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

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

2010-08-23

arxiv.org/abs/1008.3831v1

Mathematics

Classical Analysis and ODEs

24 pages, submitted

Scientific paper

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 ${\rm RBMO}(\mu)$ 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 Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$.

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