Boundedness of Multilinear Oscillatory Singular Integral on Weighted Weak Hardy Spaces

In this paper, by using the atomic decomposition of the weighted weak Hardy space WH_ω~1(R~n), the authors discuss a class of multilinear oscillatory singular integrals and obtain their boundedness from the weighted weak Hardy space WH_ω~1(R~n) to the weighted weak Lebesgue space WL_ω~1(R~n) for ω∈A_1(R~n).     

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.     

An Equivalent Characterization of $CMO(\mathbb{R}^n)$ with $A_p$ Weights

Let $1相似文献   

Lipschitz Estimates for Commutators of $N$-Dimensional Fractional Hardy Operators

In this paper, it is proved that the commutator$\mathcal{H}_{β,b}$ which is generated by the $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ and $b\in \dot{Λ}_α(\mathbb{R}^n)$ is bounded from $L^P(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$, where $0<α<1,1相似文献   

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

单位方体上沿曲面的振荡积分在Sobolev空间上的有界性

研究了欧氏空间R~2中单位方体Q~2=[0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性.     

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.     

奇异积分算子交换子在变指数空间上的特征

The main purpose of this paper is to characterize the Lipschitz space by the boundedness of commutators on Lebesgue spaces and Triebel-Lizorkin spaces with variable exponent.Based on this main purpose, we first characterize the Triebel-Lizorkin spaces with variable exponent by two families of operators. Immediately after, applying the characterizations of TriebelLizorkin space with variable exponent, we obtain that b ∈■β if and only if the commutator of Calderón-Zygmund singular integral operator is bounded, respectively, from■ to■,from■ to■ with■. Moreover, we prove that the commutator of Riesz potential operator also has corresponding results.     

双线性分数次积分算子交换子在Morrey空间上有界的充分必要条件

In this paper,we obtain that b∈ BMO(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.     

Oversampling generates super-wavelets

We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space .

     

Fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. ~~~~~~~~~~~~~~~~~(FHS)_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.     

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.     

Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.     

对带混合非线性项的基尔霍夫方程规范解问题的补充

在本文中,我们研究了带有质量约束条件的基尔霍夫方程规范解的存在性问题, $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3},$$ 其中质量约束条件为$$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}|u|^{2} \text{d} x=c\Big\},$$ 这里$a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$, 当$(p, q)$ 属于$\mathbb{R}^{2}$中的某个域时. 我们通过使用约束最小化、集中紧性原理和Minimax方法证明了规范解的存在性.我们部分地扩展了已经被研究的结果.     

Thin equivalence relations in scaled pointclasses

For ordinals α beginning a Σ₁ gap in $\mathrm{L}(\mathbb {R})$, where $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ is closed under number quantification, we give an inner model‐theoretic proof that every thin $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ equivalence relation is $\Delta _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ in a real parameter from the (optimal) hypothesis $\mathsf {AD}^{\mathrm{J}_{\alpha }(\mathbb {R})}$.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)).     

Regularity of pullback attractors and random equilibrium for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

This paper is concerned with the stochastic Fitzhugh-Nagumo system with non-autonomous terms as well as Wiener type multiplicative noises. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, the existences and upper semi-continuity of pullback attractors are proved for the generated random cocycle in $L^l(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ for any $l\in(2,p]$. The asymptotic compactness of the first component of the system in $L^p(\mathbb{R}^N)$ is proved by a new asymptotic a priori estimate technique, by which the plus or minus sign of the nonlinearity at large values is not required. Moreover, the condition on the existence of the unique random fixed point is obtained, in which case the influence of physical parameters on the attractors is analysed.     

一阶离散周期问题的多解性

Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.