Hp-bounds for spectral multipliers on Riemannian manifolds

Let M be a Riemannian manifold which satisfies the doubling volume property. Let Δ be the Laplace–Beltrami operator on M and m(λ), λ∈R, a multiplier satisfying the Mikhlin–Hörmander condition. We also assume that the heat kernel satisfies certain upper Gaussian estimates and we prove that there is a geometric constant p0<1, such that the spectral multiplier m(Δ) is bounded on the Hardy spaces Hp for all p∈(p0,1].