Bochner-Riesz means for critical magnetic Schr\"odinger operators in ${\mathbb R^2}$

TL;DR

This paper investigates the boundedness of Bochner-Riesz means for magnetic Schr"odinger operators with Aharonov-Bohm potential in two dimensions, establishing precise conditions on the order for $L^p$ boundedness.

Contribution

It provides the first characterization of $L^p$ boundedness for these operators, incorporating magnetic effects and singular kernels, extending classical results for the Laplacian.

Findings

Boundedness holds if and only if elta>\,maxig\,0, 2|1/2-1/p|-1/2ig\

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Abstract

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schr\"odinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\neq 2$, the Bochner-Riesz operator $S_{\lambda}^\delta(\mathcal{L}_{\bf A})$ of order $\delta$ is bounded on $L^p(\mathbb{R}^2)$ if and only if $\delta>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient of the proof is to obtain the localized $L^4(\mathbb R^2)$ estimate of $S_{\lambda}^\delta(\mathcal{L}_{\bf A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_{\lambda}^\delta(\Delta)$ for the Laplacian $\Delta$ in $\mathbb{R}^2$.