Is it true that $\|f_n \ast g_n\|_{L^r(\mathbb{R}^{n})} \to \|f \ast g\|_{L^r(\mathbb{R}^{n})}$ as $n \to \infty$? (Young's inequality)

Question

I'm trying to prove the following Corollary from Trèves book (page 280):

Corollary 1: If $f \in L^p$, $g \in L^q$, then $$\int_{\mathbb{R}^{n}} f(x-y)g(y)dy$$ defines an element of $L^r$ $(1/r=1/p+1/q-1)$ $(1 \leq r<\infty)$, denoted by $f \ast g$; we have $$\|f \ast g \|_{L^r}\leq \|f\|_{L^p}\|g\|_{L^q}.$$

The tip given in the book is to use the density of $C_{c}^{0}(\mathbb{R}^n)$ in $L^\alpha$ $(1 \leq \alpha<\infty)$. So, my attempt was as follows:

For all $f \in L^p$, $g \in L^q$ there exists $(f_n) \subset C_c^0(\mathbb{R}^{n})$, $(g_n) \subset C_c^0(\mathbb{R}^{n})$ such that $f_n \to f$ in $L^p$ and $g_n \to g$ in $L^q$. From Theorem 26.1 we have $$\|f_n \ast g_n \|_{L^r}\leq \|f_n\|_{L^p}\|g_n\|_{L^q}.$$ It's easy to see that the RHS of above inequality converges to $$\|f\|_{L^p}\|g\|_{L^q}.$$ So, it remains to prove that $$\|f_n \ast g_n \|_{L^r} \to \|f \ast g \|_{L^r} \hbox{ as } n \to \infty.$$ That was my difficulty.

We have that $$|(f_n*g_n)(x) - (f*g)(x)| \leq \int_{\mathbb{R}^n} |f_n(x-y)-f(x-y)||g_n(y) |dy\\ + \int_{\mathbb{R}^n} |f(x-y)| |g_n(y) -g(y)|dy $$ Then by Holder's inequality, \begin{align*} |(f_n*g_n)(x) - (f*g)(x)|&\leq \left(\int_{\mathbb{R}^n} |f_n(x-y)-f(x-y)|^pdy\right)^{1/p} \|g_n\|_q\\&\quad+ \left(\int_{\mathbb{R}^n} |f(x-y)|^pdy\right)^{1/p} \|g_n-g\|_q\\ &= \|f_n -f \|_{L^p}\|g_n\|_{L^q}+\|f\|_{L^p}\|g_n-g\|_{L^q}. \end{align*}

Therefore, $$|f_n \ast g_n(x)|^r \rightarrow |f \ast g(x)|^r.$$ By Fatou's Lemma we obtain $$\int_{\mathbb{R}^{n}}|f \ast g(x)|^r dx \leq \liminf \int_{\mathbb{R}^n}|f_n \ast g_n (x)|^r dx \leq \lim_{n \rightarrow \infty} \left(\int_{\mathbb{R}^{n}} |f_n(x)|^p \right)^{\frac{r}{p}}\left(\int_{\mathbb{R}^{n}} |g_n(x)|^q \right)^{\frac{r}{q}}=\left(\int_{\mathbb{R}^{n}} |f(x)|^p \right)^{\frac{r}{p}}\left(\int_{\mathbb{R}^{n}} |g(x)|^q \right)^{\frac{r}{q}}.$$

Answer 1

I think that I can complete your idea, multiplying both side of your last inequality by $1_{B_N(0)}(x)$ and taking limit after integrating you get that $(f_n*g_n)1_{B_N(0)}\to(f*g)1_{B_N(0)}$ in $L^r$ (the same you have with $f_n,g_n$ instead of $f,g$ but the argument for this is DCT), then again by DCT $(f*g)1_{B_N(0)}\to (f*g)$ as $N\to\infty$. Then we have that $$||(f*g)-(f_n*g_n)||\leq||(f*g)- (f*g)1_{B_N(0)} ||+||(f*g)1_{B_N(0)}-(f_n*g_n)1_{B_N(0)} ||+||(f_n*g_n)1_{B_N(0)}-(f_n*g_n)||$$ And making first $n\to\infty$ and then $N\to\infty$ you get what you wanted.