Improved local smoothing estimate for the wave equation in higher dimensions

Chuanwei Gao; Bochen Liu; Changxing Miao; Yakun Xi

doi:10.1016/j.jfa.2023.109879

Improved local smoothing estimate for the wave equation in higher dimensions

Chuanwei Gao, Bochen Liu, Changxing Miao, Yakun Xi

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

In this paper, we establish the sharp k-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions n≥3. As a byproduct, we also generalize the restriction estimates of Ou–Wang [21] to a broader class of phase functions.

Original language	English
Article number	109879
Journal	Journal of Functional Analysis
Volume	284
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2023.109879
State	Published - 1 May 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Funding

This project was supported by the National Key R&D Program of China : No. 2022YFA1005700 and 2022YFA1007200 . C. Gao was partially supported by Chinese Postdoc Foundation Grant: No. 8206300279 . B. Liu was partially supported by SUSTech start-up counterpart Y01286235 . C. Miao was partially supported by NSF of China grant: No. 11831004 . Y. Xi was partially supported by NSF of China grant: No. 12171424 and the Fundamental Research Funds for the Central Universities 2021QNA3001 . The authors would like to thank Prof. Hong Wang for some helpful discussions. The first author would like to thank Prof. Gang Tian's support and Prof. Dongyi Wei's helpful discussion. The authors want to acknowledge that David Beltran and Olli Saari realized independently [3] that it is possible to use k-broad estimates to obtain improved local smoothing estimates. The authors want to thank Prof. David Beltran for some friendly communications, and for pointing out that broad and narrow bounds for the circular cone only are not enough for proving local smoothing type estimates. The authors want to thank an anonymous referee for his or her thorough and invaluable feedback.

Funders	Funder number
Chinese Postdoc Foundation	8206300279, Y01286235
National Natural Science Foundation of China	12171424, 11831004
National Key Research and Development Program of China	2022YFA1007200, 2022YFA1005700
Fundamental Research Funds for the Central Universities	2021QNA3001

Keywords

Local smoothing
Wave equation
k-Broad “norm”

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10.1016/j.jfa.2023.109879

Cite this

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Improved local smoothing estimate for the wave equation in higher dimensions

Abstract

Bibliographical note

Funding

Keywords

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