Multiparameter Harmonic analysis and sharp geometric inequalities with applications to PDEs

摘要（陆，1301595）建议的研究项目包括解决两个主要方向的问题：尖锐的几何不等式和多参数调和分析。最近在尖锐几何不等式领域的发展包括整个Heisenberg群和更一般的Carnot群上的Moser-Trudinger不等式的最佳常数和高阶Sobolev空间上的亚当斯不等式。在这些情况下，对称化属性不成立。PI与他的博士生合作，最近成功地开发了一种无需重新排列的论点。这种新的方法表明，这种尖锐的几何不等式可以建立在更一般的情况下，包括黎曼和亚黎曼流形。 此外，PI将研究这些尖锐的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是开放的。 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，开发了一个令人满意的理论离散Littlewood-Paley平方函数在一些多参数的情况下。然而，仍然有许多其他重要的多参数设置，这样一个离散的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核心领域。在这个项目中发现的发现和新的工具可能会导致新的发展领域的经典调和分析，偏微分方程以及其他分支的数学。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（如车辆的振动和降噪），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，该项目还包括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本项目，在主要研究者的监督下接受研究培训。

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.