Commutators with Reisz potentials in one and several parameters

Abstract

Let $M_b$ be the operator of pointwise multiplication by $b$, that is $M_b f=bf$. Set $[A,B]=AB-BA$. The Reisz potentials are the operators $R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha} }, 0<\alpha<1.$ They map $L^p\mapsto L^q$, for $1-\alpha+\frac{1}{q}=\frac{1}{p}$, a fact we shall take for granted in this paper. A Theorem of Chanillo [6] states that one has the equivalence $||[M_b,R_\alpha]||_{p\to q}\simeq ||b||_{BMO}$ 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.