PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 10, Pages 3181–3189 S 0002-9939(03)06868-0 Article electronically published on February 14, 2003 HARDY SPACES OF SPACES OF HOMOGENEOUS TYPE XUAN THINH DUONG AND LIXIN YAN (Communicated by Andreas Seeger) Abstract. Let X be a space of homogeneous type, and L be the generator of a semigroup with Gaussian kernel bounds on L 2 (X). We define the Hardy spaces H p s (X) of X for a range of p, by means of area integral function associated with the Poisson semigroup of L, which is proved to coincide with the usual atomic Hardy spaces H p at (X) on spaces of homogeneous type. 1. Introduction We begin by recalling the definitions necessary for introducing Hardy spaces on spaces of homogeneous type. A quasi-metric d on a set X is a function d : X × X → [0, ∞) satisfying: (i) d(x,y) = 0 if and only if x = y; (ii) d(x,y)= d(y,x) for all x,y ∈ X ; (iii) there exists a constant A< ∞ such that for all x,y, and z ∈ X , d(x,y) ≤ A(d(x,z )+ d(z,y)). Any quasi-metric defines a topology, for which the balls B(x,r)= {y ∈ X : d(y,x) <r} form a base. However, the balls themselves need not be open when A> 1. Definition 1.1 ([CW]). A space of homogeneous type (X,d,µ) is a set together with a quasi-metric d and a nonnegative measure µ on X such that µ(B(x,r)) < ∞ for all x ∈ X and all r> 0, and there exists A ′ < ∞ such that for all x ∈ X and all r> 0, µ(B(x, 2r)) ≤ A ′ µ(B(x,r)). Here µ is assumed to be defined on a σ-algebra which contains all Borel sets and all balls B(x,r). Received by the editors January 24, 2002 and, in revised form, May 16, 2002. 2000 Mathematics Subject Classification. Primary 42B20, 42B30, 47G10. Key words and phrases. Spaces of homogeneous type, Hardyspaces, semigroup, Calder´on-type reproducing formula, atomic decomposition. Both authors were partially supported by a grant from Australia Research Council, and the second author was also partially supported by the NSF of China. c 2003 American Mathematical Society 3181 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use