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

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