$L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds

TL;DR

The paper addresses the problem of establishing $L^p$-estimates for the wave equation perturbed by a scaling-critical magnetic potential on metric cones. It develops a Hadamard parametrix for the operator $\cos(t\sqrt{\mathcal{L}_{\mathbf A}})$ on the cone and uses a holomorphic analytic family $F_{\omega,t}(\mathcal{L}_{\mathbf A})$ together with Stein interpolation to derive $L^p$-bounds for the sine propagator $\frac{\sin(t\sqrt{\mathcal{L}_{\mathbf A}})}{\sqrt{\mathcal{L}_{\mathbf A}}}$. The main result obtains sharp $L^p$-boundedness for $|\tfrac{1}{p}-\tfrac{1}{2}|<\tfrac{1}{n-1}$ under the injectivity radius condition $\rho>\pi$, extending prior works on cones and magnetic perturbations. The work provides detailed kernel estimates and a Hadamard-parametrix-based framework that could inform dispersive analysis on singular spaces and related magnetic PDEs.

Abstract

In this paper, we consider a class of conical singular spaces $Σ=(0,\infty)_r\times Y$ equipped with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$ without boundary. In this context, we aim to show that the sine wave propagator $\sin\left(t\sqrt{\mathcal{L}_{\mathbf{A}}}\right)/\sqrt{\mathcal{L}_{\mathbf{A}}}$ is bounded in $L^{p}(Σ)$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator with a scaling-critical magnetic potential on metric cone $Σ$. Our main result is the generalization of the result in \cite{L}. The novel ingredient is the construction of Hadamard parametrix for $\cos\left(t\sqrt{\mathcal{L}_{\bf A}}\right)$ on $Σ$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Proposition 3.1
• Proposition 3.2
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Lemma 3.3
• proof