Normal-Direction Energy and Fourier Restriction for Convex Planar Curves

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.