Two-weight characterization for commutators of bi-parameter fractional integrals

Abstract

Let Iα→ be the bi-parameter fractional integral operator on Rn1×Rn2, Iα→(f)(x)=∫Rn1×Rn2f(y1,y2)|x1−y1|n1−α1|x2−y2|n2−α2dy,0<αi<ni,i=1,2.In this paper, we give a characterization of two-weight norm inequality for the commutator of Iα→. We show that for μ,λ∈Ap,q(Rn→), ‖[b,Iα→]‖Lp(μp)→Lq(λq)≃‖b‖bmo(ν), where ν=μλ−1, and 1p−1q=α1n1=α1n2. It extends the recent one-parameter theory to the bi-parameter setting. We use the modern dyadic methods, in which the main idea is to represent continuous operators in terms of dyadic operators. Moreover, by introducing some new full and mixed bi-parameter dyadic paraproducts, we write the commutator as a finite linear combination of them.