Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian

TL;DR

This paper establishes the almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian in complex space, identifying the precise conditions on the summability index for various L^p spaces.

Contribution

It determines the sharp range of the summability indices ensuring convergence of Bochner--Riesz means for the twisted Laplacian across L^p spaces.

Findings

Convergence holds for all f in L^p when δ exceeds a sharp threshold.

The results extend the understanding of spectral expansions for the twisted Laplacian.

Sharp bounds on δ are established for p between 2 and infinity.

Abstract

Let $\mathcal L$ denote the twisted Laplacian in $\mathbb C^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S^\delta_{t}(\mathcal L) f$ of $f\in L^p(\mathbb C^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $\delta$ for which the convergence of $S^\delta_{t}(\mathcal L) f$ holds for all $f\in L^p(\mathbb C^d)$.