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In a previous paper the authors developed a H^1-BMO theory for unboundedmetric measure spaces $(M,\rho,m)$ of infinite measure that are locallydoubling and satisfy two geometric properties, called "approximate midpoint"property and "isoperimetric" property. In this paper we develop a similartheory for spaces of finite measure. We prove that all the results that hold inthe infinite measure case have their counterparts in the finite measure case.Finally, we show that the theory applies to a class of unbounded, completeRiemannian manifolds of finite measure and to a class of metric measure spacesof the form (R^n,\rho_\phi, m_\phi), where dm_\phi=e^{-\phi} dx and \rho_\phi$is the Riemannian metric corresponding to the length element ds^2=(1+|grad\phi|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the lasttwo authors for the Gauss space.

H1and BMO for certain locally doubling metric measure spaces of finite measure

H¹and BMO for certain locally doubling metric measure spaces of finite measure