We fill in the details of the proof of local elliptic regularity. We fix L a differential operator of order k≥1 on Rn, U⊆Rn precompact and L elliptic over Uˉ.
Lemma
If L has constant coefficients, i.e.
L=∣α∣≤k∑aαDα,
then there is an A such that for all u∈Cc∞(U),
∥u∥s+k≤A(∥u∥s+∥Lu∥s).
Proof
□
Observe that we have
Lu(ξ)=p(ξ)u^(ξ).
for some polynomial p of degree at most k. By ellipticity, for some R≫0 and constant A>0, we have
we can bound the first term by (1+R2)k∥u∥s, and we can bound the second term by
∫∣ξ∣≥R∣Lu(ξ)∣2(1+∣ξ∣2)sdξ≤∥Lu∥s.
Proof
□
Lemma
For any fixed L and x0∈U, there is some neighbourhood V⊆U of x and A>0 such that for all u∈Cc∞(V), we have
∥u∥s+k≤A(∥u∥s+∥Lu∥s).
Proof
□
Let L0 be the differential operator with constant coefficients that agree with L at x0. Then L0 is also an elliptic operator, and the above applies. So for any u, we have
∥u∥s+k≤A1(∥u∥s+∥L0u∥s)≤A′(∥u∥s+∥(L−L0)u∥s+∥Lu∥s).
So we have to control the term ∥(L−L0)u∥s. For a tiny δ≪1, pick a neighbourhood V of x0 such that the coefficients of L−L0 are bounded by δ. Then
∥(L−L0)u∥s≤δA2∥u∥s+k+A3∥u∥s+k−1,
where A2,A3 are fixed, independent of u and V. By the next lemma, for any ε>0, we can bound
We can bound the first two by a constant multiple of ∥u∥s and ∥Lu∥s. To bound the last term, we use that [L,μi] is a differential operator of order k−1, and hence