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Strichartz estimates and wave equation in a conic singular space

Zhang, Junyong and Zheng, Jiqiang 2020. Strichartz estimates and wave equation in a conic singular space. Mathematische Annalen 376 (1-2) , pp. 525-581. 10.1007/s00208-019-01892-7

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Consider the metric cone X=C(Y)=(0,∞)r×Y with metric g=dr2+r2h where the cross section Y is a compact (n−1)-dimensional Riemannian manifold (Y, h). Let Δg be the positive Friedrichs extension Laplacian on X and let Δh be the positive Laplacian on Y, and consider the operator LV=Δg+V0r−2 where V0∈C∞(Y) such that Δh+V0+(n−2)2/4 is a strictly positive operator on L2(Y). In this paper, we prove global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator LV. It verifies a conjecture in Wang (Remark 2.4 in Ann Inst Fourier 56:1903–1945, 2006) for wave equation. The range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of Δh+V0+(n−2)2/4. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension n≥3.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer
ISSN: 0025-5831
Date of First Compliant Deposit: 17 September 2019
Date of Acceptance: 5 August 2019
Last Modified: 19 Nov 2020 17:00

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