Negative Order Bochner-Riesz Operators for the Critical Magnetic Schrödinger Operator in $\mathbb{R}^2$
Huanqing Guo, Junyong Zhang, Jiqiang Zheng
TL;DR
The paper analyzes sharp $L^p$-$L^q$ bounds for the negative-order Bochner–Riesz operator $S^{−\delta}_{\lambda}(\mathcal{L}_{\mathbf{A}})$ associated with the scaling‑critical magnetic Schrödinger operator in $\mathbb{R}^2$, for $\delta\in(-3/2,0)$. By expressing the spectral-measure kernel and decomposing it into a geometry term $G$ and a diffractive term $D$, the authors reduce the problem to restricted weak-type estimates for these components. The geometry part is controlled by a stability-type oscillatory-integral argument with a dyadic decomposition (allowing an arbitrarily small $\varepsilon$-loss), while the diffractive part is handled by standard oscillatory‑integral techniques (Van der Corput, stationary phase, Morse) in a dyadic framework. The main result identifies a pentagonal region $\Delta(\delta)$ in the $(1/p,1/q)$-plane such that $\|S^{−\delta}_{\lambda}(\mathcal{L}_{\mathbf{A}})f\|_{L^q} \le C\lambda^{2(1/p-1/q)}\|f\|_{L^p}$ if and only if $(1/p,1/q)\in\Delta(\delta)$, extending prior uniform resolvent estimates and informing related electromagnetic Schrödinger operators via intertwining.
Abstract
This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^δ_λ(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schrödinger operator $\mathcal{L}_{\mathbf{A}}$ on $\mathbb{R}^2$, where $δ\in (-3/2, 0)$. 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 $Δ(δ)$ 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].