Recall that a function $\eta \in C_c^\infty(\mathbb{R}^n)$ is a mollifier if:
- $\operatorname{supp}(\eta) \subset \overline{B(0,1)}$;
- $\int_{\mathbb{R}^n}\eta = 1$.
If $\eta$ is any mollefier and $\epsilon > 0$ then we set $$\eta_\epsilon := \epsilon^{-n}\eta\left(\frac{x}{\epsilon}\right).$$
Then for any $L^1_{loc}(\mathbb{R}^n)$ function we can define the convolution $$(f \ast \eta_\epsilon)(x) := \epsilon^{-n}\int_{\mathbb{R}^n}f(t)\eta\left(\frac{x-t}{\epsilon}\right)dt.$$This is well defined because $\eta_\epsilon$ is smooth on a compact support, so attains a maximum and we can bound the previous function by $\max \eta_\epsilon\|f\|_1$ since $f \in L^1(K)$ for any $K \subset \mathbb{R}^n$ compact.
My discussion is based on mollefiers and the following fact:
- Let $j_\epsilon$ be a mollefier. If $f \in L^p$ with $1 \leq p < \infty$, then $f \star j_\epsilon \in L^p$ and $f \star j_\epsilon \rightarrow f$ strongly as $\epsilon \rightarrow 0$.
For a fixed open $\Omega \subset \mathbb{R}^n$ we use $C_c^\infty(\Omega)$ (which we shall denote as $D$ as the space of test functions. A distribution is nothing more than a functional on $D$. Then, two very useful tools about distributions are:
Two distributions $f$ and $g$ are equal if and only if $(f,\phi) = (g,\phi)$ for all $\phi \in D$. We say that $f = g$ in the distributional sense.
[The fundamental lemma of the Calculus of Variations] Suppose $\Omega \subset \mathbb{R}^n$ is open and $f \in L_{loc}^p(\Omega)$ is such that $$\int_\Omega f\phi dx = 0, \ \ \forall \phi \in C^\infty_c.$$Then $f = 0$ a.e on $\Omega$.
Since both statements consider all smooth functions with compact support, it is natural to use mollefiers. I found one proof for each statement that uses them but they start with something that I am trying to understand.
From the proof of 1: For $m = 1,2,\dots$ let $\Omega_m$ be the set of points $x \in \Omega$ such that $x + y \in \Omega$ whenever $|y| \leq 1/m$. He then defines the mollefier has: Let $j \in C^\infty_c$ with support in the unit ball and with $\int_{\mathbb{R}^n}j = 1$. Define $j_m(x) = m^nj(mx)$.
For the second one, the author considers a positive symmetric mollifier, that is a mollifier such that $\eta(x) \geq 0$ for all $x$ and $\eta(x) = \xi(|x|)$ for some $\xi : [0,\infty) \rightarrow \mathbb{R}$.
From the proof of 2: Let $\eta$ be a symmetric positive mollifier and fix $U \subset \subset \Omega$. If $\epsilon > 0$ is small enough , then for any $x \in U$ , $\eta_\epsilon(x - \cdot) \in C^\infty$ has compact support in $B(x,\epsilon)$.
It looks like both are trying to construct the mollifier in a way that is defined in a unit ball but I am not sure if that is what's happening (namely in the 1st proof). Are they in some way doing a symmetrisation of $\Omega$ to get be able to construct such a mollifier?