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We denote by $L^p (\Omega) (1 \leq p < \infty)$ the space $L^p (X, \mu)$, where $(X, \mu)$ is the Lebesgue measure space corresponding to an open set $\Omega$ in $R^n$. Let $u \in L^p (\Omega)$, and define

$$(J_\epsilon u)(x) = \epsilon^{-n} \int_\Omega \rho \left( \frac{x-y}{\epsilon} \right) u(y) dy$$ With $\epsilon > 0$.

We call $J_\epsilon u$ a mollifier of u. Prove that

(i) $J_\epsilon u$ is in $C^\infty (R^n)$

(ii) If u vanishes outside a subset $A$ of $\Omega$, then $J_\epsilon u$ vanishes outside an $\epsilon-neighborhood$ of $A$ [that is, outside the set $\{ x ; \rho (x,A) < \epsilon\} ]$

(iii) If $K$ is a closed subset of $\Omega$ with $\rho (K, R^n - \Omega ) \geq \delta > 0$, then

$$(J_\epsilon u)(x) = \epsilon^{-n} \int_{|y-x|<\epsilon} \rho \left( \frac{x-y}{\epsilon} \right) u(y) dy = \int_{|z|<1} \rho (z) u(x- \epsilon z)dz$$

for any $x \in K$, provided $\epsilon < \delta$

It is an exercise of the book "Foundation of modern analysis, Avner Friedman" exercise (3.3.2). Could you give me tips on how to prove this? I still can't get anywhere. Thanks you in advance

Papi
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  • What is $\rho$? – Mittens Jul 06 '22 at 00:20
  • Your question is a case of what is call approximation to the identity. The posting here deals with a specific type of approximation to the identity, it also gives general bounds. – Mittens Jul 06 '22 at 00:22
  • I still can't understand how to apply it to this particular case. – Papi Jul 06 '22 at 00:51
  • Let's start with (i). The function $J_{\varepsilon} u $ is defined as a mollification of $u$ with some $\rho_{\varepsilon} \in C^{\infty}$. What do you know about interchanging the operator $\partial^{\alpha}$ and the mollification $*$, where $\alpha \in \mathbb{N}_0$? – Mike Jul 06 '22 at 12:36

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