Commutators with Reisz Potentials in One and Several Parameters

Details Commutators with Reisz Potentials in One and Several Parameters Commutators with Reisz Potentials in One and Several Parameters

2005-02-16

arxiv.org/abs/math/0502336v2

Mathematics

Classical Analysis and ODEs

To appear in Hokkaido Math J. This is the final version of the paper. Several typos corrected

Scientific paper

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.

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