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

TL;DR

This paper investigates the boundedness of localized singular integral operators and their commutators with functions in bmo and lmo spaces on local Hardy spaces, establishing equivalences and boundedness results.

Contribution

It introduces local analogues of atomic Hardy spaces and proves boundedness of commutators with bmo and lmo functions, extending classical results to localized settings.

Findings

Boundedness of localized singular integral operators on h^1 is equivalent under mild conditions.

Commutators with bmo functions are bounded from h^1 to L^1.

Additional cancellation conditions ensure boundedness from h^1 to itself.

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 $[b,T](a)$. 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 $h^1$.