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## Elliptic pseudodifferential operator estimate [closed]

If $P$ is an elliptic pseudodifferential operator of order 1 in the sense that its principal symbol is invertible, then we have the a priori estimate

$\|u\|_{H^1(U)} \le C (\|Pu\|_{L^2(W)} + \|u\|_{L^2(W)}), \quad (C>0 \text{ a constant})$

for any $\overline{U}\subset W\subset \mathbb{R}^n$ with $W$ bounded. Say I'm only interested in an upper bound of $\|u\|_{L^2(U)}$. Then the term $\|u\|_{L^2(W)}$ on the right hand side above feels a bit redundant. Can I obtain

$\|u\|_{L^2(U)} \le C\|Pu\|_{L^2(W)} \quad \text{?}$

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## closed as no longer relevant by Ryan Budney, Willie Wong, Andy Putman, Deane Yang, Bill JohnsonDec 2 2011 at 17:32

The answer to your question is no. Take any non-injective operator $P.$

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I forgot to mention I work semi-classically, so for a small parameter elliptic operators are injective. Anyway, I think I solved the problem. Thank you. – Alex A Dec 2 2011 at 12:41
If your operator is injective then you can even estimate the H^1 -norm of u in terms of the L^2-norm of u. This is true because any injective operator with closed image satisfies an injectivity estimate. – Orbicular Dec 2 2011 at 12:50