h¹ boundedness of Localized Operators and Commutators with bmo and lmo

2023/02/01 by Galia Dafni, Dafni, Galia, Chun Ho Lau +1
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2302.00542

openalex publication_date 2023/02/01 · openalex created_date 2023/02/04 · openalex updated_date 2026/07/28

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 h¹(ℝⁿ) is equivalent. We then study the boundedness on h¹(ℝⁿ) of the commutator [b,T] of an inhomogeneous singular integral operator with b in bmo(ℝⁿ), the nonhomogeneous space of functions of bounded mean oscillation. We define local analogues of the atomic space H¹_b(ℝⁿ) introduced by Pérez 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(ℝⁿ), this gives h¹ to L¹ boundedness of [b,T]. Finally, under additional approximate cancellation conditions on T, we show boundedness to h¹.

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