# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 36 (2007) pp.175-191

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### Abstract

Let $\operatorname M_b$ be the operator of pointwise multiplication by $b$, that is $\operatorname M_b f=bf$. Set $[\operatorname A,\operatorname B]={} \operatorname A \operatorname B-\operatorname B \operatorname A$. The Reisz potentials are the operators \begin{equation*} \operatorname R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha} },\qquad 0<\alpha<1. \end{equation*} They map $L^p\mapsto L^q$, for $1-\alpha+\frac1q=\frac1p$, a fact we shall take for granted in this paper. A Theorem of Chanillo [6] states that one has the equivalence \begin{equation*} \norm [\operatorname M_b,\operatorname R_\alpha].p\to q.\simeq \norm b.\operatorname{BMO}. \end{equation*} with the later norm being that of the space of functions of bounded mean oscillation. We discuss a proof of this result in a discrete setting, and extend part of the equivalence above to the higher parameter setting.

MSC(Primary) 42B35 42B25, 42B20 Reisz potential, fractional integral, paraproduct, commutator, multiparameter, bounded mean oscillation