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Register a new userIterated commutators under a joint condition on the tuple of multiplying functions
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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.
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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.
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.
Let $fin L_{2pi}$ be a real-valued even function with its Fourier series $ frac{a_{0}}{2}+sum_{n=1}^{infty}a_{n}cos nx,$ and let $S_{n}(f,x), ngeq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence ${a_{n}}$ is decreasing and $limlimits_{nto infty}a_{n}=0$, then $$ limlimits_{nto infty}Vert f-S_{n}(f)Vert_{L}=0 {if and only if} limlimits_{nto infty}a_{n}log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $fin L_{2pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $fin L_{2pi}$ under the $% MVBV$ condition.
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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