I have a problem in an exercise and I would be grateful for hints. I have to show that if $ f \in L^1(\mathbb{R}^n)$, the integral operator $$\begin{array}{rcl} T_f: L^p(\Omega)& \to& L^p(\Omega) \\ g &\mapsto& f *g \end{array}$$ is compact for all $1\leq p < \infty $, where $\Omega$ is a bounded open set in $\mathbb{R}^n$. Here, $f*g$ denotes the convolution $$(f*g)(x)=\int_{\mathbb{R}^n} f(y)g(x-y) dy.$$
My attemp: By the definition of a compact operator, we have to show, that the image of the unit ball in $L^p(\mathbb{R}^n)$ under $T_f$ is relatively compact in $L^p(\mathbb{R}^n)$. Now, by the Fréchet–Kolmogorov theorem, I have to show the equicontinuity and equitightness, i.e., for all $g \in B_1(L^p(\mathbb{R}^n))$ the following holds $$\begin{array}{rrcll} (\text{equicontinuity})& \lim_{|h|\to 0} ||\tau_hT_fg-T_fg||_{L^p(\mathbb{R}^n)}&=&0 & \text{uniformly}, \\ (\text{equitightness})&\lim_{r \to \infty} \int_{|x|>r} |T_fg|^p&=&0& \text{uniformly}. \end{array}$$ Here, $\tau_hu(x)=u(x-h)$ denotes the translation of $u$. For the first property it remains to show $$\begin{array}{rcl} \lim_{|h|\to 0} ||\tau_hT_fg-T_fg||_{L^p(\mathbb{R}^n)}&=& \lim_{|h|\to 0} \left( \int_{\mathbb{R}^n}\left|\int_{\mathbb{R}^n}f(y)(g(x-h-y)-g(x-y))\, dy\right|^p \, dx\right)^{\frac{1}{p}} \overset{?}{=} 0. \end{array}$$ But here, I am struggeling with the theory. How can I show that this integral converges to $0$ for $|h|\to 0$? And for the second property, equitightness, I am completely clueless.