Commit 753db117 by Jannes Bantje

### added local version of Friedrichs mollifiers

parent c5bbf7b3
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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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