Commit 753db117 authored by Jannes Bantje's avatar Jannes Bantje

added local version of Friedrichs mollifiers

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......@@ -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 $\phi \ge 0$, $\int \phi(x) \intmathd x = 1$ 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 $\le 1$,
\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 $u \in W^s$, $F_\varepsilon u \grenzw{\varepsilon \to 0} u$ in the $W^s$-norm.
\end{enumerate}
\end{propositiondef}
\begin{proof}
See \cite[Prop.~3.4.2]{ebert_index_lec}.
\end{proof}
\begin{theorem}{Local Regularity Theorem}\label{thm:local_regularity}
Let $D$ be a differential operator of order $k$, that is elliptic over $\overline{U}$, $U \subseteq \mathbb{R}^n$ relatively compact.
Let $l,r$ be integers, $f \in W^l$, $u \in W^r$.
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