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Register a new userIterated Riesz Commutators: A Simple Proof of Boundedness
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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.
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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.
In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr{o}dinger operator $P=-Delta+V(x)$ on $mathbb{R}^n, ngeq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q geq n/2$. Let $T_1=(-Delta+V)^{-1}V, T_2=(-Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-Delta+V)^{-1/2} abla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(mathbb{R}^n)$ when $p$ ranges in a interval, where $b in mathbf{BMO}(mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.
In this paper we characterise the pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where the two metrics are not equivalent. We further establish a suitable version of the pointwise lower bound via the Euclidean metric and then characterise boundedness of commutator of the Dunkl Riesz transform via the BMO space associated with the Euclidean metric and the Dunkl measure. This shows that BMO space via the Euclidean metric is the suitable one associated to the Dunkl setting but not the one via the Dunkl metric.
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.
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