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Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in R n with the property that sup{ T a Y : a is a (1, q)-atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H 1 (R n ) to Y . We show that the same is true if we replace (1, q)-atoms by continuous (1, ∞)-atoms. This is known to be false for (1, ∞)-atoms.

satisfy suitable Ho rmander type conditions, then the spectral operator M(L) extends to a bounded operator on L p (#) and hence on L q (#) for all q such that |1Âq&1Â2| |1Âp&1Â2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than S ,* p . 2001Academic Press

In a previous paper the authors developed a H 1 −BM O theory for unbounded metric measure spaces (M, ρ, µ) of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and "isoperimetric" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R d , ρϕ, µϕ), where dµϕ = e −ϕ dx and ρϕ is the Riemannian metric corresponding to the length element ds 2 = (1+|∇ϕ|) 2 ( dx 2 1 +· · ·+ dx 2 d ). This generalizes previous work of the last two authors for the Gauss space.2000 Mathematics Subject Classification. 42B20, 42B30, 46B70, 58C99 .