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

Abstract

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\LL_{\A}$ in ${\mathbb{R}}^2$, which involve the {physical} Aharonov-Bohm potential. We show that for $1\le p\le +\infty$ and $p\ne 2$, the Bochner-Riesz operator $S_λ^δ(\LL_{\A})$ of order $\delta$ is bounded on $L^p({\mathbb{R}}^2)$ if and only if $\delta >\max \{0,2|1/2-1/p|-1/2\}$. The new ingredient {in} the proof is to obtain the localized $L^4({\mathbb{R}}^2)$ estimate of $S_λ^δ(\LL_{\A})$, 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 ${\mathbb{R}}^2$.

Comments are welcome! 36 pages. We corrected some typos and updated the proof