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   

Initial Boundary Value Problem for One Class of System of Multi- Dimensional Inhomogeneous GBBM Equations

This paper studies the following initial-boundary value problem for the system of multidimensional inhomogeneous GBBM equations $[\begin{array}{l} {u_r} - \Delta {u_i} + \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}} grad\varphi (u) = f(u),{\rm{ (1}}{\rm{.1)}}\u{|_{t = 0}} = {u_0}(x),x \in \Omega ,{\rm{ (1}}{\rm{.2)}}\u{|_{\partial \Omega }} = 0,t \ge 0,{\rm{ (1}}{\rm{.3)}} \end{array}\]$ The existence and uniqueness of the global solution for the problem(l.l) (1.2) (1.3) are proved. The asymptotic behavior and “blow up” phenomenon of the solution for the problem (1.1) (1.2) (1.3) are investigated under certain conditions.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

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。     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

MEAN APPROXIMATION OF A PERIODIC FUNCTION BY THE PARTIAL SUMS OF ITS FOURIER SERIES

Given a sequence of positive real numbers \[{\varepsilon _0},{\varepsilon _1},...,{\varepsilon _n},...\] which satisfy the conditions \[{\varepsilon _v} \to 0,{\varepsilon _v} - {\varepsilon _{v + 1}} \ge 0,{\varepsilon _v} - 2{\varepsilon _{v + 1}} + {\varepsilon _{v + 2}} \ge 0\] for v =0, 1, 2, ..., and a class L(s) of 2pi-periodic, L-integrable functions f(x) such that \[{E_n}{(f)_L} \le {\varepsilon _n}(n = 0,1,2,...)\], where \[{E_n}{(f)_L}\] is the best mean approximation of f(x) by trigonometrical polynomials of degree ≤n Let \[{S_n}(f)\] be the n-th partial sum of the Fourier series of f(x). It’s known that Oskolkov has proved \[\mathop {\sup }\limits_{f \in L(\varepsilon )} ||f - {S_n}{(f)_L}|| = \sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} \] where \[||f|{|_L} = \int_0^{2\pi } {|f(x)|} dx\] Oskolkov asked whether there is a single function \[{f_0}(x) \in L(s)\] for which the above relation is satisfied for all n, In this paper the following result is obtained. Theorem Let \[L(\varepsilon )\] be a class of 2pi-periodic, L-integrable functions as giyen above, then there exists a funotion \[{f_0}(x) \in L(\varepsilon )\] such that \[{{\tilde f}_0}(x) \in L(\varepsilon )\] and \[\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{f_0} - {S_n}({f_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0\\overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{{\tilde f}_0} - {S_n}({{\tilde f}_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0 \end{array}\] where C is an absolute constant. Some generalizations of the theorem are given.     

ON MILIN-LEBDEV INEQUALITIES

Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\] \[\begin{gathered} \frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\ \end{gathered} \] Milin-Lebedey proved that \[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \] where p>l and \[\lambda \]>0. In this paper, we have proved the following theorems； Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and \[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\] then F(x) is a decreasing function of x on [0, 1]. This theorem is stronger than the result (1). Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and \[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \] then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result.     

从$L^{\infty}$空间到Bloch型空间一类奇异积分算子的范数

本文研究了单位圆盘上从$L^{\infty}(\mathbb{D})$空间到Bloch型空间 $\mathcal{B}_\alpha$ 一类奇异积分算子$Q_\alpha, \alpha>0$的范数, 该算子可以看成投影算子$P$ 的推广,定义如下$$Q_\alpha f(z)=\alpha \int_{\mathbb{D}}\frac{f(w)}{(1-z\bar{w})^{\alpha+1}}\d A(w),$$ 同时我们也得到了该算子从 $C(\overline{\mathbb{D}})$空间到小Bloch型空间$\mathcal{B}_{\alpha,0}$上的范数.     

ON MILIN-LEBDEV INEQUALITIES

Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\] \[\begin{gathered} \frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\ \end{gathered} \] Milin-Lebedey proved that \[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \] where p>l and \[\lambda \]>0. In this paper, we have proved the following theorems； Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and \[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\] then F(x) is a decreasing function of x on [0, 1]. This theorem is stronger than the result (1). Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and \[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \] then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result.     

一类多线性奇异积分算子的加权估计

In this paper, weighted estimates with general weights are established for the multilinear singular integral operator defined by TAf(x) = p. v.RnΩ(x- y)|x- y|n+1 A(x)- A(y)- A(y)(x- y) f(y)dy,where Ω is homogeneous of degree zero, has vanishing moment of order one, and belongs to Lipγ(Sn-1) with γ∈(0, 1], A has derivatives of order one in BMO(Rn).     

沿超曲面的强奇异积分算子在调幅空间上的有界性

设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面.     

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

ＯＮＡＭＵＬＴＩＬＩＮＥＡＲＯＳＣＩＬＬＡＴＯＲＹＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲ（Ｉ）ＣＨＥＮＷＥＮＧＵＨＵＧＵＯＥＮＬＵＳＨＡＮＺＨＥＮＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１８，１９９４．ＲｅｖｉｓｅｄＤｅｃｅ...     

带粗糙初始值向列型液晶流的适定性

考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.     

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral

A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ（f）（x）=（∫0^∞│∫│1-y│≤t Ω（x,x-y）/│x-y│^n-p f（y）dy│^2dt/t1＋2p）^1/2 are investigated.It is proved that if Ω∈ L∞（R^n） × L^r（S^n-1）（r〉（n-n1p＇/n） is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p（R^n） to L^p（R^n） for 1 〈 p ≤ max{（n＋1）/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type（p,p） for 1 〈 p ≤ 2,of the weak type（1,1） and bounded from H1 to L1.     

THE FIRST BOUNDARY VALUE PROBLEM FOR SOLUTIONS OF DEGENERATE QUASUJNEAR PARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities

In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type $$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$ Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter ${\varepsilon}$ is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents.