# Stein extension Theorem in Sobolev spaces involving time

Let $n>0$ and $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$. The Stein extension Theorem states that there exists a total extension operator $E$ for $\Omega$ (see Theorem 5.24 in [1]).

As a consequence, for $u \in W^{m,p}(\Omega)$, with $m,p>0$, $Eu$ belongs to $W^{m,p}(\mathbb{R}^n)$ and there exists a constant $C>0$ such that,

• $Eu(x) = u(x)$ a.e. in $\Omega$,
• $\|Eu\|_{W^{m,p}(\mathbb{R}^n)} \leq C \|u\|_{W^{m,p}(\Omega)}$.

Question: Let $T>0$. Is there a similar result for Sobolev spaces involving time, i.e. for spaces such as $W^{r,s}([0,T],W^{m,p}(\Omega))$, where $m,p,r$ and $s$ are positive integers ?

To be more specific, for $u\in W^{r,s}([0,T],W^{m,p}(\Omega))$, is there an extension operator $E$ such that $Eu$ belongs to $W^{r,s}([0,T],W^{m,p}(\mathbb{R}^n))$ and verifies,

• $Eu(x) = u(x)$ a.e. in $\Omega$,
• $\|Eu\|_{W^{r,s}([0,T],W^{m,p}(\mathbb{R}^n))} \leq C \|u\|_{W^{r,s}([0,T],W^{m,p}(\Omega))}$ ?

[1] R. A. Adams and J. J. Fournier. Sobolev spaces, volume 140. Academic press, 2003

• Look at the "time-expansion" for $E$, so $(\mathcal{E}u)(t) := E(u(t))$ with $E$ being the $W^{m,p}(\Omega) \to W^{m,p}(\mathbb{R}^n)$ extension operator. Commented Feb 23, 2018 at 13:40