Topics in Harmonic Analysis

数学分析的方法在理解自然科学和工程中的物理现象方面有着广泛的应用。该项目关注的是谐波分析的主题，旨在为这些学科提供有效的数学工具，并可能有助于统一看似无关的领域。该项目的主要目标是沿着沿着这些路线扩展谐波分析的数学工具箱。该项目包括指导研究生和博士后研究人员。主要研究者将从事谐波分析方面的几个项目。第一个项目是关于自伴椭圆和次椭圆偏微分算子的泛函微积分。研究将集中在一个模型的例子，在亚椭圆的情况下，海森堡群的拉普拉斯算子。我们的目标是获得新的乘子结果勒贝格空间，特别是模型的情况下，Bochner-Riesz手段。在欧几里德的情况下，这样的乘数结果可以基于傅立叶限制理论，这在海森堡群上是不可用的。基于海森堡群上波核的精细结构，提出了一种新的方法。第二个项目的动机是一个问题的成本混合的不可压缩流所产生的时间依赖的矢量场。结果表明，这个问题与某些双线性奇异积分算子的有界性问题有关。一个一般的理论将开发，包括多线性的推广，是独立的利益。第三个项目是有关问题的逼近理论，涉及函数空间的低阶光滑。在许多情况下，Besov和Sobolev空间范畴的已知结果之间存在差距。作为一个例子，Haar小波展开的无条件收敛的范围在Besov空间是很好理解的，但在Sobolev空间仍然是开放的。这个项目的一个目标是确定Sobolev空间的正确范围。

Methods from mathematical analysis have found wide applications in understanding physical phenomena in the natural sciences and engineering. This project is concerned with topics in harmonic analysis that are designed to provide effective mathematical tools for these disciplines and that could well contribute to the unification of seemingly unrelated areas. The main goal of the project is to expand the mathematical toolbox in harmonic analysis along these lines. The project involves mentoring of graduate students and postdoctoral researchers.The principal investigator will to work on several projects in harmonic analysis. The first project is concerned with the functional calculus for selfadjoint elliptic and subelliptic partial differential operators. The research will focus on a model example in the subelliptic case, the Laplacian on the Heisenberg group. The goal is to derive new multiplier results on Lebesgue spaces, in particular for the model case of Bochner-Riesz means. In the Euclidean case such multiplier results can be based on Fourier restriction theory, which is not available on the Heisenberg group. A new approach is proposed based on the fine structure of the wave kernel on the Heisenberg group. A second project is motivated by a problem on the cost of mixing for incompressible flows that are generated by time-dependent vector fields. It turns out that this problem is related to boundedness questions on certain bilinear singular integral operators. A general theory will be developed that incorporates multilinear generalizations and is of independent interest. A third project is concerned with problems in approximation theory that involve function spaces of low order of smoothness. In many cases there are gaps between the known results for the categories of Besov and Sobolev spaces. As an example, the range for unconditional convergence of Haar wavelet expansions is well understood for Besov spaces but remains open for Sobolev spaces. One goal for this project is to determine the correct range for Sobolev spaces.