Why the properties of $W^{m,p}(\Omega)$ cannot be extended to $W^{m,p}(\mathbb{R}^d)$ The text books on the topic of Sobolev Spaces and PDE etc., they treate the case $W^{m,p}(\Omega)$ with $\Omega\subset \mathbb{R}^d$ and $\Omega = \mathbb{R}^d \text{ or }\mathbb{R}^d_+$ separately.
And some results cannot be extended from the first case to the latter one.
Could anyone summarize the reasons why the extension is not generally valid?
I always find it difficult to wrap up the main result of big theorems after going through many proofs of lemmas, propositions.

One of the reasons could be that you should be carefull when you deal with extentions of functions in Sobolev spaces: consider $f(x) = \chi_{(0,1)}(x)$. $f \in W^{1,1}((0,1))$ for example, but $f \notin W^{1,1}(\mathbb{R})$ because it has two jump discontinuities.
