Back to elliptic regularity. We have a constant-coefficient partial differential operator which is elliptic, i.e. the polynomial

satisfies for large. We used this last property to find a near-fundamental solution to . That is, we chose such that , where was our arbitrary cut-off function equal to one in some neighborhood of the origin. The point of all this was that

In other words, is near the fundamental solution. So given that , we can use to “almost” obtain from by convolution —if this were exact, we’d have the fundamental solution itself.

We now want to show that isn’t all that badly behaved.

**The singular locus of the parametrix **

We are going to show that . The basic lemma we need is the following. Fix . Consider a smooth function such that, for each , there is a constant with

then this is a distribution, but it is not necessarily a Schwarz function. And cannot be expected to be one, thus. Nevertheless:

Lemma 1is regular outside the origin. (more…)