在本项目里，我们将主要研究几类向量值时滞微分方程的最大正则性，问题所带有的时滞项可以是有限时滞项也可以是无限时滞项。我们考虑的微分方程包括一阶方程、二阶方程、三阶方程以及更加一般的带有分数导数方程，包括退化情形也包括某些非退化情形。我们考虑最大正则性的函数空间是Lebesgue-Bochner空间、周期Besov空间、周期Triebel-Lizorkin空间、周期Hardy空间以及Holder连续函数空间。我们将把这些方程的最大正则性问题自然地转化成为相应函数空间上的算子值傅里叶乘子问题，再利用已有的算子值傅里叶乘子定理得到这些方程具有最大正则性的充分条件、必要条件或充要条件。本项目将要得到的结果将推广之前已知非时滞情形的已有结果。

In this project, we will study the maximal regularity of some class of vector-valued differential equations with delay, the delay term may be finite or infinite. The differential equations we will consider include first order equations, second order equations, third order equations and more generally fractional order equations. We will consider both the nondegenerate case and the degenerate case, and the function spaces concerning the maximal regularity are Lebesgue-Bochner function spaces, periodic Besov spaces, periodic Triebel-Lizorkin spaces, periodic Hardy spaces and Holder continuous function spaces. We will transform our maximal regularity problems to an operator-valued Fourier multiplier problems. The main results will be necessary condition, sufficient condition or necessary and sufficient condition to ensure the maximal regularity on the corresponding function spaces. Our results will extend known results in the case without delay term.