共查询到20条相似文献,搜索用时 46 毫秒
1.
Let μ be a nonnegative Radon measure on R d which satisfies the growth condition μ(B(x,r)) ≤ C0rn for all x ∈Rd and r >0,where C0 is a fixed constant and 0相似文献
2.
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 $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$. 相似文献
3.
证明了拟线性次椭圆方程组-X_α~*(a_(ij)~(αβ)(x,u)X_βu~j)=-X_α~*f_i~α+g_i,i=1,2,…,N,x∈Ω的弱解广义梯度Xu在Morrey空间L_x~(p,λ)(Ω,R~(mN))(p2)上的部分正则性,其中光滑实向量场族X=(X_1,X_2,…,X_m)满足H(o|¨)rmander有限秩条件,X_α~*是X_α的共轭;而且主项系数a_(ij)~(αβ)(x,u)关于x一致VMO(Vanishing Mean Oscillation的缩写,消失平均震荡)间断,且关于u为一致连续. 相似文献
4.
洪勇 《数学年刊A辑(中文版)》2011,32(5):599-606
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. 相似文献
5.
Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem 下载免费PDF全文
Elsayed M. E. Zayed 《偏微分方程(英文版)》2016,29(1):59-70
In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach. 相似文献
6.
We prove the absence of positive eigenvalues of Schrödinger operators
$ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough
potentials $V$. To describe our class of potentials fix an exponent
$q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$.
For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$
(or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and$\begin{equation*}$$\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$$\end{equation*}$Under these assumptions we prove that the operator $H$ does not
admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K].
The absence of positive eigenvalues follows from a uniform Carleman
inequality of the form$\begin{equation*}$$||W_m u||_{l^a(L^{p(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$$\end{equation*}$for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$,
where $p(q)$ and $p(q)$ are dual exponents with the property that $1/p(q)-1/p(q)=1/q$. 相似文献
7.
In this article, we investigate a nonlinear system of differential equations with two parameters
$$\left\{ \begin{array}{l} x"(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y"(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$
where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result. 相似文献
8.
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. 相似文献
9.
Chen Yunmei 《数学年刊B辑(英文版)》1987,8(4):498-522
This paper deals with the following IBV problem of nonlinear parabolic equation:
$$\[\left\{ {\begin{array}{*{20}{c}}
{{u_t} = \Delta u + F(u,{D_x}u,D_x^2u),(t,x) \in {B^ + } \times \Omega ,}\{u(0,x) = \varphi (x),x \in \Omega }\{u{|_{\partial \Omega }} = 0}
\end{array}} \right.\]$$
where $\[\Omega \]$ is the exterior domain of a compact set in $\[{R^n}\]$ with smooth boundary and F satisfies $\[\left| {F(\lambda )} \right| = o({\left| \lambda \right|^2})\]$, near $\[\lambda = 0\]$. It is proved that when $\[n \ge 3\]$, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property $\[{\left\| {u(t)} \right\|_{{L^\infty }(\Omega )}} = o({t^{ - \frac{n}{2}}})\]$, as $\[t \to + \infty \]$. 相似文献
10.
Liu Quan-sheng 《数学年刊B辑(英文版)》1989,10(2):214-220
The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence. 相似文献
11.
赵俊燕 《数学年刊A辑(中文版)》2017,38(4):405-418
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. 相似文献
12.
Zhen-yueZhang Tiang-weiQuyang 《计算数学(英文版)》2003,21(5):657-670
It is well-known that if we have an approximate eigenvalue λ- of a normal matrix A of order n,a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position,say kmax,of the largest component of u is known.In this paper we give a detailed theoretical analysis to show relations between the eigenvecor u and vector xk,k=1,…,n,obtained by simple inverse iteration,i.e.,the solution to the system(A-λI)x=ek with ek the kth column of the identity matrix I.We prove that under some weak conditions,the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Frobenius norm.We also prove that the normalized absolute vector v=|u|/||u||∞ of u can be approximated by the normalized vector of (||x1||2,…||xn||2)^T,We also give some upper bounds of |u(k)| for those “optimal“ indexeds such as Fernando‘s heuristic for kmax without any assumptions,A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u. 相似文献
13.
In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form
$$
du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t
$$
with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation. 相似文献
14.
15.
Consider a class of nonlocal problems
$$
\left \{\begin{array}{ll}
-(a-b\int_{\Omega}|\nabla u|^2dx)\Delta u= f(x,u),& \textrm{$x \in\Omega$},\u=0, & \textrm{$x \in\partial\Omega$},
\end{array}
\right.
$$
where $a>0, b>0$,~$\Omega\subset \mathbb{R}^N$ is a bounded open domain, $f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R $ is a
Carath$\acute{\mbox{e}}$odory function. Under suitable conditions, the equivariant link theorem without the $(P.S.)$ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $a^2/(4b)$, and they are neither large nor small. 相似文献
16.
研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤q
p~*/p~*-q,其中当N≤p时,p~*=+∞,而当1
相似文献
17.
V.I. Kolyada 《Constructive Approximation》2005,22(2):149-191
We consider the uniformly bounded orthonormal system of functions
$$
u_n^{(\l)}(x)= \varphi_n^{(\lambda)}(\cos x)(\sin x)^\lambda, \qquad x\in [0,\pi],
$$
where $\{\varphi_n^{(\lambda)}\}_{n=0}^\infty \,\, (\lambda > 0)$
is the normalized system of ultraspherical polynomials. R. Askey and S. Wainger proved
that the $L^p$-norm $(1 < p < \infty)$ of any linear combination of the first $N+1$
functions $u_n^{(\lambda)}(x)$
is equivalent to the $L^p$-norm of the even trigonometric polynomial
of degree $N$ with the same coefficients. This theorem fails if $p=1 $ or $p=\infty.$
Studying these limiting cases, we prove (for $0 < \lambda < 1$) similar transplantation theorems
in $\mbox{Re } H^1$ and $\mbox{BMO}.$ 相似文献
18.
Hu Ke 《数学年刊B辑(英文版)》1983,4(2):187-190
AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2 相似文献
19.
The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$. 相似文献
20.
Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed. 相似文献