Say I have an invertible partial differential operator P(adsbygoogle = window.adsbygoogle || []).push({}); ^{1}(R^{n}) -> L^{2}(R^{n}) where H^{1}denotes the first order L^{2}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 R^{n}where V is only "slightly" larger than U (e.g. U is compactly contained in V)?

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# Local operator estimate

Can you offer guidance or do you also need help?

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