XMO and Weighted Compact Bilinear Commutators
XMO and Weighted Compact Bilinear Commutators
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DOI：
10.1007/s00041-021-09854-x
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发表时间：
2019-09
影响因子：
1.2
通讯作者：
Jin Tao;Qingying Xue;Dachun Yang;Wen Yuan
中科院分区：
文献类型：
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作者：
Jin Tao;Qingying Xue;Dachun Yang;Wen Yuan
为了研究某些双线性Calderón-Zygmund算子的双线性交换子的紧性，其中包括(非齐次)Coifman-Meyer双线性傅立叶乘子和双线性伪微分算子作为特例，Torres和Xue(Rev Mat Iberoam 36：939-956,2020)引入了BMO的一个新的子空间，记为XMO，并猜想它就是D.Sarason引入的VMO空间。本文通过建立XMO的一个等价刻划，否定地回答了这一猜想，进一步阐明了XMO是VMO的真子空间。XMO的这个等价刻划在形式上类似于A.Uchiyama得到的CMO的等价刻划，但它的证明需要一些必要的关于并元立方体的新技巧和一些精细的几何观察。作为应用，我们还得到了这类双线性交换子的一个加权紧性结果，它优化了未加权设置下的相应结果。
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, denoted by XMO, and conjectured that it is just the space VMOintroduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO, which further clarifies that XMOis a proper subspace of VMO. This equivalent characterization of XMOis formally similar to the corresponding one of CMOobtained 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.