# Problem from Brezis's book (mollifiers)

Any ideas on how to get started with this?

Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in $L^p(\mathbb{R}^N)$.

The proof of Theorem 4.22 would almost go through for this case. The problem is that I don't know anything about the support of the $\rho_n$'s.This seems to be crucial for the proof of Proposition 4.21 which is used to prove the theorem.

-
Could you give a link to google book's or write down the cited theorems? – martini Jun 29 '12 at 14:34
It's Exercise (4.28), on page 127, of the Springer edition. Just found it. – Siminore Jun 29 '12 at 14:39
You know that $\int_{\mathbb{R}^n\backslash B_R(0)} \rho$ becomes arbitrily small if $R$ is sufficiently large, that should suffice. I don't think this is true if $p=\infty$. – user20266 Jun 29 '12 at 14:42
You can prove that $$\text{support}(\rho_n)=\frac{1}{n}\text{support}(\rho).$$ – Mercy King Jun 29 '12 at 14:53
I don't know your approach of proof, but you should note that, unlike in the case of smooth $\rho$ with compact support, you will only get convergence in $L^p$, $1 \le p < \infty$. In particular, as pointed out by an example in Stein's book on Harmonic analysis (Chapter II, § 5.16), $\lim_{n\rightarrow\infty} \rho_n \star f(x)$ may fail to exist for almost every $x$. – user20266 Jun 29 '12 at 15:45

Define $$\bar \rho = \frac{I_{B(R)} \: \rho}{\|I_{B(R)} \: \rho\|_1},$$ where $R>0$ is such that $\int_{\mathbb{R}^N \setminus B(R)} \rho < \epsilon$. Then I write
$$\| \rho_n \star f - \bar \rho_n \star f + \bar \rho_n \star f -f \|_p \leq \| \rho_n \star f - \bar \rho_n \star f \|_p + \| \bar \rho_n \star f -f \|_p$$ $$\leq \| \rho_n - \bar \rho_n\|_1 \|f\|_p + \| \bar \rho_n \star f -f \|_p$$
The text passage is a discussion about a mathematical problem from Brezis's book on mollifiers, including a problem statement and several comments or suggestions for solving it.
no_question

the passage does not contain a direct mathematical question that requires a specific answer. instead, it is a discussion or problem statement with suggestions and comments.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Problem from Brezis's book (mollifiers)

Any ideas on how to get started with this?

Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in $L^p(\mathbb{R}^N)$.

The proof of Theorem 4.22 would almost go through for this case. The problem is that I don't know anything about the support of the $\rho_n$'s.This seems to be crucial for the proof of Proposition 4.21 which is used to prove the theorem.

-
Could you give a link to google book's or write down the cited theorems? – martini Jun 29 '12 at 14:34
It's Exercise (4.28), on page 127, of the Springer edition. Just found it. – Siminore Jun 29 '12 at 14:39
You know that $\int_{\mathbb{R}^n\backslash B_R(0)} \rho$ becomes arbitrily small if $R$ is sufficiently large, that should suffice. I don't think this is true if $p=\infty$. – user20266 Jun 29 '12 at 14:42
You can prove that $$\text{support}(\rho_n)=\frac{1}{n}\text{support}(\rho).$$ – Mercy King Jun 29 '12 at 14:53
I don't know your approach of proof, but you should note that, unlike in the case of smooth $\rho$ with compact support, you will only get convergence in $L^p$, $1 \le p < \infty$. In particular, as pointed out by an example in Stein's book on Harmonic analysis (Chapter II, § 5.16), $\lim_{n\rightarrow\infty} \rho_n \star f(x)$ may fail to exist for almost every $x$. – user20266 Jun 29 '12 at 15:45

Define $$\bar \rho = \frac{I_{B(R)} \: \rho}{\|I_{B(R)} \: \rho\|_1},$$ where $R>0$ is such that $\int_{\mathbb{R}^N \setminus B(R)} \rho < \epsilon$. Then I write
$$\| \rho_n \star f - \bar \rho_n \star f + \bar \rho_n \star f -f \|_p \leq \| \rho_n \star f - \bar \rho_n \star f \|_p + \| \bar \rho_n \star f -f \|_p$$ $$\leq \| \rho_n - \bar \rho_n\|_1 \|f\|_p + \| \bar \rho_n \star f -f \|_p$$
user
The text passage is a discussion about a mathematical problem from Brezis's book on mollifiers, including a problem statement and several comments or suggestions for solving it.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question that requires a specific answer. Instead, it is a discussion or problem statement with suggestions and comments.