  We investigate the problem of the equivalence of $L^q$-Sobolev norms in
Malliavin spaces for $q\in [1,\infty)$, focusing on the graph norm of the
$k$-th Malliavin derivative operator and the full Sobolev norm involving all
derivatives up to order $k$, where $k$ is any positive integer. The case $q=1$
in the infinite-dimensional setting is challenging, since at such extreme the
standard approach involving Meyer's inequalities fails. In this direction, we
are able to establish the mentioned equivalence for $q=1$ and $k=2$ relying on
a vector-valued Poincar\'e inequality that we prove independently and that
turns out to be new at this level of generality, while for $q=1$ and $k>2$ the
equivalence issue remains open, even if we obtain some functional estimates of
independent interest. With our argument (that also resorts to the Wiener chaos)
we are able to recover the case $q\in (1,\infty)$ in the infinite-dimensional
setting; the latter is known since the eighties, however our proof is more
direct than those existing in the literature, and allows to give explicit
bounds on all the multiplying constants involved in the functional
inequalities. Finally, we also deal with the finite-dimensional case for all
$q\in [1,\infty)$ (where the equivalence, without explicit constants, follows
from standard compactness arguments): our proof in such setting is much
simpler, relying on Gaussian integration-by-parts formulas and an adaptation of
Sobolev inequalities in Euclidean spaces, and it provides again quantitative
bounds on the multiplying constants, which however blow up when the dimension
diverges to $\infty$ (whence the need for a different approach in the
infinite-dimensional setting).
