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 $\mathrm{\mid }f{\mathrm{\mid }}^2\mathrm{d}\sigma$ through the Gauss map. The pushforward measure $${\nu }_f=N_{\mathrm{\#}}\left(\mathrm{\mid }f{\mathrm{\mid }}^2\mathrm{d}\sigma \right)$$ records its distribution in normal directions. The turning measure $\mathrm{d}{\mu }_{\kappa }=\kappa \mathrm{d}\sigma$ determines an intrinsic terminal tangential scale $r_R(\xi )$ through $$r_R(\xi ){\mu }_{\kappa }(B_{\mathrm{\Gamma }}(\xi ,r_R(\xi )))\asymp R^{-1},$$ and hence a position-dependent angular resolution ${\rho }_R(\xi )=R^{-1}\mathrm{/}{r}_R(\xi )$. Under a doubling hypothesis on ${\mu }_{\kappa }$, $r_R$ 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 ${\gamma }_k(t)=(t,t^k)$, $k\ge 3$ real, the terminal scales are explicit and the energy admits a multiscale representation in terms of angular correlations of ${\nu }_f$. Under an $s$-dimensional Frostman condition on ${\nu }_f$, 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.