Let $\rho \in L^1(R^N)$ with $\int_{}^{} \rho=1$ .Set $\rho_n(x)=n^N\rho(nx)$. Let $f\in L^p(R^N)$ with $1\leq p<\infty$. Prove that $\rho_n \star f \to f$ in $L^p(R^N)$.
My try:
Since $f \in L^{p}(\mathbb{R^N})$, there exists $f_1 \in C_{c}(\mathbb{R^N})$ such that $|f-f_1|_{L^{p}} \lt \epsilon$ for a given $\epsilon \gt 0$. I add and subtract $\rho_n *f_1$. The problem is to show that $\rho_n *f_1$ and $f$ can be arbitrarily small either by adding and subtracting terms or directly. I am trying to bring in mollifiers somehow since for a sequence of mollifiers $\mu_n$, I know that $\mu_n*f \to f$ uniformly as $n \to \infty$. I am unable to proceed from here.
Thanks for the help!!
You'll want to start by showing that $\|\rho\star f\|_{p}\le \|\rho\|_1\|f\|_{\rho}$, which holds for $1 \le p \le \infty$, but you don't need this for $p=\infty$. For $p=1$, this is fairly standard. If you're not sure how to estimate that, let $q$ be conjugate to $p$ so that $\frac{1}{p}+\frac{1}{q}=1$ and write $$ \left|\int \rho fdx\right| \le \int |\rho|^{\frac{1}{q}}\{|\rho|^{\frac{1}{p}}|f|\}dx $$ For any $R > 0$, \begin{align} \|(\chi_{|x|\ge R}\rho_n)\star f\|_p & \le \|\chi_{|x|\ge R}\rho_n\|_1\|f\|_p \\ & = \int_{|x|\ge R}|\rho_n|dx\|f\|_p \\ & = \int_{|x| \ge nR}|\rho|dx\|f\|_p\rightarrow 0 \mbox{ as } n\rightarrow\infty. \end{align} The remaining piece is $(\chi_{|x| \le R}\rho_n)\star f$, which can be studied for continuous $f$.