H1 and BMO for certain locally doubling metric measure spaces

Suppose that (M,ρ,μ) is a metric measure space, which possesses two "geometric" properties, called "isoperimetric" property and approximate midpoint property, and that the measure μ is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure μ is not globally doubling. In this paper we define an atomic Hardy space H1(μ), where atoms are supported only on "small balls", and a corresponding space BMO(μ) of functions of "bounded mean oscillation", where the control is only on the oscillation over small balls. We prove that BMO(μ) is the dual of H1(μ) and that an inequality of John-Nirenberg type on small balls holds for functions in BMO(μ). Furthermore, we show that the Lp(μ) spaces are intermediate spaces between H1(μ) and BMO(μ), 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 H1(μ)-L1(μ) and L∞(μ)-BMO(μ) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L1 (μ) and on L ∞(μ).