XMO and Weighted Compact Bilinear Commutators

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