  We first consider two types of localizations of singular integral operators
of convolution type, and show, under mild decay and smoothness conditions on
the auxiliary functions, that their boundedness on the local Hardy space
$h^1(\mathbb{R}^n)$ is equivalent. We then study the boundedness on
$h^1(\mathbb{R}^n)$ of the commutator $[b,T]$ of an inhomogeneous singular
integral operator with $b$ in $bmo(\mathbb{R}^n)$, the nonhomogeneous space of
functions of bounded mean oscillation. We define local analogues of the atomic
space $H^1_b(\mathbb{R}^n)$ introduced by P\'erez in the case of the
homogeneous Hardy space and $BMO$, including a variation involving atoms with
approximate cancellation conditions. For such an atom $a$, we prove
integrability of the associated commutator maximal function and of $[b,T](a)$.
For $b$ in $lmo(\mathbb{R}^n)$, this gives $h^1$ to $L^1$ boundedness of
$[b,T]$. Finally, under additional approximate cancellation conditions on $T$,
we show boundedness to $h^1$.
