Analysis of sums of vector fields

本研究计划的主要目的是研究简并椭圆型偏微分算子在欧氏空间上的整体定性性质。这些算子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，但直到新千年开始，它们的全局类似物才得到了深入的研究。本研究项目的重点包括但不限于以下主题。(1)建立相应的单参数椭圆算子族在Schwartz空间和Gelfand-Shilov空间中的全局亚椭圆性。(2)得到单参数椭圆算子族控制的偏微分方程解的估计。(3)利用第2项的结果推导出平方和控制偏微分方程解的Sobolev估计。(4)显式计算向量场平方和的谱。虽然第2项解的显式公式可用于本建议中处理的某些平方和，但它们通常不适合获得所需的估计，并且肯定不是计算平方和谱的正确工具。本建议采用的方法是对所讨论的平方和的退化子空间使用傅里叶变换，以便将算子转换为单参数椭圆算子族，可以使用简谐振子的变分和摄动详细分析。利用Hermite函数、Hermite函数的Wigner变换以及伪微分算子和/或Weyl变换的估计，可以得到关于简谐振子突变的非常详细的信息。然后，参数化椭圆算子的详细信息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。历史是参数化的椭圆算子，它们在量子力学中是重要的，设想目标和潜在的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扩展到流形。

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.