{"ID":23646987,"CreatedAt":"2026-09-18T07:39:37.512926072Z","UpdatedAt":"2026-09-18T07:39:37.512926072Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20643","arxiv_id":"2609.20643","title":"Normal-Direction Energy and Fourier Restriction for Convex Planar Curves","abstract":"We study Fourier extension from compact convex $C^2$ planar curves by organizing the mass $|f|^2\\,\\mathrm dσ$ through the Gauss map. The pushforward measure \\[ ν_f= N_\\#\\bigl(|f|^2\\,\\mathrm dσ\\bigr) \\] records its distribution in normal directions. The turning measure $\\mathrm dμ_κ=κ\\,\\mathrm dσ$ determines an intrinsic terminal tangential scale $r_R(ξ)$ through \\[ r_R(ξ)\\, μ_κ\\bigl(B_Γ(ξ,r_R(ξ))\\bigr) \\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 $γ_k(t)=(t,t^k)$, $k\\geq3$ 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)=\\frac{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.","short_abstract":"We study Fourier extension from compact convex $C^2$ planar curves by organizing the mass $|f|^2\\,\\mathrm dσ$ through the Gauss map. The pushforward measure \\[ ν_f= N_\\#\\bigl(|f|^2\\,\\mathrm dσ\\bigr) \\] records its distribution in normal directions. The turning measure $\\mathrm dμ_κ=κ\\,\\mathrm dσ$ determines an intrinsi...","url_abs":"https://arxiv.org/abs/2609.20643","url_pdf":"https://arxiv.org/pdf/2609.20643v1","authors":"[\"Vicente Vergara\"]","published":"2026-09-17T16:25:37Z","proceeding":"math.FA","tasks":"[\"math.FA\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}