Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates
We consider an abstract non-negative self-adjoint operator $L$ acting on $L^2(X)$ which satisfies Davies–Gaffney estimates. Let $H^p_L(X)$ $(p>0)$ be the Hardy spaces associated to the operator $L$. We assume that the doubling condition holds for the metric measure space $X$. We show that a sharp Hörmander-type spectral multiplier theorem on $H^p_L(X)$ follows from restriction-type estimates and Davies–Gaffney estimates. We also establish a sharp result for the boundedness of Bochner–Riesz means on $H^p_L(X)$.