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

2025/04/04 by Gao, Xiaofen, Wang, Jialu, Xu, Chengbin +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.03124

Abstract

In this paper, we consider a class of conical singular spaces Σ=(0,∞)_r× Y equipped with the metric g=dr²+r²h, 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(t√L_A)/√L_A is bounded in L^p(Σ), where L_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 \citeL. The novel ingredient is the construction of Hadamard parametrix for cos(t√L_{\bf A}) on Σ.

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