THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERóN-ZYGMUND OPERATORS

Let T be an anisotropic Calderón-Zygmund operator and φ:R~n×[0,∞)→[0,∞) be an anisotropic Musielak-Orlicz function with φ(x,·) being an Orlicz function andφ(·,t) being a Muckenhoupt A_∞(A) weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMO_w(R~n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R~n,A)),the commutator [b,T] is bounded from anisotropic weighted Hardy space H_ω~1(R~n,A) to weighted Lebesgue space L_ω~1(R~n) and when b∈BMO(R~n)(bounded mean oscillation space),the commutator [b,T] is bounded on Musielak-Orlicz space L~φ(R~n),which are extensions of the isotropic setting.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

Musielak–Orlicz BMO-type Spaces Associated with Generalized Approximations to the Identity

Let X be a space of homogenous type and ψ: X × [0,∞) → [0,∞) be a growth function such that ψ(·,t) is a Muckenhoupt weight uniformly in t and ψ(x,·) an Orlicz function of uniformly upper type 1 and lower type p ∈(0,1].In this article,the authors introduce a new Musielak–Orlicz BMO-type space BMOψA(X) associated with the generalized approximation to the identity,give out its basic properties and establish its two equivalent characterizations,respectively,in terms of the spaces BMOψA,max(X) and BMOψA(X).Moreover,two variants of the John–Nirenberg inequality on BMOψA(X) are obtained.As an application,the authors further prove that the space BMOψΔ1/2(Rn),associated with the Poisson semigroup of the Laplace operator Δ on Rn,coincides with the space BMOψ(Rn) introduced by Ky.     

Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

COMMUTATORS OF MULTIPLIERS ON HARDY SPACES

Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（R^n） into the weak L^1 （R^n） space and from certain atomic Hardy space Hb^1 （R^n） into the Lebesgue space L^1 （R^n）, when the multiplier m satisfies the conditions of Hoermander type.     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

More on Fixed Point Theorem of {a, b, c}-Generalized Nonexpansive Mappings in Normed Spaces

Let X be a weakly Cauchy normed space in which the parallelogram law holds, C be a bounded closed convex subset of X with one contracting point and T be an {a, b, c}-generalized-nonexpansive mapping from C into C. We prove that the infimum of the set { ||x-T(x)|| } on C is zero, study some facts concerning the {a,b,c}-generalized-nonexpansive mapping and prove that the asymptotic center of any bounded sequence with respect to C is singleton. Depending on the fact that the {a,b,0}-generalized-nonexpansive mapping from C into C has fixed points, accordingly, another version of the Browder ’s strong convergence theorem for mappings is given.     

Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

Let X be a ball quasi-Banach function space on R~n. In this article, we introduce the weak Hardytype space W H_X(R~n), associated with X, via the radial maximal function. Assuming that the powered HardyLittlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space W X and the associated space, we then establish several real-variable characterizations of W H_X(R~n), respectively, in terms of various maximal functions,atoms and molecules. As an application, we obtain the boundedness of Calderón-Zygmund operators from the Hardy space H_X(R~n) to W H_X(R~n), which includes the critical case. All these results are of wide applications.Particularly, when X := M_q~p(R~n)(the Morrey space), X := L~p(R~n)(the mixed-norm Lebesgue space) and X :=(E_Φ~q)_t(R~n)(the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasiBanach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.     

L~p-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on R~n

Let L=-div(A▽) be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L~(1/2)]of the Kato square root L~(1/2) and b with ▽b∈L~n(R~n)(n 2),is bounded from the homogenous Sobolev space L_1~p(R~n) to L~p(R~n)(p-(L) pp+(L)).     

数学学报英文版Vol．23(2007)，No．9论文摘要

<正>Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces Yong DING Shan Zhen LU Qing Ying XUE In this paper,the authors prove that ifΩsatisfies a class of thc integral Dini condition,then the parametrized area integralμ_(Ω,S)~ρis a bounded operator from the Hardy space H~1(R~n)to L~1(R~n)and from the weak Hardy space H~(1,∞)(R~n)to L~(1,∞)(R~n),respectively.As corollaries of the above results,it is shown thatμ_(Ω,S)~ρis also an operator of weak type(1,1)and of type(p,p)for 1相似文献   

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.     

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).     

变指标Herz-Morrey空间上的Marcinkiewicz积分算子高阶交换子

In the case of Ω∈ Lipγ(Sn-1)(0 γ≤ 1), we prove the boundedness of the Marcinkiewicz integral operator μΩon the variable exponent Herz-Morrey spaces. Also, we prove the boundedness of the higher order commutators μmΩ,bwith b ∈ BMO(Rn) on both variable exponent Herz spaces and Herz-Morrey spaces, and extend some known results.     

与强奇异 Calder\''{o}n-Zygmund算子相关的Toeplitz算子的双权估计

研究了与强奇异Calder\'{o}n-Zygmund算子和加权 Lipschitz函数${\rm Lip}_{\beta_0,\omega}$相关的Toeplitz算子$T_b$的sharp极大函数的点态估计,并证明了Toeplitz算子是从 $L^p(\omega)$到$L^q(\omega^{1-q})$上的有界算子.此外, 建立了与强奇异Calder\'{o}n-Zygmund算子和加权 BMO函数${\rm BMO}_{\omega}$相关的Toeplitz算子$T_b$的sharp极大函数的点态估计,并证明了Toeplitz算子是从 $L^p(\mu)$到$L^q(\nu)$上的有界算子.上述结果包含了相应交换子的有界性.     

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)).     

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 $\alpha =1$.     

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

Let $1相似文献   

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 $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

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   

The q-Variation of Functions and Spectral Integration from Dominated Ergodic Estimates

For $1\leq q < \infty$, let $\mathfrak{M}_{q}\left( \mathbb{T}\right)$, (respectively, $\mathfrak{M}_{q}\left( \mathbb{R}\right) $) denote the Banach algebra consisting of the bounded complex-valued functions having uniformly bounded $q$-variation on the dyadic arcs of the unit circle, (respectively, on the dyadic intervals of the real line). Suppose that $(\Omega,\mu)$ is a $\sigma$-finite measure space, $1< p < \infty$, and $T:L^{p}(\mu)\rightarrow L^{p}(\mu)$ is a bounded, invertible, separation-preserving linear operator such that the two-sided ergodic means of the linear modulus of $T$ are uniformly bounded in norm. We show that there is a real number $q_{_{0}} > 1$ such that whenever $1\leq q < q_{_{0}}$, $T $ has a norm-continuous functional calculus associated with $\mathfrak{M}_{q}\left(\mathbb{T}\right) $. Our approach is rooted in a dominated ergodic theorem of Mart\{\i}n--Reyes and de la Torre which assigns $T$ a canonical family of bilateral $A_{p}$ weight sequences. We first establish the relevant multiplier properties of $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ classes in weighted settings, transfer the outcome to $\mathfrak{M}_{q}\left(\mathbb{T}\right) $, and then apply the consequent $\mathfrak{M}_{q}\left(\mathbb{T}\right) $ multiplier theorem for weighted settings to the spectral decomposition of $T$. The desired $\mathfrak{M}_{q}\left(\mathbb{T}\right)$-functional calculus for $T$ then results from an extension criterion for spectral integration obtained in the general Banach space setting. The multiplier result for $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ shown at the outset of this process expands the scope of the weighted Marcinkiewicz multiplier theorem from $q=1$ to appropriate values of $q > 1$