Let $M=[0,\infty) \times S^2$. We have the regular regular Sobolev space $H^1(M)$. We also have the space $H^1\bigg([0,\infty); H^1(S^2)\bigg)$. Are those two spaces the same? Does one contain the other?
If $f \in H^1(M)$, we know that $f|_{\partial M}$ is in $H^{\frac{1}{2}}(S^2)$ by the trace theorem. If $f \in H^1\bigg([0,\infty); H^1(S^2)\bigg)$, do we get the same thing? Or is $f|_{\partial M} \in H^1(S^2)$?
