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

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


1 Answer 1


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

  • $\begingroup$ 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. $\endgroup$
    – flavio
    Dec 2, 2011 at 12:41
  • 3
    $\begingroup$ 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. $\endgroup$
    – Orbicular
    Dec 2, 2011 at 12:50

Not the answer you're looking for? Browse other questions tagged or ask your own question.