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

TL;DR

This paper establishes sharp spectral multiplier theorems for Hardy spaces linked to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, under doubling conditions, with applications to Bochner-Riesz means.

Contribution

It proves a sharp H"ormander-type spectral multiplier theorem for Hardy spaces associated with such operators, extending previous results.

Findings

Sharp spectral multiplier theorem for Hardy spaces

Restriction type estimates derived from Davies-Gaffney estimates

Results on boundedness of Bochner-Riesz means

Abstract

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 $H^p_L(X)$. We assume that doubling condition holds for the metric measure space $X$. We show that a sharp H\"ormander-type spectral multiplier theorem on $H^p_L(X)$ follows from restriction type estimates and the Davies-Gaffney estimates. We also describe the sharp result for the boundedness of Bochner-Riesz means on $H^p_L(X)$.