Let $I \in \mathbb R^d$. A result I need, states that a certain property holds weakly in $BV(I)$, and holds strictly in $W^{1,1}(I)$ (which is a subspace of $BV(I)$).
I would actually need this property to hold in the Sobolev space $H^1(I)$, hence the following questions: 1) Is $H^1(I)$ a subspace of $W^{1,1}(I)$? If so, the property would hold in the strict sense.
2) If not, is it at least $H^1(I)$ a proper subspace of $BV(I)$? In this case, I'd have at least the weak property.
Does it depend on $d$? (The case $d=2$ would actually be enough in my case)
$H^1=W^{1,2}$ is a subspace of $W^{1,1}$ if and only if the domain is bounded. This is just a manifestation of the fact that $L^2$ is a subspace of $L^1$ if and only if the space has finite measure.