Bounds on the maximal Bochner-Riesz means for elliptic operators

TL;DR

This paper establishes sharp $L^p$ bounds for maximal Bochner-Riesz means of elliptic operators, extending results to Schrödinger operators on manifolds, harmonic oscillators, and their perturbations.

Contribution

It provides the first sharp $L^p$ boundedness results for maximal Bochner-Riesz means for a broad class of elliptic operators under finite speed propagation assumptions.

Findings

Sharp $L^p$ bounds for maximal Bochner-Riesz means for elliptic operators.

Extension of bounds to Schrödinger operators on asymptotically conic manifolds.

Results applicable to harmonic oscillators and elliptic operators on compact manifolds.

Abstract

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp $L^p$ maximal bounds for the Schr\"odinger operators on asymptotically conic manifolds, the harmonic oscillator and its perturbations or elliptic operators on compact manifolds.