Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces

Abstract

Denote by T and \(\mathcal {I}_{\alpha }\) the bilinear Calderon-Zygmund operator and bilinear fractional integrals, respectively. In this paper we give the boundedness and compactness of the commutators [T,b] i , maximal operator T ∗,b,i and \([\mathcal {I}_{\alpha },b]_{i}\) on Morrey spaces. More precisely, we prove that [T,b] i , T ∗,b,i and \([\mathcal {I}_{\alpha },b]_{i}\) are all the bounded operators (if b∈B M O) and compact operators (if b ∈ C M O, the BMO-closure of \(C_{c}^{\infty }\)) from \(L^{p_{1},\lambda _{1}}\times L^{p_{2},\lambda _{2}}\) to L p, λ for some suitable indexes λ, λ 1, λ 2 and p, p 1, p 2. As an application of our results, we give also the boundedness and compactness of the commutators formed by the bilinear pseudodifferential operators on Morrey spaces.