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Search: All articles in the CMB digital archive with keyword commutator

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1. CMB Online first

Chen, Jiecheng; Hu, Guoen
Compact Commutators of Rough Singular Integral Operators
Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier transform estimates and approximation to the operator $T_{\Omega}$ by integral operators with smooth kernels, it is proved that if $b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain minimal size condition, then the commutator generated by $b$ and $T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for appropriate index $p$. The associated maximal operator is also considered.

Keywords:commutator,singular integral operator, compact operator, maximal operator
Category:42B20

2. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
Under the assumption that $\mu$ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$
Categories:42B25, 47B47, 42B20, 47A30

3. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$.

Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates
Categories:42B20, 42B25, 35S05, 47G30

4. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials.

Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional
Categories:46L05, 47B99

5. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
Commutators Estimates on Triebel--Lizorkin Spaces
In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$.

Keywords:commutators, Triebel--Lizorkin spaces, paraproduct
Categories:42B, 46F

6. CMB 1999 (vol 42 pp. 463)

Hofmann, Steve; Li, Xinwei; Yang, Dachun
A Generalized Characterization of Commutators of Parabolic Singular Integrals
Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1, \dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots \le\az_n$. Denote $|\az|=\az_1+\cdots+\az_n$. We characterize those functions $A(x)$ for which the parabolic Calder\'on commutator $$ T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n} K(x-y)[A(x)-A(y)]f(y)\,dy $$ is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$, $K$ is smooth away from the origin and satisfies a certain cancellation property.

Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1-theorem, symbol
Category:42B20

7. CMB 1999 (vol 42 pp. 198)

Guadalupe, José J.; Pérez, Mario; Varona, Juan L.
Commutators and Analytic Dependence of Fourier-Bessel Series on $(0,\infty)$
In this paper we study the boundedness of the commutators $[b, S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$-th partial sum of the Fourier-Bessel series on $(0,\infty)$. Perturbing the measure by $\exp(2b)$ we obtain that certain operators related to $S_n$ depend analytically on the functional parameter $b$.

Keywords:Fourier-Bessel series, commutators, BMO, $A_p$ weights
Category:42C10

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