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

Let I_α→ be the bi-parameter fractional integral operator on R^n₁×R^n₂, I_α→(f)(x)=∫_{Rⁿ¹×Rⁿ²}[Formula presented]dy,0<α_i<n_i,i=1,2.In this paper, we give a characterization of two-weight norm inequality for the commutator of I_α→. We show that for μ,λ∈A_p,q(R^n→), ‖[b,I_α→]‖_{L^p(μ^p)→L^q(λ^q)}≃‖b‖_bmo(ν), where ν=μλ⁻¹, and [Formula presented]−[Formula presented]=[Formula presented]=[Formula presented]. 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.