A sequence of mollifiers $\left(\rho_n\right)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^N$ such that $$ \rho_n \in C_c^{\infty}\left(\mathbb{R}^N\right), \quad \operatorname{supp} \rho_n \subset \overline{B(0,1 / n)}, \quad \int \rho_n=1, \quad \rho_n \geq 0 \text { on } \mathbb{R}^N . $$
Notation (shift of function). We set $\left(\tau_h f\right)(x)=f(x+h), x \in \mathbb{R}^N, h \in \mathbb{R}^N$.
For $p \in [1, \infty]$, let $p' \in [1, \infty]$ be the exponent conjugate of $p$, i.e., $$ \frac{1}{p} + \frac{1}{p'} = 1. $$
I'm reading the proof of Kolmogorov–Riesz theorem in Brezis's Functional Analysis, i.e.,
Theorem 4.26 (Kolmogorov-M. Riesz-Fréchet). Let $\mathcal{F}$ be a bounded set in $L^p\left(\mathbb{R}^N\right)$ with $1 \leq p<\infty$. Assume that $$ \lim _{|h| \rightarrow 0}\left\|\tau_h f-f\right\|_p=0 \quad \text { uniformly in } f \in \mathcal{F}, $$ i.e., $\forall \varepsilon>0 \exists \delta>0$ such that $\left\|\tau_h f-f\right\|_p<\varepsilon \forall f \in \mathcal{F}, \forall h \in \mathbb{R}^N$ with $|h|<\delta$. Then the closure of $\mathcal{F}_{\mid \Omega}$ in $L^p(\Omega)$ is compact for any measurable set $\Omega \subset \mathbb{R}^N$ with finite measure. [Here $\mathcal{F}_{\mid \Omega}$ denotes the restrictions to $\Omega$ of the functions in $\mathcal{F}$.]
The first two steps of the proof are as follows, i.e.,
- Step 1: We claim that $$ \left\|\left(\rho_n \star f\right)-f\right\|_{L^p\left(\mathbb{R}^N\right)} \leq \varepsilon \quad \forall f \in \mathcal{F}, \quad \forall n>1 / \delta . \tag{23} $$ Indeed, we have $$ \begin{aligned} \left|\left(\rho_n \star f\right)(x)-f(x)\right| & \leq \int|f(x-y)-f(x)| \rho_n(y) d y \\ & \leq\left[\int|f(x-y)-f(x)|^p \rho_n(y) d y\right]^{1 / p} \end{aligned} $$ by Hölder's inequality. Thus we obtain $$ \begin{aligned} \int\left|\left(\rho_n \star f\right)(x)-f(x)\right|^p d x & \leq \iint|f(x-y)-f(x)|^p \rho_n(y) d x d y \\ & =\int_{B(0,1 / n)} \rho_n(y) d y \int|f(x-y)-f(x)|^p d x \leq \varepsilon^p, \end{aligned} $$ provided $1 / n<\delta$.
- Step 2: We claim that $$ \left\|\rho_n \star f\right\|_{L^{\infty}\left(\mathbb{R}^N\right)} \leq C_n\|f\|_{L^p\left(\mathbb{R}^N\right)} \quad \forall f \in \mathcal{F} \tag{24} $$ and $$ \begin{array}{r} \left|\left(\rho_n \star f\right)\left(x_1\right)-\left(\rho_n \star f\right)\left(x_2\right)\right| \leq C_n\|f\|_p\left|x_1-x_2\right| \\ \forall f \in \mathcal{F}, \quad \forall x_1, x_2 \in \mathbb{R}^N \end{array} \tag{25} $$ where $C_n$ depends only on $n$. Inequality $(24)$ follows from Hölder's inequality with $C_n=\left\|\rho_n\right\|_{p^{\prime}}$. On the other hand, we have $\nabla\left(\rho_n \star f\right)=\left(\nabla \rho_n\right) \star f$ and therefore $$ \left\|\nabla\left(\rho_n \star f\right)\right\|_{L^{\infty}\left(\mathbb{R}^N\right)} \leq\left\|\nabla \rho_n\right\|_{L^{p^{\prime}}\left(\mathbb{R}^N\right)}\|f\|_{L^p\left(\mathbb{R}^N\right)} $$ Thus we obtain $(25)$ with $C_n=\left\|\nabla \rho_n\right\|_{L^{p^{\prime}}\left(\mathbb{R}^N\right)}$.
My attempt We have $$ \begin{aligned} \|\rho_n \star f\|_{L^p (\mathbb R^N)} &\le \bigg\{ \int \bigg [ \int |f(x-y)| \rho_n (y) dy \bigg ]^p dx \bigg \}^{1/p}. \end{aligned} $$
By Hölder's inequality, $$ \begin{aligned} \int |f(x-y)| \rho_n (y) dy &\le \bigg [ \int |f(x-y)|^p dy \bigg ]^{1/p} \bigg [ \int |\rho_n (y)|^{p'} dy \bigg ]^{1/p'} \\ &= \|f\|_{L^p (\mathbb R^N)} \|\rho_n\|_{p'}. \end{aligned} $$
So $$ \|\rho_n \star f\|_{L^p (\mathbb R^N)} \le \bigg [ \int \|f\|^p_{L^p (\mathbb R^N)} \|\rho_n\|^p_{p'} dx \bigg ]^{1/p} = \infty. $$
Could you elaborate how the author obtains the inequality $(24)$, i.e., $$ \left\|\rho_n \star f\right\|_{L^{\infty}\left(\mathbb{R}^N\right)} \leq C_n\|f\|_{L^p\left(\mathbb{R}^N\right)} $$ ?
It’s a very obvious estimate with Holder: for each $x\in\Bbb{R}^n$, \begin{align} |(\rho_n*f)(x)|&=\left|\int_{\Bbb{R}^n}f(x-y)\rho_n(y)\,dy\right|\\ &\leq \int_{\Bbb{R}^n}|f(x-y)|\cdot|\rho_n(y)|\,dy\\ &\leq \|y\mapsto f(x-y)\|_{L^p}\cdot\|\rho_n\|_{L^{p’}}\tag{Holder}\\ &=\|f\|_{L^p}\cdot\|\rho_n\|_{L^{p’}}, \end{align} where the last line used of course translation-invariance of the Lebesgue-measure. Since this is true for all $x$, it follows $\|\rho_n*f\|_{L^{\infty}}\leq \|\rho_n\|_{L^{p’}}\cdot\|f\|_{L^p}$.
By the way, this is nothing but (part of) Young’s inequality for convolution, showing that it is a bounded bilinear map $L^p\times L^{p’}\to L^{\infty}$.