Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means

We establish estimates for the Poisson kernel, the heat kernel, and Bochner–Riesz means defined in terms of $H=-\mathrm{\Delta }+V$, where $V$ is a possibly large rough real-valued scalar potential and $H$ can have negative eigenvalues. All results are in three space dimensions. We eliminate several unnecessary conditions on $V$, leaving just $V\in {\mathcal{K}}_0$, meaning that $V$ is locally integrable and $(-\mathrm{\Delta }{)}^{-1}\mathrm{\mid }V\mathrm{\mid }$ is bounded. For the spectral multiplier bounds, we assume that $H$ has no zero or positive energy bound states. For $V\in {\mathcal{K}}_0$, we prove that $H$ has at most a finite number of negative bound states. If in addition $V\in {\dot{W}}^{-1\mathrm{/}4,4\mathrm{/}3}$, then by [GoSc] and [KoTa] there are no positive energy bound states.