$h^1$ boundedness of Localized Operators and Commutators with bmo and lmo

Abstract: 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 $b,T$. 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 $h^1$.