Regularity of Evolutionary Problems via Harmonic Analysis and Operator Theory

该提案由四个密切相关的部分组成。第一个项目涉及运算符-值傅立叶乘法器。在这里，我们的中心目标是证明具有Muckenoupt权的空间上的Marcinkiewicz(周期情形)和Mikhlin乘子定理。进一步的目的是关于一般Schauder分解的乘子及其在和的闭性问题上的应用。第二部分是最大正则性结构理论的发展，其中正性起着至关重要的作用。在第三部分所论述的非自治情形中，广泛开放仍然是最大的规则。它是由拟线性抛物问题强烈驱动的。最后，所有结果都将针对具有不同边界条件的椭圆算子进行测试，这是提案的最后一部分。

The proposal consists of four closely related parts. A first project concerns operator-valued Fourier multipliers. Here the proof of the Marcinkiewicz (periodic case) and Mikhlin Multiplier Theorem on spaces with Muckenhoupt weights form our central goal. Further aims concern multipliers with respect to general Schauder decompositions with applications to the closedness-of-the-sum-problem. The second part is the development of a structure theory for maximal regularity where positivity plays an essential role. Wide open is still maximal regularity in the non-autonomous case to which the third part is devoted. It is strongly motivated by quasilinear parabolic problems. Finally, all results will be tested for elliptic operators with diverse boundary conditions, the last part of the proposal.