Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type

Abstract

AbstractLet (𝒳, d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that μ satisfies certain estimates from below and there exists a suitable Calderón reproducing formula in L 2(𝒳), the authors establish a Lusin‐area characterization for the atomic Hardy spaces H p at(𝒳) of Coifman and Weiss for p ∈ (p 0, 1], where p 0 = n /(n + ε 1) depends on the “dimension” n of 𝒳 and the “regularity” ε 1 of the Calderón reproducing formula. Using this characterization, the authors further obtain a Littlewood–Paley g *λ ‐function characterization for Hp (𝒳) when λ > n + 2n /p and the boundedness of Calderón–Zygmund operators on Hp (𝒳). The results apply, for instance, to Ahlfors n ‐regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in ℂN . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)