@@ -282,6 +282,24 @@ In particular one proves, that this multiplication operator extends to a bounded

This is \cite[Prop.~3.3.3]{ebert_index_lec}.

The proof of Gårding's inequality is performed in three stages: firstly it is proven for differential operators with contants coefficients, then extended to variable coefficients with small support and from that the general case is deduced.\todo{how does the ellipticity come in?}

Another important tool, which will be useful in other places as well, are \Index{Friedrichs' mollifiers}:

Let $\phi\in C^\infty_c(\mathbb{R}^n)$ be a function with $\phi\ge0$, $\int\phi(x)\intmathd x =1$ and $\phi(-x)=\phi(x)$.

For $\varepsilon>0$ we let $\phi_\varepsilon(x)=\frac{1}{\varepsilon^n}\phi\enbrace*{\frac{x}{\varepsilon}}$.

\begin{propositiondef}

The \Index{Friedrichs' mollifier} is the operator $F_\varepsilon\colon\Schwartz\to\Schwartz$, $u \mapsto\phi_\varepsilon* u$.

It has the following properties:

\begin{enumerate}[(i)]

\item$F_\varepsilon$ extends to a bounded operator $W^s \to W^s$ with operator norm $\le1$,

\item$F_e$ commutes with all differential operators with constant coefficients,

\item for each $u \in W^s$, $F_\varepsilon u$ is in $C^\infty\cap W^s$

\item for each $u \in W^s$, $F_\varepsilon u \grenzw{\varepsilon\to0} u$ in the $W^s$-norm.