# Local operator estimate

1. Dec 7, 2011

### 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)?