3.1 Local elliptic regularity
The main theorem of elliptic regularity is the following:
(Local elliptic regularity)
Let be a differential operator of order on , precompact and elliptic over . Then
There is a constant such that for all , we have
If for some is such that for some , then for every .
The first part requires getting our hands dirty and proving explicit estimates. A proof sketch will be given here, with the details carried out in the Appendix.
Proof sketch of (1)
First prove it if has constant coefficients, which is easy since
for some polynomial .
Next, for arbitrary , we show that for any , there is a neighbourhood such that the estimate holds for all supported in . To do so, let be the differential operator with constant coefficients that agree with at . We then have to bound on the assumption that the coefficients of can be made arbitrarily small near the support of .
Finally, use compactness and partitions of unity to glue these results together.
The second part follows from the first formally.
Observe the first part tells us if we already knew that were in , then would be very well-behaved. This is an a priori estimate. To show that is actually in , we apply appropriate smooth approximations.
Fix a such that
For , we define
We then define the mollifier
The main theorem about mollifiers, which I will not prove, is that
extends to a bounded operator with norm . Moreover
commutes with all differential operators with constant coefficients.
If is a differential operator with compact support, then extends to a map for all , and has uniformly bounded operator norm.
For any , we have .
For any , we have in .
If is precompact and for some , and is uniformly bounded in , then .□
The last point is how we are going to show that
, while the others are needed to establish the required bounds.
Proof of (2) from (1)
Inductively, we may assume
. We then bound
where for the third term, we used that
is a differential operator of degree