Analysis of sums of vector fields

本文的主要目的是研究欧氏空间上退化椭圆型偏微分算子的全局定性性质。这些算子是欧氏空间上向量场的平方和。虽然这些运营商的本地hypoellipticity是众所周知的，全球类似物一直在深入研究以来，新千年的开始。本研究项目的重点包括但不限于以下主题。(1)建立了相应的单参数椭圆算子族在Schwartz空间和Gelfand-Shilov空间中的整体亚椭圆性。(2)获得由单参数椭圆算子族控制的偏微分方程的解的估计。(3)利用第2项中的结果推导出平方和控制的偏微分方程解的Sobolev估计。(4)明确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，但它们通常不适合于获得所需的估计值，并且肯定不是计算平方和谱的正确工具。在这个建议中采取的方法是使用傅立叶变换相对于子空间的退化的平方和的问题，以便将运营商转换为一个单参数家庭的椭圆运营商，这可以详细分析使用变种和扰动的简谐振子。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Wigner变换和伪微分算子和/或Weyl变换的估计来获得。参数化椭圆算子的详细信息，然后可以转移回正在研究的平方和。这种方法类似于理查德·费曼在量子力学中的历史总和。历史是参数化椭圆算子，它们在量子力学中是重要的，据设想，目标和基本方法将为量子场论提供新的见解。一个长期的项目是将结果推广到流形。

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.