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

Abstract

This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^{\delta}_{\lambda}(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schr\"odinger 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].