Can local operator estimate be applied to subsets of Rn?

[jiku1797]

Say I have an invertible partial differential operator P¹(Rⁿ) -> L²(Rⁿ) where H¹ denotes the first order L² Sobolev space. I know

|u|_H¹(Rⁿ) ≤ |(P-z)u|_L²(Rⁿ)

for certain z. Can I somehow obtain

|u|_H¹(U) ≤ |(P-z)u|_L²(V)

for subsets U, V of Rⁿ where V is only "slightly" larger than U (e.g. U is compactly contained in V)?

 

[gtoguy97]

Yes, you can obtain this result. The result follows from the fact that the Sobolev norm is equivalent to the L2 norm on compactly contained sets. This means that there exists a constant C such that |u|H1(U) ≤ C|u|L2(U) for all u in H1(U). Moreover, since P is an invertible partial differential operator, we also have |(P-z)u|L2(V) ≤ C'|(P-z)u|L2(V) for some constant C'. Combining these two inequalities gives us the desired result.