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.     

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.     

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)).     

Bochner-Riesz算子的多线性交换子的有界性

In this paper,the boundedness for the multilinear commutators of Bochner-Riesz operator is considered.We prove that the multilinear commutators generated by Bochner-Riesz operator and Lipschitz function are bounded from Lp（Rn）into ∧˙（β-np）（Rn）and from Lnβ（Rn）into BMO（Rn）.     

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Let L_2=(-?)~2+ V~2 be the Schr?dinger type operator, where V■0 is a nonnegative potential and belongs to the reverse H?lder class RH_(q1) for q_1 n/2, n ≥5. The higher Riesz transform associated with L_2 is denoted by ■and its dual is denoted by ■. In this paper, we consider the m-order commutators [b~m, R] and [b~m, R*], and establish the(L~p, L~q)-boundedness of these commutators when b belongs to the new Campanato space Λ_β~θ(ρ) and 1/q = 1/p-mβ/n.     

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

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

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 $|x|→∞.$     

一类Marcinkiewicz积分交换子的Hardy空间估计

In this paper, we obtain the （H^1,L^n/（n-β） and （HKq1^n（1-1/q2）,p,Kq2^n（1-1/q1）,p） type estimates for the commutator of Marcinkiewicz integral with the kernel satisfying the logarithmic type Lipschitz conditions.     

Compact operators whose real and imaginary parts are positive

Let be a compact operator on a Hilbert space such that the operators and are positive. Let be the singular values of and the eigenvalues of , all enumerated in decreasing order. We show that the sequence is majorised by . An important consequence is that, when is less than or equal to , and when this inequality is reversed.

     

The Boundedness of Littlewood-Paley Operators with Rough Kernels on Weighted $(L^q,L^p)^{\alpha}(\mathbf{R}^n)$ Spaces

In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral $\mu_{\Omega,s}$ and Littlewood-Paley functions $\mu_{\Omega}$ and $\mu^{*}_{\lambda}$ on the weighted amalgam spaces $(L^{q}_\omega,L^{p})^{\alpha}(\mathbf{R}^{n})$ as $1 < q\leq \alpha < p\leq \infty$.     

$\mathbb{C}^{n}$中$\mu$-Bergman空间的刻画和微分复合算子

设$p>0$, $\mu$和$\mu_{1}$是$[0,1)$上的正规函数. 本文首先给出了$\mathbb{C}^{n}$中单位球上$\mu$-Bergman空间$A^{p}(\mu)$的几种等价刻画; 然后 分别刻画了$A^{p}(\mu)$到$A^{p}(\mu_{1})$的 微分复合算子$D_{\varphi}$为有界算子以及紧算子的充要条件, 同时给出了当$p>1$时$D_{\varphi}$为 $A^{p}(\mu)$到$A^{p}(\mu_{1})$上紧算子的一种简捷充分条件和必要条件.     

单位球上的加权Bergman空间上的紧复合算子

在一定条件下,证明了(?)~n中单位球上的加权Bergman空间A~p(φ)上的复合算子C_φ是紧算子的充要条件是当|z|→1~-时(1-|x|~2)/(1-|φ(z)|~2)→0.     

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 the absence of positive eigenvalues of Schrödinger operators with rough potentials

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 $1/p(q)-1/p(q)=1/q$.     

多圆柱上Bloch型空间之间加权复合算子的本性范数

设φ(z)=(φ1(z),…,φ_n(z))是D~n到自身的一个全纯映射,ψ(z)是D~n上的全纯函数,其中D~n是C~n中的单位多圆柱.研究了单位多圆柱上Bloch型空阊之间的加权复合算子ψC_φ的本性范数,并给出了其上下界估计.     

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     

一类Hankel算子与Toeplitz算子的Sp性质

对于L~(α,2)(D))的两类Moebius不变子空间A~(α,2)(D)和A~(β,2)(D),我们定义了它们之间的Toeplitz算子T_f~s与其乘积空间上的Hankel算子H_f~r,并且研究了它们的有界性、紧性及Schatten-von Neumann性质。     

ON ADMISSffilLITY OF VARIANCE COMPONENTS ESTIMATES

Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.