Analysis of sums of vector fields

矢量场和的分析

基本信息

批准号：
8562-2011
负责人：
Wong, ManWah
金额：
$ 0.8万
依托单位：
York University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2014
资助国家：
加拿大
起止时间：
2014-01-01 至 2015-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=544533
关键词：
Analysis sums vector fields

项目摘要

The main objective of the proposed reseach program is to investigate the global qualitative properties of degenerate elliptic partial differential operators on Euclidean spaces by looking at concrete operators. These operators are sums of squares of vector fields on Euclidean spaces. While the local hypoellipticity of these operators are well-known, the global analogs have been studied intensely only since the beginning of the new millennium. The focus of this research project includes but is not limited to the following topics. (1) Establish the global hypoellipticity in the Schwartz space and Gelfand-Shilov spaces of the corresponding one-parameter family of elliptic operators. (2) Obtain estimates for the solutions of PDEs governed by the one-parameter family of elliptic operators. (3) Use results in item 2 to derive Sobolev estimates for solutions of PDEs governed by sums of squares. (4) Compute explicitly the spectrum of a sum of squares of vector fields. While explicit formulas for the solutions in item 2 are available for some sums of squares dealt with in this proposal, they are in general not suitable for obtaining the estimates as desired and are certainly not the right tools for computing the spectra of sums of squares. The approach taken in this proposal is to use the Fourier transform with respect to the subspace of degeneracy for a sum of squares in question in order to convert the operator to a one-parameter family of elliptic operators, which can be analyzed in detail using variants and perturbations of simple harmonic oscillators. Very detailed information about these mutations of simple harmonic oscillators can be obtained using Hermite functions, Wigner transforms of Hermite functions and estimates for pseudo-differential operators and/or Weyl transforms. The detailed information on the parametrized elliptic operators can then be transferred back to the sum of squares being studied. This approach is akin to Richard Feynman's sum of histories in quantum mechanics. The histories are the parametrized elliptic operators and they are of import in quantum mechanics and it is envisaged that the objective and the underlying approach will provide new insight into quantum field theory. A long-term project is to extend the results to manifolds.

拟议研究计划的主要目标是通过研究具体算子来研究欧几里得空间上简并椭圆偏微分算子的全局定性特性。这些运算符是欧几里德空间上向量场的平方和。虽然这些算子的局部亚椭圆性是众所周知的，但从新千年开始以来，人们才对全局类似物进行了深入的研究。本研究项目的重点包括但不限于以下主题。 (1) 在相应的一参数椭圆算子族的Schwartz空间和Gelfand-Shilov空间中建立全局亚椭圆性。 (2) 获得由单参数椭圆算子族控制的偏微分方程解的估计。 (3) 使用第 2 项中的结果导出由平方和控制的偏微分方程解的 Sobolev 估计。 (4) 显式计算向量场平方和的谱。虽然第 2 项中的解的显式公式可用于本提案中处理的某些平方和，但它们通常不适合获得所需的估计，并且当然不是计算平方和谱的正确工具。该提案中采用的方法是使用有关平方和的简并子空间的傅里叶变换，以便将算子转换为椭圆算子的单参数族，可以使用变体对其进行详细分析和简谐振子的扰动。关于简谐振子的这些突变的非常详细的信息可以使用 Hermite 函数、Hermite 函数的维格纳变换以及伪微分算子和/或 Weyl 变换的估计来获得。然后可以将有关参数化椭圆算子的详细信息转移回正在研究的平方和。这种方法类似于理查德·费曼（Richard Feynman）对量子力学历史的总结。这些历史是参数化椭圆算子，它们在量子力学中很重要，预计目标和基本方法将为量子场论提供新的见解。一个长期项目是将结果扩展到流形。