, &; x\in \partial\Omega. \end{array} \right.$ 其中$\lambda \in \mathbb{R}, \mu \in \mathbb{R}$. 这些解由不同的节点来区分.     

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

由$q$-积分算子定义的某些亚纯函数类的Fekete-Szeg\"{o}问题

本文首先引入满足如下条件$$-\frac{qzD_{q}f(z)}{f(z)}\prec \varphi (z)$$和$$\frac{-(1-\frac{\alpha }{q})qzD_{q}f(z)+\alpha qzD_{q}[zD_{q}f(z)]}{(1-\frac{\alpha}{q})f(z)-\alpha zD_{q}f(z)}\prec \varphi (z)~(\alpha \in\mathbb{C}\backslash (0,1],\ 0相似文献   

ON PRIME IDEALS OF THE RING OF DIFFERENTIAL OPERATORS

For given linear differential operators \[{P_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{P_n}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\], let \[({P_1},...,{P_m})\] be the left ideal generated by \[{P_1},...,{P_m}\], and F be the space of the common solutions of the operators \[{P_i}(i = 1,...,m),i,e.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F = \{ f|{P_i}f = 0,i = 1,...,m\} \].Inspired by the Hilbert's Nullstailerisatz, we introduce another ideal \[H({P_1},...,{P_m})\] related to \[{P_1},...,{P_m}\], \[H({P_1},...,{P_m}) = \{ P|Pf = 0,\forall f \in F\} \], Obviously, \[({P_1},...,{P_m}) \subseteq H({P_1},...,{P_m})\]. Definition The ideal \[({P_1},...,{P_m})\] is prime if \[H({P_1},...,{P_m}) = ({P_1},...,{P_m})\]. The aim of this paper is to stndy under what conditions the ideal \[({P_1},...,{P_m})\] is prime. We have the following results： Theorem 1 \[(\Delta )\]is prime, where \[\Delta \] is the Laplace operator of \[{{\cal R}^n},i,e\], \[\Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \] Theorem 2 \[{\Delta _S}\] is prime, where \[\Delta \] is the Laplace-Beltrami operator of \[B = \{ ({Z_1},...,{Z_n}) \in {{\cal L}^n}|\sum\limits_{i = 1}^n {|{z_i}} {|^2} < 1\} \] it is well hmwn that \[{\Delta _S} = \sum\limits_{i,j = 1}^n {({\delta _{ij}} - {z_i}{{\bar z}_j})\frac{{{\partial ^2}}}{{\partial {z_i}\partial {{\bar z}_j}}}} \] Theorem 3 Suppose \[{T_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{T_m}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\] are differential operators with constant coefficients, then \[({T_1},...,{T_m})\] is prime if the corresponding polynomial ideal \[({T_1}({X_1},...,{X_n}),...,{T_m}({X_1},...,{X_n}))\] is prime in the usual sense. For general linear differential operators with variable coeffcients, we have given some sufficient conditions for which \[({P_1},...,{P_m})\] is prime (see Th. 4). The two important ones among these sufficient conditions are that \[{P_1},...,{P_m}\] possess a Poisson kernel and they satisfy group invariance in some sense. Finally, as an example, we study in detail the case of the classical domain \[R(2) = \{ Z = \left( {\begin{array}{*{20}{c}} {{z_1}}&{{z_2}}\{{z_3}}&{{z_4}} \end{array}} \right)|I - Z{Z^'} > 0\} \] We show that \[H({\Delta _{R(2)}}),i,e,({\Delta _{R(2)}})\] is not prime, and we also give the basis of the ideal \[H({\Delta _{R(2)}})\], where \[H({\Delta _{R(2)}})\] is the Laplaoe-Beltrami operator of R(2).     

On an Extension of Hadamard Inequalities for Convex Functions(in English)

设f是区间[a,b]上连续的凸函数，我们证明了Hadamard的不等式 $[f(\frac{{a + b}}{2}) \le \frac{1}{{b - a}}\int_a^b {f(x)dx \le \frac{{f(a) + f(b)}}{2}}$ 可以拓广成对[a,b]中任意n+1个点x_0,\cdots,x_n和正数组p_0,\cdots,p_n都成立的下列不等式 $f(\frac{\sum\limits_{i=0}^n p_ix_i}{\sum\limits_{i=0}^n p_i}) \leq |\Omega|^-1 \int_\Omega f(x(t))dt \leq \frac{\sum\limits _{i=0}^n {p_if(x_i)}}{\sum\limits_{i=0}^n p_i}$ 式中\Omega是一个包含于n维单位立方体的n维长方体，其重心的第i个坐标为$\sum\limits _{j=i}^n p_j /\sum\limits_{j=i-1}^n p_i$,|\Omega|为\Omega的体积，对\Omega中的任意点$t=(t_1,\cdots,t_n)$, $w(t)=x_0(1-t_1)+\sum\limits _{i=1}^{n-1} x_i(1-t_{i+1})\prod\limits_{j = 1}^i {{t_j}} +x_n \prod\limits _{j=1}^n t_j$ 不等式中两个等号分别成立的情形亦已被分离出来。 此不等式是著名的Jensen 不等式的精密化。     

算子代数的性质∏σ和张量积

本文引入算子代数的性质${\Pi}_\sigma$这一概念,证明了任一 vonNeumann代数中的套子代数和有限宽度CSL子代数都具有性质$\Pi_\sigma.$最后得到张量积公式$\mbox{alg}_{\cal M}{\cal L}_1\overline{\otimes}\mbox{alg}_{\cal N}{\cal L}_2= \mbox{alg}_{{\cal M}\overline{\otimes}{\cal N}}({\cal L}_1\otimes{\cal L}_2)$成立,这里${\cal L}_1$和 ${\cal L}_2$分别是von Neumann代数${\cal M}$和${\cal N}$中的有限宽度CSL.     

有界完全Reinhardt域上推广的Roper-Suffridge算子

在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在C~n中的有界完全Reinhardt域Ω上推广的Roper-Suffridge算子Φ(f)定义为Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)(z)=(rf(z_1/r),((rf(z_1/r))/z_1)~(β_2)(f′(z_1/r))~γ_2_(z_2,…,)((rf(z_1/r))/z_1)~(β_n)(f′(z_1/r))~(γ_n)_(z_n),其中n≥2,(z_1,z_2,…,z_n)∈Ω,r=r(Ω)=sup{|z_1|:(z_1,z_2,…,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1,这里选取幂函数的单值解析分支,使得((f(z_1))/z_1)~(β_j)|_(z_1=0)=1和(f′(z_1))~(γ_j)|_(z_1=0)=1,j= 2,…,n.证明了Ω上的算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)是将S_α~*(U)的子集映入S_α~*(Ω)(0≤α<1),且对于一些合适的常数β_j,γ_j,p_j,D_p上的这个算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)保持α阶星形性或保持β型螺形性,其中(?) U是复平面C上的单位圆,S_α~*(Ω)是Ω上所有正规化α阶星形映射所成的类.也得到:对于某些合适的常数β_j,γ_j,p_j和0≤α<1,Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)∈S_α~*(D_p)当且仅当f∈S_α~*(U).     

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献