XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderon-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 (R-n), denoted by XMO (R-n), and conjectured that it is just the space VMO (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 (R-n), which further clarifies that XMO (R-n) is a proper subspace of VMO(R-n). This equivalent characterization of XMO (R-n) is formally similar to the corresponding one of CMO (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.