I need some help with this exercise:
Let $f \in L^p(\Bbb R)$ and let $g$ be a continuous function with compact support. Prove that:
a) If $p=1$ or $p=+\infty$, then $f*g$ is continuous.
b) If $1\lt p \lt+\infty$, then $f*g$ is continous.
I have no idea how to solve this, but I know that if $1\le p \lt +\infty$ then the set of continuous functions with compact support is dense in $L^p(\Bbb R)$ so I guess that first I can suppose that $f$ is continuous with compact support and then I'll have to use this density property.
Also, given that the exercise is separated in two parts, I imagine that the solution is different in each case, is that right?
Best Answer
Here are the ideas. Try to work out the details on your own. It is enough to assume that $1\leq p <\infty$. By H\"older's inequality and translation invariance of Lebesgue in $\mathbb{R}^n$ \begin{aligned} |(f*g)(x+h)-(f*g)(x+k)|&\leq \int|(f(x+h-y)-f(x+k-y)||g(y)|\,dy\\ &\leq \|\tau_{-(k-h)}f-f\|_p\|g\|_q. \end{aligned} where $\tau_h$ is the translation operator $\tau_hf(x)=f(x-h)$. The result follows once you show that $\|\tau_hf-f\|_p\rightarrow0$ as $h\rightarrow0$. This is typically done by using approximations with continuous functions of compact support.
If $1<p<\infty$, by using continuous functions of compact support you can actually prove that $f*g\in\mathcal{C}_0(\mathbb{R}^n)$.