本项目的主要内容是研究Calderón型交换子的多线性有界性理论，并且进一步利用该有界性理论来研究流体混合问题。Calderón型交换子作为一种典型的奇异积分算子，在其它数学学科中（如复分析、偏微分方程）起着至关重要的作用，一直以来都是非常活跃的研究领域。另一方面，流体混合是自然界中常见现象，对其研究不仅在理论上有重要的意义，在实际生产也有广泛的应用。目前在研究关于流体混合的程度随时间变化过程的Bressan猜想中，Seeger等人发现一类双线性的Calderón型交换子在其中起着至关重要的作用。在本项目中，我们将从多线性算子有界性方面出发，综合运用调和分析的现代工具及发展一些适当的新工具，研究Calderón型交换子在Hardy空间、Besov空间、Triebel–Lizorkin空间及其相应的乘积空间上的有界性问题。并且利用这些建立的理论来研究关于流体混合问题的Bressan猜想。

The main purpose of this project is to study the boundedness theory of Calderón commutator and apply it to the mixing flows problem. As a typical example of singular integral operator, Calderón commutator plays an important role in other mathematics (such as complex analysis, partial differential equation), so it is a quite active research area all the time. On the other hand, mixing flows is ubiquitous in nature, which has application not only in theory but also in practice. Currently when studying the Bressan conjecture which concerns about how to quantify mixing as time goes, Seeger et al found that a kind of bilinear Calderón commutator plays a key role there. In this project, basing on the multilinear boundedness theory, we will apply the modern tools in harmonic analysis and develop some new tools, to study the boundedness of Calderón commutator on Hardy space, Besov space, Triebel–Lizorkin space and their product space. And then we use this new theory to study the Bressan conjecture of the mixing flows problem.