  Let $ M_b$ be the operator of pointwise multiplication by $b$, that is
$\operatorname M_b f=bf$. Set $[ A,B]={} AB- BA$. The Reisz potentials are the
operators $$
  R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha}},\qquad 0<\alpha<1. $$
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 \cite{MR84j:42027} states that
one has the equivalence $$ \norm [ M_b, R_\alpha].p\to q.\simeq \norm
b.\operatorname{BMO}. $$ with the later norm being that of the space of
functions of bounded mean oscillation. We discuss a new proof of this result in
a discrete setting, and extend part of the equivalence above to the higher
parameter setting.
