## $H^1$ and BMO for certain locally doubling metric measure spaces of finite measure

### Volume 118 / 2010

#### Abstract

In a previous paper the authors developed an $H^1\hbox{-BMO}$ 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 $(\mathbb R^d,\rho_\varphi, \mu_\varphi)$, where ${\rm d}\mu_\varphi = {\rm e}^{-\varphi}\,{\rm d} x$ and $\rho_\varphi$ is the Riemannian metric corresponding to the length element ${\rm d} s^2=(1+|\nabla\varphi|)^2 ({\rm d} x_1^2 +\cdots+{\rm d} x_d^2)$. This generalizes previous work of the last two authors for the Gauss space.