Normal-Direction Energy and Fourier Restriction for Convex Planar Curves

Abstract: We study Fourier extension from compact convex $C<sup>2$ planar curves by organizing the mass $|f|^2\,\mathrm dσ$ through the Gauss map. The pushforward measure [ νf= N#\bigl(|f|^2\,\mathrm dσ\bigr) ] records its distribution in normal directions. The turning measure $\mathrm dμ<em>κ=κ\,\mathrm dσ$ determines an intrinsic terminal tangential scale $r_R(ξ)$ through [ r_R(ξ)\, μκ\bigl(B_Γ(ξ,r_R(ξ))\bigr) \asymp R^{-1}, ] and hence a position-dependent angular resolution $ρ<em>R(ξ)=R<sup>{-1}/r_R(ξ)$. Under a doubling hypothesis on $μ</em>κ$, $r_R$ is, up to structural constants, the largest scale on which the curve can be linearized to precision $R<sup>{-1}$. These scales define a normal-direction energy, and the associated terminal decomposition and transverse geometry yield local $L<sup>4$ Fourier extension estimates and weighted variants. For the monomial curves $γ_k(t)=(t,t<sup>k)$, $k\geq3$ 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.