Analysis of sums of vector fields

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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.