Endpoint weighted estimate for a fixed finite-type curve

Establish the weighted Fourier extension inequality at the endpoint exponent \(\iota=\beta_{\max}(\Gamma)=(m_{\max}(\Gamma)+1)/(m_{\max}(\Gamma)+2)\) for a fixed finite-type convex planar curve \(\Gamma\), where \(m_{\max}(\Gamma)\) is the largest order of curvature vanishing attained on the curve.

Background

The paper proves that, for a fixed finite-type curve, the weighted estimate holds whenever $\iota>\beta_{\max}(\Gamma)$, while a counterexample rules it out whenever $\iota<\beta_{\max}(\Gamma)$. Thus only the endpoint remains unresolved. Resolving it would complete the sharp weighted-threshold theory for each fixed curve under consideration.