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 1 is regular outside the origin. (more…)