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

TL;DR

This paper establishes spectral multiplier estimates for elliptic and parabolic operators with rough potentials in three dimensions, focusing on Bochner-Riesz means and kernel bounds, under minimal conditions on the potential.

Contribution

It proves new spectral multiplier bounds and kernel estimates for operators with rough potentials, relaxing previous conditions and handling negative eigenvalues.

Findings

Finite negative bound states for $H$ with $V \

No positive energy bound states under additional regularity conditions

Spectral multiplier bounds for rough potentials

Abstract

We establish estimates for the Poisson kernel, the heat kernel, and Bochner--Riesz means defined in terms of $H=-\Delta+V$, 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 $(-\Delta)^{-1}|V|$ 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 $V \in \dot W^{-1/4, 4/3}$, then by [GoSc] and [KoTa] there are no positive energy bound states.