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

We consider the abstract non-negative self-adjoint operator $L$ acting on $L^2(X)$ which satisfies Davies-Gaffney estimates and the corresponding Hardy spaces $H_L^p(X)$. We assume that doubling condition holds for the metric measure space $X$. We show that a sharp H"ormander-type spectral multiplier theorem on $H_L^p(X)$ follows from restriction type estimates and the Davies-Gaffney estimates. We also describe the sharp result for the boundedness of Bochner-Riesz means on $H_L^p(X)$.