Generalized square function estimates for curves and their conical extensions

TL;DR

This paper establishes sharp square function estimates for planar curves with degenerate curvature and extends these results to conical surfaces, including cones over complex parabolas, using advanced harmonic analysis techniques.

Contribution

It extends classical biorthogonality to finite type curves and analyzes wave envelope estimates for cones over degenerate curves, advancing harmonic analysis methods.

Findings

Sharp square function estimates for degenerate curves in the plane.

Endpoint estimates for cones over these curves.

Extension of techniques to cones over complex parabolas.

Abstract

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 C\'ordoba--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.