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ℓ2 Decoupling for Certain Surfaces of Finite Type in ℝ3

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Abstract

In this article, we establish an ℓ2 decoupling inequality for the surface

$$F_4^2: = \{ ({\xi _1},{\xi _2},\xi _1^4 + \xi _2^4):({\xi _1},{\xi _2}) \in {[0,1]^2}\} $$

associated with the decomposition adapted to finite type geometry from our previous work [Li, Z., Miao, C., Zheng, J.: A restriction estimate for a certain surface of finite type in ℝ3. J. Fourier Anal. Appl., 27(4), Paper No. 63, 24 pp. (2021)]. The key ingredients of the proof include the so-called generalized rescaling technique, an ℓ2 decoupling inequality for the surfaces

$$\{ ({\xi _1},{\xi _2},{\phi _1}({\xi _1}) + \xi _2^4):({\xi _1},{\xi _2}) \in {[0,1]^2}\} $$

with ϕ1 being non-degenerate, reduction of dimension arguments and induction on scales.

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Acknowledgements

We would like to thank the associated editor and anonymous referee for their invaluable comments and suggestions which helped improve the paper greatly. We also thank Jianhui Li and Tongou Yang for sharing their paper [12] and point out that our main theorem has some overlap with their result after we submit our paper in the arXiv.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, P. R. China

    Zhuo Ran Li

  2. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, P. R. China

    Ji Qiang Zheng

Authors
  1. Zhuo Ran Li
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  2. Ji Qiang Zheng
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Correspondence to Ji Qiang Zheng.

Additional information

Supported by National key R&D program of China (Grant No. 2021YFA1002500), NSFC (Grant No. 12271051), PFCAEP project (Grant No. YZJJLX201901)

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Li, Z.R., Zheng, J.Q. ℓ2 Decoupling for Certain Surfaces of Finite Type in ℝ3. Acta. Math. Sin.-English Ser. 39, 1442–1458 (2023). https://doi.org/10.1007/s10114-023-1374-9

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  • DOI: https://doi.org/10.1007/s10114-023-1374-9

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