Multiparameter Harmonic analysis and sharp geometric inequalities with applications to PDEs

摘要(Lu，1301595)提出的研究项目包括解决两个主要方向的问题：尖锐几何不等式和多参数调和分析。在尖锐几何不等式领域的最新进展包括整个Heisenberg群上的Moser-Trudinger不等式的最佳常数，以及更一般的卡诺群和欧氏空间中无界区域上的高阶Soblev空间上的Adams不等式。这些都是对称化性质不成立的情况。最近，PI与他的博士生合作，成功地提出了一个没有重排的论点。这一新方法表明，这种尖锐的几何不等式可以在更一般的情形下建立，包括黎曼流形和次黎曼流形。此外，PI将研究这些尖锐几何不等式的极值函数的存在性，其中许多具有挑战性的问题仍然悬而未决。另一个主要的研究方向是在几个复杂但重要的多参数设置下发展多参数调和分析函数空间理论。PI与其他人合作，在许多多参数情况下发展了一个令人满意的离散Littlewood-Paley平方函数理论。然而，仍然有许多其他重要的多参数设置，其中这样的离散Littlewood-Paley理论尚未建立。多参数调和分析和非线性偏微分方程是现代数学的中心领域。在这个项目中发现的发现和新的工具可能会导致经典调和分析、偏微分方程式以及其他数学分支领域的新发展。它们在科学和工程中有许多应用。拟议项目的解决方案将对许多其他学科产生影响，包括机械工程(如车辆减振和降噪)、医学科学中的成像处理和模式识别、随机控制和优化、博弈论、化学燃烧、人类视觉以及生命和医学科学中的其他主题。此外，该项目还包含大量的培训和教育内容。它将研究与教育完美地结合在一起。许多研究生将积极参与这个项目，在首席研究员的指导下接受研究培训。

Abstract (Lu, 1301595)The proposed research project includes solving problems in two main directions: sharp geometric inequalities and multiparameter harmonic analysis. Recent developments in the area of sharp geometric inequalities include best constants for Moser-Trudinger inequalities on the entire Heisenberg group and more general Carnot groups and Adams inequalities on high order Sobolev spaces on unbounded domains in Euclidean spaces. These are circumstances where symmetrization properties do not hold. The PI, in collaboration with his PhD students, have very recently succeeded in developing a rearrangement-free argument. This new method suggests that such sharp geometric inequalities can be established in more general scenarios including Riemannian and sub-Riemannian manifolds. Moreover, the PI will investigate the existence of extremal functions for these sharp geometric inequalities where many challenging problems still remain open. Another main direction of research is to develop multiparameter harmonic analysis function space theory in several complicated but important multiparameter settings. The PI, in collaboration with others, has developed a satisfactory theory of discrete Littlewood-Paley square functions in a number of multiparameter scenarios. However, there are still many other important multiparameter settings where such a discrete Littlewood-Paley theory is yet to be established. Multi parameter Harmonic analysis and nonlinear partial differential equations are central areas of modern mathematics. Findings and new tools discovered in this project may lead to new development in the area of classical harmonic analysis, partial differential equations as well as other branches of mathematics. They have many applications in sciences and engineering. The solution to the proposed project will have impact on many other disciplines, including mechanics engineering (such as vibration and noise reduction for vehicles), imaging processing and pattern recognitions in medical sciences, stochastic control and optimization, game theory, chemical combustion, human vision and other topics in the life and medical sciences. Moreover, this project has a substantial training and educational component. It finely integrates research together with education. Many graduate students will actively participate in this project by receiving research training under the supervision of the principal investigator.