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Register a new userA revisit on the compactness of commutators
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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.
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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.
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.
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.
We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubins theorem for functions on $mathbb{S}^{n}$ with zero first order moments of the area element to higher order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher order Sobolev spaces.
We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates.
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