Sharp spherical L^p→ L^4 extension estimates over finite fields in odd dimensions

Abstract

Let q be an odd prime power, let d≥3 be odd, and let S_j={ξ∈ 𝔽_q^d:ξ₁²+⋯+ξ_d²=j}, where j≠ 0. We prove the sharp extension estimate L^{4(d+1)/(3d+1)}(S_j, dσ)→ L⁴(𝔽_q^d,dc) whenever η_q((−1)^(d−1)/2j)=−1, where η_q is the quadratic character. This confirms the odd-dimensional spherical extension conjecture of Koh, Pham, and Vinh over every finite field of odd characteristic. Together with the known estimate in the complementary sign class, this determines the sharp L^p→L⁴ extension range for all nonzero-radius spheres in odd dimensions. We also apply the endpoint estimate to obtain sufficient conditions for the unique recovery of sparse signals from incomplete Fourier data.

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BibTeX RIS

BibTeX

@misc{pham2026sharp,
  title = {{Sharp spherical $L^p\to L^4$ extension estimates over finite fields in odd dimensions}},
  author = {Thang Pham and Boqing Xue},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29664},
  eprint = {2610.29664},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29664},
}

RIS

TY  - UNPB
TI  - Sharp spherical L^p→ L^4 extension estimates over finite fields in odd dimensions
AU  - Thang Pham
AU  - Boqing Xue
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29664
UR  - https://arxiv.org/abs/2610.29664
AB  - Let q be an odd prime power, let d≥3 be odd, and let S_j={ξ∈ 𝔽_q^d:ξ_1^2+⋯+ξ_d^2=j}, where j≠ 0. We prove the sharp extension estimate L^4(d+1)/(3d+1)(S_j, dσ)→ L^4(𝔽_q^d,dc) whenever η_q((−1)^(d−1)/2j)=−1, where η_q is the quadratic character. This confirms the odd-dimensional spherical extension conjecture of Koh, Pham, and Vinh over every finite field of odd characteristic. Together with the known estimate in the complementary sign class, this determines the sharp L^p→L^4 extension range for all nonzero-radius spheres in odd dimensions. We also apply the endpoint estimate to obtain sufficient conditions for the unique recovery of sparse signals from incomplete Fourier data.
ER  -

Sharp spherical L^p→ L⁴ extension estimates over finite fields in odd dimensions

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