What is the importance of the density of $C_c^\infty(\mathbb R^n)$ and $L_c^\infty(\mathbb R^n)$ on $L^p(\mathbb R^n)$, for every $1 \leq p < \infty$?

Question

Consider the usual Lebesgue spaces $L^p(\mathbb R^n)$, for $1 \leqslant p < \infty$.

It is well known that both the spaces $L_c^\infty(\mathbb R^n)$ of essentially bounded functions with compact support on $\mathbb R^n$ and $C_c^\infty(\mathbb R^n)$ of infinitely differentiable functions with compact support on $\mathbb R^n$ are dense in $L^p(\mathbb R^n)$.

There are obviously many others dense subsets of $L^p(\mathbb R^n)$: for example, the space $C_c(\mathbb R^n)$ of continuous functions with compact support in $\mathbb R^n$ is also known to be dense in $L^p(\mathbb R^n)$. In this answer, the author provides a proof of a property on $L^p(\mathbb R)$ by using a density argument, using the set $C_c(\mathbb R)$. Clearly, the same proof works if we replace $C_c(\mathbb R)$ with $C_c^\infty(\mathbb R)$, for example.

Essentially, I would like to know examples of other results about $L^p(\mathbb R^n)$ spaces that can be proved using density arguments, specially using the spaces $L_c^\infty(\mathbb R^n)$ and $C_c^\infty(\mathbb R^n)$ that I mentioned above.

Thanks for any help in advance.

Answer 1

See The boundedness of classical operators on variable $L^p$ spaces.