Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The Hardy space H1 on non-homogeneous metric spaces


a1 Department of Mathematics and Statistics, P.O.B. 68, (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland. e-mail:

a2 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People's Republic of China. e-mail:

a3 School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China. e-mail:


Let (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

(Received September 01 2010)

(Revised September 01 2011)

(Online publication December 08 2011)


c1 Corresponding author


† Supported by the Academy of Finland (Grant Nos. 130166, 133264, 218148).

‡ Supported by the National Natural Science Foundation (Grant No. 11171027) of China and the Program for Changjiang Scholars and Innovative Research Team at the University of China.

§ Supported by the National Natural Science Foundation (Grant No. 11101339) of China.