Paraproducts With Flag Singularities

摘要在过去的几年里，我们一直致力于进一步发展多元线性奇异积分理论及其carleson型极大类似物，并与一维薛定谔算子的谱理论相联系。我们了解了它的许多美丽的特性，但仍有很多工作需要做。目前的项目重点是为这些多线性算子创建一个一致的多参数理论。这种研究的动机不仅是我们以前的工作，而且也是分析人员希望拥有一个通用的加藤庞塞理论，这在PDE中非常有用。其中一些算符具有非常特殊的多参数结构，这在以前研究的调和分析算符中是没有的。理解它们的组合学是一项深刻而富有挑战性的研究。我们的计划是不仅要处理大量具有“标志奇点”的副积，而且要处理这些算子的“解析级数”。我们相信这些新技术将对非线性分析中的一些问题产生广泛的影响。

DMS - 0355360Camil Muscalu Cornell UniversityParaproducts with Flag SingularitiesAbstractOver the last few years we have been working on developing furtherthe theory of multilinear singular integrals and their Carleson-type maximal analogs, in connection with the spectral theory of one-dimensionalSchroedinger operators. We understood many of its beautiful features, but still much more work needs to be doneThe present Project is focused on creating a consistent multi-parametertheory for these multilinear operators. The motivation for such a studyis not only our previous work, but also the desire of analystsof having a general Kato-Ponce theory, so useful in PDE.Some of these operators have a very special multi-parameterstructure not seen among the previously studied operators inharmonic analysis. Understanding their combinatorics is a deep and challenging study. Our plan is to handle not only large classes ofsuch paraproducts with "flag singularities", but also "analytic series"of such operators. We believe that these new techniques will have a broadimpact to several problems in nonlinear analysis.