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 $H=-\Delta +V$, where V is an arbitrarily 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 $V\in K_0$, where $K_0$ is the closure of the space of test functions in the global Kato class $K$, meaning$\underset{y\in R^3}{\sup }\underset{R^3}{\int }\frac{\left|V\right(x\left)\right|dx}{|x-y|}<\infty .$

For the spectral multiplier bounds, we assume that H has no zero or positive energy bound states. For $V\in K_0$, we prove that H has at most a finite number of negative bound states. If in addition $V\in {\dot{W}}^{-1/4,4/3}$, then by [16] and [20] there are no positive energy bound states.