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Restriction Estimates in a Conical Singular Space: Wave Equation

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Abstract

We study the restriction estimates in a class of conical singular space \(X=C(Y)=(0,\infty )_r\times Y\) 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). Let \(\Delta _g\) be the Friedrichs extension positive Laplacian on X, and consider the operator \(\mathcal {L}_V=\Delta _g+V\) with \(V=V_0r^{-2}\), where \(V_0(\theta )\in \mathcal {C}^\infty (Y)\) is a real function such that the operator \(\Delta _h+V_0+(n-2)^2/4\) is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with \(\mathcal {L}_V\). The smallest positive eigenvalue of the operator \(\Delta _h+V_0+(n-2)^2/4\) plays an important role in the result. As an application, for independent of interests, we prove local energy estimates and Keel–Smith–Sogge estimates for the wave equation in this setting.

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Notes

  1. The assumption here is weaker than the hypothesis in [45] where one needs \(\Delta _h+V_0(\theta )\ge 0\).

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Acknowledgements

The authors were supported by National Natural Science Foundation of China (12171031, 11771041, 11831004, 11901041,11671033) and H2020-MSCA-IF-2017(790623).

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Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Xiaofen Gao

  2. Department of Mathematics, Beijing Key Laboratory on Mathematical Characterization, Analysis, and Applications of Complex Information, Beijing Institute of Technology, Beijing, 100081, China

    Junyong Zhang

  3. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Jiqiang Zheng

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  1. Xiaofen Gao
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  2. Junyong Zhang
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  3. Jiqiang Zheng
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Correspondence to Junyong Zhang.

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Communicated by Luis Vega.

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Gao, X., Zhang, J. & Zheng, J. Restriction Estimates in a Conical Singular Space: Wave Equation. J Fourier Anal Appl 28, 44 (2022). https://doi.org/10.1007/s00041-022-09941-7

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  • DOI: https://doi.org/10.1007/s00041-022-09941-7

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