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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.
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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.
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.
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.
We study an extension of the Falconer distance problem in the multiparameter setting. Given $ellgeq 1$ and $mathbb{R}^{d}=mathbb{R}^{d_1}timescdots timesmathbb{R}^{d_ell}$, $d_igeq 2$. For any compact set $Esubset mathbb{R}^{d}$ with Hausdorff dimension larger than $d-frac{min(d_i)}{2}+frac{1}{4}$ if $min(d_i) $ is even, $d-frac{min(d_i)}{2}+frac{1}{4}+frac{1}{4min(d_i)}$ if $min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.
Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. An decomposition of the commutator [H,b] into paraproducts is presented.
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