Bochner-Riesz means for the Hermite and special Hermite expansions

TL;DR

This paper investigates the boundedness of Bochner-Riesz means for Hermite and special Hermite expansions, establishing optimal $L^p$ bounds in various dimensions and challenging existing conjectures about summability indices.

Contribution

It provides new sharp $L^p$ boundedness results for Hermite Bochner-Riesz means and introduces a lower bound that refutes previous conjectures.

Findings

Established $L^p$ boundedness on the optimal range in 2D.

Extended the known $L^p$ range in higher dimensions.

Proved a new lower bound on the summability index that contradicts prior conjectures.

Abstract

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.