Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

TL;DR

This paper extends spectral multiplier estimates from the free Laplacian to perturbed Hamiltonians with potential, providing sharp bounds, characterizations of Hardy spaces, and Strichartz estimates for these operators.

Contribution

It introduces new bounds and characterizations for spectral multipliers and Hardy spaces associated with perturbed Hamiltonians, advancing understanding of these operators.

Findings

Sharp bounds for Mihlin multipliers on perturbed Hamiltonians

Partial confirmation of a conjecture on intertwining operators

Characterization of twisted Hardy spaces for perturbed operators

Abstract

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $H=-\Delta+V$, where $V$ is a scalar real-valued potential. Results include sharp bounds for Mihlin multipliers, partial confirmation for a conjecture made in [BeGo3] about intertwining operators, a characterization of the twisted Hardy spaces that correspond to these perturbed Hamiltonians, Strichartz estimates, and maximum principles.