Commutators with Reisz Potentials in One and Several Parameters

الأصل · EN

Let Mb be the operator of pointwise multiplication by b, that is Mb f=bf. Set [A,B]= AB- BA. The Reisz potentials are the operators Rαf(x)=∫ f(x-y)dy/ y α, 0<α<1. They map Lᵖ Lq, for 1-α+1q=1p, a fact we shall take for granted in this paper. A Theorem of Chanillo MR84j:42027 states that one has the equivalence [Mb, Rα].p→ q.≃ b.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.