If there is a bounded linear extension $W^{1,p}(Λ)\to W^{1,p}(ℝ^d)$, do the Sobolev inequalities for $W^{k,p}(ℝ^d)$ transfer over to $W^{k,p}(Λ)$?

Question

Let

• $d\in\mathbb N$
• $\Lambda\subseteq\mathbb R^d$ be open
• $k\in\mathbb N$
• $p\in[1,\infty)$

We say that $\Lambda$ has the $(k,p)$-extension property $:\Leftrightarrow$ There is a bounded linear operator $W^{k,\:p}(\Lambda)\to W^{k,\:p}(\mathbb R^d)$ with $$\left.Eu\right|_\Lambda=u\;\;\;\text{for all }u\in W^{k,\:p}(\Lambda)\;.\tag1$$ If $\Lambda$ has the $(k,p)$-extension property, it's easy to see that the Sobolev inequalities for $W^{k,\:p}(\mathbb R^d)$ hold for $W^{k,\:p}(\Lambda)$ too.

My question is: Are we able to show that the Sobolev inequalities hold for $W^{k,\:p}(\Lambda)$, even when $\Lambda$ only has the $(1,p)$-extension property?