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