Generalized square function estimates for curves and their conical extensions

We show sharp square function estimates for curves in the plane whose curvature degenerates at a point and estimates sharp up to endpoints for cones over these curves. To this end, for curves of finite type we extend the classical Córdoba–Fefferman biorthogonality. For cones over degenerate curves, we analyze wave envelope estimates proved via High-Low-decomposition. The arguments are subsequently extended to the cone over the complex parabola.