AI 中文总结
该研究证明双线性形式的稀疏控制蕴含凸体控制，通过随机化论证对两类稀疏族建立结论，并将其应用于推导带符号局部振荡的迭代交换子的稀疏控制。
AI 中文摘要
我们证明了双线性形式的稀疏控制蕴含凸体控制。更确切地说，若双线性形式具有$(r,s)$-稀疏界，则其对$\boldsymbol{\text{C}}^n$值函数的逐坐标扩展具有$(r,s)$-凸体稀疏界。该证明依赖于随机化论证，我们针对固定二进格中的稀疏族及任意立方体的稀疏族均建立了此结果。作为应用，我们推导出迭代交换子的稀疏控制，其中符号的局部振荡出现在稀疏形式中。
英文摘要
We prove that sparse domination of a bilinear form implies convex body domination. More precisely, if a bilinear form admits an $(r,s)$-sparse bound, then its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound. The proof relies on a randomization argument. We establish the result both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application, we deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.
Comments15 pages
