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PreprintarXiv licencearXiv

Stability of Fourier Restriction for a Class of Finite Type Surfaces in ℝ3

  • Wenjuan Li
  • Hongqiang Wang
  • Huiju Wang
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Abstract

We establish the stability of the optimal non-endpoint Fourier restriction range under smooth flat perturbations of curved-root surfaces in ℝ3. Specifically, we consider graphs of ϕ(x1,x2)=(x2−x1m)n+R(x1,x2), where m,n≥2 are integers and R is smooth and flat at the origin. On a sufficiently small neighborhood depending on the perturbation, we prove the local Lp(ℝ3)→ L2(S,dσ) restriction estimate for 1≤ p<pc, where pc=qc/(qc−1), qc=max{2+2mn/(m+1),2n}. If the nonnegative localizing density is positive at the origin, an adapted-box example shows that the estimate fails for p>pc. Thus arbitrary smooth flat perturbations preserve the optimal restriction range away from the critical endpoint. For n≥3, the main local estimate combines planar curve decoupling with a one-dimensional finite-type extension estimate, whose principal scale factors cancel exactly. For n=2, we prove a uniform C4 perturbation estimate using a weighted convolution bound and an anisotropic Whitney decomposition. Anisotropic rescaling and summation complete the proof.

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Cite

BibTeXRIS
BibTeX
@misc{li2026stability,
  title = {{Stability of Fourier Restriction for a Class of Finite Type Surfaces in $\mathbb R^3$}},
  author = {Wenjuan Li and Hongqiang Wang and Huiju Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31389},
  eprint = {2610.31389},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31389},
}
RIS
TY  - UNPB
TI  - Stability of Fourier Restriction for a Class of Finite Type Surfaces in ℝ^3
AU  - Wenjuan Li
AU  - Hongqiang Wang
AU  - Huiju Wang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31389
UR  - https://arxiv.org/abs/2610.31389
AB  - We establish the stability of the optimal non-endpoint Fourier restriction range under smooth flat perturbations of curved-root surfaces in ℝ^3. Specifically, we consider graphs of ϕ(x_1,x_2)=(x_2−x_1^m)^n+R(x_1,x_2), where m,n≥2 are integers and R is smooth and flat at the origin. On a sufficiently small neighborhood depending on the perturbation, we prove the local L^p(ℝ^3)→ L^2(S,dσ) restriction estimate for 1≤ p<p_c, where p_c=q_c/(q_c−1), q_c=max{2+2mn/(m+1),2n}. If the nonnegative localizing density is positive at the origin, an adapted-box example shows that the estimate fails for p>p_c. Thus arbitrary smooth flat perturbations preserve the optimal restriction range away from the critical endpoint. For n≥3, the main local estimate combines planar curve decoupling with a one-dimensional finite-type extension estimate, whose principal scale factors cancel exactly. For n=2, we prove a uniform C^4 perturbation estimate using a weighted convolution bound and an anisotropic Whitney decomposition. Anisotropic rescaling and summation complete the proof.
ER  -