Analysis of sums of vector fields

所提出的研究方案的主要目的是通过考察具体算子来研究欧氏空间上退化椭圆偏微分算子的整体定性性质。这些算子是欧氏空间上向量场的平方和。虽然这些算子的局部亚椭圆性是众所周知的，但直到新千年开始，才对全局相似算子进行了深入研究。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族的Schwartz空间和Gelfand-Shilov空间的整体亚椭圆性。(2)得到了单参数椭圆族偏微分方程解的估计。(3)利用项2中的结果推导出由平方和支配的偏微分方程解的Soblev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建议中涉及的某些平方和，但它们一般不适合于获得所需的估计，并且肯定不是计算平方和谱的正确工具。在这个方案中采取的方法是使用关于所讨论的平方和的简并子空间的傅里叶变换，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种和扰动来详细分析。使用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，可以获得关于简谐振子这些突变的非常详细的信息。然后，关于参数化椭圆算子的详细信息可以被传递回正在研究的平方和。这种方法类似于理查德·费曼的量子力学历史总和。这些历史是参数化的椭圆算符，它们在量子力学中是重要的，预计其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多种形式。

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.