Let $1\leq p\leq\infty$. There exists a bounded linear operator $P:W^{1,\,p}(I)\rightarrow W^{1,\,p}(\mathbb{R})$, called an extension operator, satisfying the following properties:
(i) $Pu|_{I}=u\quad\forall u\in W^{1,\,p}(I)$,
(ii) $\|Pu\|_{L^{p}(\mathbb{R})}\leq C\|u\|_{L^{p}(I)}\quad\forall u\in W^{1,\,p}(I)$,
(iii) $\|Pu\|_{W^{1,p}(\mathbb{R})}\leq C\|u\|_{W^{1,\,p}(I)}\quad\forall u\in W^{1,\,p}(I)$,
where $C$ depends only on $|I|\leq\infty$.
I understand the majority of the proof except for finding the constants $C$. The proof goes as follows:
Let $I=(0,1)$ we show that extension by reflection, \begin{align} (Pu)(x)=u^{*}(x)=\begin{cases} u(x)\quad &x\geq 0\\ u(-x)\quad &x<0 \end{cases} \end{align} works. We have, \begin{align} \|u^{*}\|_{p}^{p}=\int_{\mathbb{R}}|u^{*}|^{p}dx&=\int_{-\infty}^{0}|u^{*}|^{p}dx+\int_{0}^{\infty}|u^{*}|^{p}dx\\ &=\int_{-\infty}^{0}|u(-x)|^{p}dx+\int_{0}^{\infty}|u(x)|^{p}dx\\ &=\int_{0}^{\infty}|u(x)|^{p}dx+\int_{0}^{\infty}|u(x)|^{p}dx\\ &=2\int_{0}^{\infty}|u(x)|^{p}dx=2\|u\|_{p}^{p}, \end{align} which implies $\|u^{*}\|_{p}\leq 2^{1/p}\|u\|_{p}$.
However something must be going wrong with my working because the correct inequality is $\|u^{*}\|_{p}\leq 2\|u\|_{p}$. Can anyone identify where I am going wrong? Thank you.