I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution.
Some context, I am working with weighted Sobolev spaces of the form $W^{m,2}(I,\omega)$ were $m=1,2$, $I$ is an interval of the form $(0,a)$ and the weight function $\omega$ is given by $\lbrace 1, x^{2},x^{2} \rbrace$, so in the case of $W^{2,2}(I,\omega)$ we have the norm
$ \Vert u \Vert_{W^{2,2}(I,\omega)} = \int_{I} \vert u \vert^{2} + \int_{I} x^{2} \vert D^{(1)}[u] \vert^{2} + \int_{I} x^{2} \vert D^{(2)}[u] \vert^{2} .$
I am trying to determine if there is a difference bwteen the spaces $W^{m,2}(I,\omega)$ and $W^{m,2}(\bar{I},\omega)$ and I realised that I cannot determine thisven for the unweighted case of $W^{m,2}(I)$ and $W^{m,2}(\bar{I})$. I realise for more complecated domains these spaces are not equivalent, however for this simple case I cannot seem to find a counter example.
Clearly the norms are equivalent, but due to the definition of the weak derivative $D^{(i)}$, it seems as though the functions in the two spaces may differ. Since we have that $C_{0}^{\infty}(I) \subset C_{0}^{\infty}(\bar{I})$, we should have that $W^{m,2}(\bar{I})$ is embedded in $W^{m,2}(I)$, but that is as far as I can get.
