Even reflection on the half-line

Take $k=1$ in [L1]. The moment system for $k=1$ is the single equation ${\sum }_{j=1}^1{a}_j(-j{)}^0=1$, so $a_1=1$ is the unique coefficient, and the extension operator of [L1] is $E_{1,p}u(x)=u(x)$ for $x>0$ and $E_{1,p}u(x)=a_{1}u(-x)=u(-x)$ for $x<0$; this is the even reflection $Eu(x)=u(\mathrm{\mid }x\mathrm{\mid })$. Hence $Eu\in W^{1,p}(\mathbb{R})$ for every $1\le p\le \mathrm{\infty }$, and its weak derivative satisfies $(Eu{)}^{\mathrm{\prime }}=u^{\mathrm{\prime }}$ on $(0,\mathrm{\infty })$ and $(Eu{)}^{\mathrm{\prime }}(x)=-u^{\mathrm{\prime }}(-x)$ on $(-\mathrm{\infty },0)$, as the $k=1$ instance of the reflection formula.

Finite $p$. By [L3] and step 1.1, $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{L^p(\mathbb{R})}^p=2\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(0,\mathrm{\infty })}^p$ and $\mathrm{\parallel }(Eu{)}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(\mathbb{R})}^p={\int }_0^{\mathrm{\infty }}\mathrm{\mid }{u}^{\mathrm{\prime }}(x){\mathrm{\mid }}^{p}dx+{\int }_{-\mathrm{\infty }}^0\mathrm{\mid }{u}^{\mathrm{\prime }}(-x){\mathrm{\mid }}^{p}dx=2\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(0,\mathrm{\infty })}^p$; adding the two components and using [L2] gives $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{W^{1,p}(\mathbb{R})}^p=2\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(0,\mathrm{\infty })}^p$, that is, $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{W^{1,p}(\mathbb{R})}=2^{1\mathrm{/}p}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(0,\mathrm{\infty })}$.

Case $p=\mathrm{\infty }$ and the source comparison. Both $x\mapsto \mathrm{\mid }Eu(x)\mathrm{\mid }$ and $x\mapsto \mathrm{\mid }(Eu{)}^{\mathrm{\prime }}(x)\mathrm{\mid }$ are even and agree on $(0,\mathrm{\infty })$ with $\mathrm{\mid }u\mathrm{\mid }$, respectively $\mathrm{\mid }{u}^{\mathrm{\prime }}\mathrm{\mid }$, so their essential suprema coincide with those of $u$ and $u^{\mathrm{\prime }}$; by [L2] the maximum norm is unchanged: $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}(\mathbb{R})}=\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}(0,\mathrm{\infty })}$. The displayed identity in the cited reflection example, which prints factor $2$ for $1\le p<\mathrm{\infty }$ and factor $1$ for $p=\mathrm{\infty }$, agrees with the computation at $p=1$ and $p=\mathrm{\infty }$; the correct finite-$p$ factor is the $2^{1\mathrm{/}p}$ of step 2.1.