Up until now I have assiduously avoided the use of many of the modern analytic techniques which have become essential tools for experts working in partial differential equations. In particular, nearly all my reasoning has been based on the minimum principle, and I have made no use so far of either Sobolev spaces or the theory of pseudodifferential operators. In this concluding chapter, I will attempt to correct this omission by first giving a brief review of the basic theories of Sobolev spaces and pseudodifferential operators and then applying them to derive significant extensions of the sort of hypoellipticity results proved in §3.4.

Because the approach taken in this chapter is such a dramatic departure from what has come before, it may be helpful to explain the origins of the analysis which follows and of the goals toward which it is directed. For this purpose, consider the Laplace operator Δ for ℝN, and ask yourself what you can say about u on the basis of information about Δu. In particular, what can you say about ∂i∂ju? One of the most bedeviling facts with which one has to contend is that (except in one dimension) u need not be twice continuous differentiable just because Δu ∈ C(ℝN;ℂ) in the sense of Schwartz distributions. On the other hand, if one replaces continuity by integrability, this uncomfortable fact disappears.