# When does weakly elliptic $\Rightarrow$ strongly elliptic?

While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions which imply that weakly elliptic operators are strongly elliptic?

Context: $$M$$: closed smooth manifold; $$E,F\rightarrow M$$ smooth, $$\mathbb{C}$$-vector bundles over $$M$$; $$P:\Gamma(E)\rightarrow \Gamma(F)$$.

Def. 1: $$P$$ is called weakly elliptic if, for each $$m\in M$$ and $$\xi\neq 0\in T^{\ast}_mM$$, its symbol $$\sigma( P)(m,\xi)$$ is invertible.

Def. 2: $$P$$ is called strongly elliptic if there exists a constant $$C>0$$ such that

$$(\sigma( P)(m,\xi)v,v)\geq C\|v \|^2$$

for all $$v\in E$$ and $$\|\xi\|=1$$.

So, again, I'm wondering:

$$\underline{\textbf{Question:}} \text{ Under what conditions do the above notions coincide?}$$

Or, at least some references?

• The passage from (1) to (2) is the passage from pointwise positivity of $(\sigma( P)(m,\xi)v,v)$ to having a uniform positive lower bound for it. It is possibly under the combination of continuity and compactness. I don't think there is anything else going on.
– user147263
Commented Jun 14, 2014 at 1:50
• @wordsthatendinGRY Ah, yeah. That sounds reasonable to me. Thanks! Commented Jun 14, 2014 at 2:48