Theorem in Adams Sobolev spaces book requires $u\in L^p(\Omega) \cap L^r(\Omega)$ but we only have $u \in C^\infty$ so how can theorem be applied?

Question

Let $\Omega$ be a (open) domain in $\mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1\le p < q < r$ and $u\in L^p(\Omega) \cap L^r(\Omega)$, then we have that $u\in L^q(\Omega)$ and $$ ||u||_q \le ||u||_p^\theta ||u||_r^{1-\theta}, $$

for $0 < \theta < 1$.

However in Theorem 4.19 we only have $u\in C^\infty(\Omega)$ so how can he apply Theorem 2.11?

I don't know if it makes any difference, but he also states that $u$ and all its derivatives are extended by zero outside $\Omega$ in Theorem 4.19. Is it this extension by zero that lets him have know that $u\in L^p(\Omega) \cap L^r(\Omega)$ and thus apply Theorem 2.11?

Answer 1

When Adams states that $u \in C^\infty(\Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).

He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^\infty$ functions in $L^p$ spaces.