Elliptic pseudodifferential operator estimate - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T06:40:00Z http://mathoverflow.net/feeds/question/82444 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82444/elliptic-pseudodifferential-operator-estimate Elliptic pseudodifferential operator estimate Alex A 2011-12-02T10:36:09Z 2011-12-02T11:39:52Z <p>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 </p> <p><code>$\|u\|_{H^1(U)} \le C (\|Pu\|_{L^2(W)} + \|u\|_{L^2(W)}), \quad (C&gt;0 \text{ a constant})$</code></p> <p>for any $\overline{U}\subset W\subset \mathbb{R}^n$ with $W$ bounded. Say I'm only interested in an upper bound of <code>$\|u\|_{L^2(U)}$</code>. Then the term $\|u\|_{L^2(W)}$ on the right hand side above feels a bit redundant. Can I obtain </p> <p><code>$\|u\|_{L^2(U)} \le C\|Pu\|_{L^2(W)} \quad \text{?}$</code> </p> http://mathoverflow.net/questions/82444/elliptic-pseudodifferential-operator-estimate/82449#82449 Answer by Orbicular for Elliptic pseudodifferential operator estimate Orbicular 2011-12-02T11:39:52Z 2011-12-02T11:39:52Z <p>The answer to your question is no. Take any non-injective operator $P.$</p>