Weighted Norm Inequalities for Spectral Multipliers without Gaussian Estimates

Abstract

Let $L$ be a nonnegative self-adjoint operator on $L^2(\mathbb{R}^n)$ satisfying the full off-diagonal estimates $L^{q_0}-L^2$ for some $q_0\in [1,2)$. In this paper, we study the sharp weighted $L^p$ estimates for the spectral multipliers of the operator $L$ and their commutators with BMO functions $b$. As an application, we study the weighted norm inequalities for spectral multipliers of Schr\"odinger operators with negative potentials.