To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

为了研究某些双线性 Calderón-Zygmund 算子的双线性换向器的紧凑性，这些算子包括（非齐次的）Coifman-Meyer 双线性傅立叶乘法器和双线性伪微分算子作为特例，Torres 和 Xue（Rev Mat Iberoam 36:939-956, 2020）介绍了一个BMO \(\,(\mathbb {R}^n)\) 的新子空间，记为 XMO \(\,(\mathbb {R}^n)\)，并推测它只是空间 VMO \( \,(\mathbb {R}^n)\)由 D. Sarason 介绍。在这篇文章中，作者通过建立 XMO \(\,(\mathbb {R}^n)\)的等价表征，对这一猜想给出了否定的答案，这进一步阐明了 XMO \(\,(\mathbb {R }^n)\)是 VMO 的真子空间\(\,(\mathbb {R}^n)\)。XMO \(\,(\mathbb {R}^n)\) 的这种等效表征形式上类似于A. Uchiyama 获得的 CMO \(\,(\mathbb {R}^n)\) 的相应特征，但它的证明需要一些关于二元立方体的基本新技术以及一些精美的几何观察。作为应用，作者还获得了此类双线性换向器的加权紧凑性结果，从而优化了未加权设置中的相应结果。