Compactness for the Commutator of the Multilinear Fourier Multiplier on the Morrey Space

Given s 1,…, s m ∈ (n/2, n], let T σ be a multilinear Fourier multiplier operator associated with a multilinear multiplier σ satisfying a Sobolev regularity condition $\sup _{\ell \in \mathbb {Z}}\|\sigma _{\ell }\|_{W^{s_{1},\ldots ,s_{m}}(\mathbb {R}^{mn})}<\infty .$ By the strongly precompactness of Banach space, the authors prove that if $b_{1},\ldots ,b_{m}\in CMO(\mathbb {R}^{n})$ , then the commutator T σ,Σb is a compact operator from the product Morrey space $L^{p_{1},\lambda }(\mathbb {R}^{n})\times \cdots \times L^{p_{m},\lambda }(\mathbb {R}^{n})$ to the Morrey space $L^{p,\lambda }(\mathbb {R}^{n})$ . As an application, the compactness of the commutator T σ,Σb from the product Morrey space $L^{p_{1},\lambda }(\mathbb {R}^{n})\times \cdots \times L^{p_{m},\lambda }(\mathbb {R}^{n})$ to the Morrey space $L^{p,\lambda }(\mathbb {R}^{n})$ is also obtained under the Sobolev regularity condition $\sup _{\ell \in \mathbb {Z}}\|\sigma _{\ell }\|_{W^{s}(\mathbb {R}^{mn})}<\infty $ for s∈(m n/2, m n].