Let $I = (a, b)$ be a (possibly unbounded) open interval of $\mathbb R$. I would like to verify Remark 16(ii) at page 218 in Brezis' Functional Analysis
Let $p \in (1, \infty)$. Then $u \in W^{1, p}_0 (I)$ IFF there is a constant $C>0$ such that $$ \left | \int_I u \varphi' \right | \le C \| \varphi\|_{L^{p'} (I)} \quad \forall \varphi \in C^1_c (\mathbb R). $$
Above,
- $W^{1, p}_0 (I)$ is the closure of $C^1_c (I)$ in $W^{1, p} (I)$.
- $p'$ the Hölder conjugate of $p$.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
We need two results from the same book
- Proposition 8.3 Let $u \in L^p (I)$ with $p \in (1, \infty]$. Then $u \in W^{1, p} (I)$ IFF there is a constant $C>0$ such that $$ \left|\int_I u \varphi' \right| \leq C\|\varphi\|_{L^{p'}(I)} \quad \forall \varphi \in C_c^1(I) . $$
- Remark 16(i) For $u:I \to \mathbb R$, we define $\bar u:\mathbb R \to \mathbb R$ by $$ \bar u (x) := \begin{cases} u (x) &\text{if} \quad x \in I, \\ 0 &\text{if} \quad x \in \mathbb R \setminus I. \end{cases} $$ Let $p \in [1, \infty)$. Then $u \in W^{1, p}_0 (I)$ IFF $\bar u \in W^{1, p} (\mathbb R)$.
By Remark 16(i), $u \in W^{1, p}_0 (I)$ IFF $\bar u \in W^{1, p} (\mathbb R)$. By Proposition 8.3, $\bar u \in W^{1, p} (\mathbb R)$ IFF there is a constant $C>0$ such that $$ \left| \int_\mathbb R \bar u \varphi' \right| \leq C\|\varphi\|_{L^{p'}(I)} \quad \forall \varphi \in C_c^1( \mathbb R) . $$
The proof then follows by the fact that $$ \int_\mathbb R \bar u \varphi' = \int_I u \varphi'. $$
