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Register a new userOn the compactness of oscillation and variation of commutators
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In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.
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A new characterization of CMO(R^n) is established by the local mean oscillation. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.
Let $alphain (0, 1]$, $betain [0, n)$ and $T_{Omega,beta}$ be a singular or fractional integral operator with homogeneous kernel $Omega$. In this article, a CMO type space ${rm CMO}_alpha(mathbb R^n)$ is introduced and studied. In particular, the relationship between ${rm CMO}_alpha(mathbb R^n)$ and the Lipchitz space $Lip_alpha(mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{Omega,beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_alpha(mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{Omega,beta})^m_b$ via functions in ${rm CMO}_alpha(mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.
This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-H{o}rmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such variation operators.
We present a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2in operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calderon-Zygmund operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.
It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert transforms. The method of proof requires the real-variable methods throughout, which is new in the multi-parameter context.
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