Normal-Direction Energy and Fourier Restriction for Convex Planar Curves
Abstract
We study Fourier extension from compact convex C2 planar curves by organizing the mass |f|2 dσ through the Gauss map. The pushforward measure νf= N#(|f|2 dσ) records its distribution in normal directions. The turning measure dμκ=κ dσ determines an intrinsic terminal tangential scale rR(ξ) through rR(ξ) μκ(BΓ(ξ,rR(ξ))) asymp R−1, and hence a position-dependent angular resolution ρR(ξ)=R−1/rR(ξ). Under a doubling hypothesis on μκ, rR is, up to structural constants, the largest scale on which the curve can be linearized to precision R−1. These scales define a normal-direction energy, and the associated terminal decomposition and transverse geometry yield local L4 Fourier extension estimates and weighted variants. For the monomial curves γk(t)=(t,tk), k≥3 real, the terminal scales are explicit and the energy admits a multiscale representation in terms of angular correlations of νf. Under an s-dimensional Frostman condition on νf, this yields a growth diagram with critical threshold sc(k)=(k−2)/(3k−4), separating the flat-point and nondegenerate regimes. The resulting rates, including the critical logarithmic correction, are sharp at the energy level.
Related
BibTeX
@misc{vergara2026normaldirection,
title = {{Normal-Direction Energy and Fourier Restriction for Convex Planar Curves}},
author = {Vicente Vergara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20643},
eprint = {2609.20643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20643},
note = {CC0},
}
RIS
TY - UNPB
TI - Normal-Direction Energy and Fourier Restriction for Convex Planar Curves
AU - Vicente Vergara
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.20643
UR - https://arxiv.org/abs/2609.20643
AB - We study Fourier extension from compact convex C^2 planar curves by organizing the mass |f|^2 dσ through the Gauss map. The pushforward measure ν_f= N_#(|f|^2 dσ) records its distribution in normal directions. The turning measure dμ_κ=κ dσ determines an intrinsic terminal tangential scale r_R(ξ) through r_R(ξ) μ_κ(B_Γ(ξ,r_R(ξ))) asymp R^−1, and hence a position-dependent angular resolution ρ_R(ξ)=R^−1/r_R(ξ). Under a doubling hypothesis on μ_κ, r_R is, up to structural constants, the largest scale on which the curve can be linearized to precision R^−1. These scales define a normal-direction energy, and the associated terminal decomposition and transverse geometry yield local L^4 Fourier extension estimates and weighted variants. For the monomial curves γ_k(t)=(t,t^k), k≥3 real, the terminal scales are explicit and the energy admits a multiscale representation in terms of angular correlations of ν_f. Under an s-dimensional Frostman condition on ν_f, this yields a growth diagram with critical threshold s_c(k)=(k−2)/(3k−4), separating the flat-point and nondegenerate regimes. The resulting rates, including the critical logarithmic correction, are sharp at the energy level.
ER -