$H^{\mathbf{1}}$ and $BMO$ for certain locally doubling metric measure spaces

Suppose that $(M,\rho ,\mu )$ is a metric measure space, which possesses two “geometric” properties, called “isoperimetric” property and approximate midpoint property, and that the measure $\mu$ is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure $\mu$ is not globally doubling. In this paper we define an atomic Hardy space $H^1\left(\mu \right)$, where atoms are supported only on “small balls”, and a corresponding space $BMO\left(\mu \right)$ of functions of “bounded mean oscillation”, where the control is only on the oscillation over small balls. We prove that $BMO\left(\mu \right)$ is the dual of $H^1\left(\mu \right)$ and that an inequality of John–Nirenberg type on small balls holds for functions in $BMO\left(\mu \right)$. Furthermore, we show that the $L^p\left(\mu \right)$ spaces are intermediate spaces between $H^1\left(\mu \right)$ and $BMO\left(\mu \right)$, and we develop a theory of singular integral operators acting on function spaces on $M$. Finally, we show that our theory is strong enough to give $H^1\left(\mu \right)$-$L^1\left(\mu \right)$ and $L^{\infty }\left(\mu \right)$-$BMO\left(\mu \right)$ estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on $L^1\left(\mu \right)$ and on $L^{\infty }\left(\mu \right)$.