Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Let (X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a non-negative self-adjoint operator on L²(X). Assume that the semigroup e-tL generated by L satisfies the Davies-Gaffney estimates. Let HLᵖ(X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier theorem for L on HLᵖ(X) for 0 < p <∞: the operator $m(L)$ is bounded from HLᵖ(X) to HLᵖ(X) if the function m possesses s derivatives with suitable bounds and $s > n(1/p - 1/2)$ where n is the "dimension" of X. By interpolation, $m(L)$ is bounded on HLᵖ(X) for all 0 < p < ∞ if m is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on Lᵖ spaces with appropriate weights in the reverse Hölder class.