H1 and BMO for certain locally doubling metric measure spaces of finite measure

In a previous paper the authors developed an HI1-BMO theory for unbounded metric measure spaces (M, rho, mu) 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, rho(phi), mu(phi)), where d mu(phi) = e(-phi) dx and rho(phi) is the Riemannian metric corresponding to the length element ds(2) = (1 + del(phi))(2)(d(x1)(2) |...| + dx(d)(2)). This generalizes previous work of the last two authors for the Gauss space.