Let be a metric measure space endowed with a distance and a nonnegative Borel doubling measure . Let be a non-negative selfadjoint operator on and the spectral resolution of . For any bounded Borel function , define
Assume that the semigroup generated by satisfies the Davies-Gaffney estimates, that is, there exist constants , such that, for any open subset ,
for every with supp, .
Denote by the Hardy space associated with . The authors establish a Hörmander-type spectral multiplier theorem for on for . Precisely, let be a non-negative function on such that
If and the bounded measurable function satisfies
for some , then is bounded from to . If for all , then is bounded on for all .
They also obtain a spectral multiplier theorem on spaces with appropriate weights in the reverse Hölder class.