The Hardy Space $H^1$ on Non-homogeneous Metric Spaces

Abstract

Let $(\mathcal{X},d,\mu )$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu )$ and prove that its dual space is the known space $\mathrm{RBMO}(\mu )$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu )$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón--Zygmund operators from $H^1(\mu )$ to $L^1(\mu )$.

This paper has been withdrawn by the authors, since it has already been published

References in corpus (1)

• Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces

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• Spaces of Type BLO on Non-homogeneous Metric Measure Spaces
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• Non-homogeneous T1 theorem for bi-parameter singular integrals
• An Interpolation Theorem for Sublinear Operators on Non-homogeneous Metric Measure Spaces
• Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function
• Multilinear fractional integral operators on non-homogeneous metric measure spaces
• Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type
• Boundedness of fractional integral operators on non-homogeneous metric measure spaces
• Commutators of multilinear singular integral operators on non-homogeneous metric measure spaces