Negative Order Bochner-Riesz Operators for the Critical Magnetic Schrödinger Operator in ${\mathbb{R}}^2$

Abstract

This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S_{\lambda }^{\delta }({\mathcal{L}}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schrödinger operator ${\mathcal{L}}_{\mathbf{A}}$ on ${\mathbb{R}}^2$, where $\delta \in (-3/2,0)$. We determine the conditions on the exponents $p$ and $q$ under which the operator is bounded from $L^p({\mathbb{R}}^2)$ to $L^q({\mathbb{R}}^2)$. Our main result characterizes the boundedness region as a pentagonal subset $\Delta (\delta )$ of the $(1/p,1/q)$-plane, extending previous uniform resolvent result in Fanelli, Zhang and Zheng[Int. Math. Res. Not., 20(2023), 17656-17703].

14 pages