A Proof of local elliptic regularity
We fill in the details of the proof of local elliptic regularity. We fix a differential operator of order on , precompact and elliptic over .
If has constant coefficients, i.e.
then there is an such that for all ,
Observe that we have
for some polynomial of degree at most . By ellipticity, for some and constant , we have
So in the decomposition
we can bound the first term by , and we can bound the second term by
For any fixed and , there is some neighbourhood of and such that for all , we have
be the differential operator with constant coefficients that agree with
is also an elliptic operator, and the above applies. So for any
, we have
So we have to control the term . For a tiny , pick a neighbourhood of such that the coefficients of are bounded by . Then
where are fixed, independent of and . By the next lemma, for any , we can bound
We then deduce that
Picking and to be small enough, we are done.
For any and , there exists such that
for all . Hence
The claim is the same as
Observe that for any , we always have
Then take .
For any , there exists such that
is elliptic on
, and cover
) with finitely
where we have a bound as above, say
for any supported in the 's. Now pick a partition of unity subordinate to . Then
We can bound the first two by a constant multiple of and . To bound the last term, we use that is a differential operator of order , and hence