一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

带有负指数 Kirchhoff 方程稳定解的Liouville型定理

In this paper, we consider the Liouville-type theorem for stable solutions of the following Kirchhoff equation ■,where M(t) = a + bt~θ, a 0, b, θ≥ 0, θ = 0 if and only if b = 0. N ≥ 2, q 0 and the nonnegative function g(x) ∈ L_(loc)~1(R~N). Under suitable conditions on g(x), θ and q, we investigate the nonexistence of positive stable solution for this problem.     

Projektive Skalen,Tschebyscheff-Polynome und Fibonacci-Zahlen

Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where $\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or $x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.

     

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   

齐型空间上的Toeplitz型算子

设X是齐型空间.设T_(j,1)和T_(j,2)是具有非光滑核的奇异积分算子,或者是±II(I是恒等算子).令Toeplitz型算子T_b=■T_(j,1)M_T_(j,2),其中M_bf(x)=b(x)f(x).研究了当b∈BMO(X)时,T_b(f)在加权情况下的有界性,以及当b∈BMO(X)时,与经典Carderon-Zygmund算子相联的T_b(f)在Morrey空间上的有界性.     

Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

In this work we consider the following class of elliptic problems $\begin{cases} −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N), \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

New Sobolev-Weinstein Spaces and Applications

In this paper, we consider the generalized Weinstein operator $\Delta_{W}^{d,\alpha,n}$, we introduce new Sobolev-Weinstein spaces denoted $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1}),$ $s\in\mathbb{R},$ associated with the generalized Weinstein operator and we investigate their properties. Next, as application, we study the extremal functions on the spaces $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1})$ using the theory of reproducing kernels.     

一类带齐次核的奇异重积分算子的范数及其应用

设核函数K(u,v)具有对称性和齐次性,对如下定义的奇异重积分算子T:(Tf)(y)=∫R_+~n K(‖x‖α,‖y‖α)f(x)dx,y∈R_+~n,其中‖x‖α=(x_1~α+…+x_n~α)~1/α(α>0),研究了T的范数及其应用.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

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.     

Homoclinic orbits for an unbounded superquadratic

We consider the following nonperiodic diffusion systems

Existence of entire positive solutions for semilinear elliptic systems with gradient term

Under the Keller?COsserman condition on ${\Sigma_{j=1}^{2}f_{j}}$ , we show the existence of entire positive solutions for the semilinear elliptic system ${\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}$ , where ${p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}$ are continuous functions.     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

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).     

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.