Square function estimates for cones over quadratic manifolds

TL;DR

This paper generalizes $L^4$-square function estimates from classical parabola and cone cases to higher-dimensional quadratic manifolds and their cones, using transversality and wave envelope techniques.

Contribution

It introduces new $L^4$-square function estimates for quadratic manifolds and their cones in higher dimensions, expanding the scope of previous results.

Findings

Established $L^4$-square function estimates for quadratic manifolds.

Proved biorthogonality for systems of quadratic equations.

Derived wave envelope estimates for conical extensions.

Abstract

We extend the $L^4$-square function estimates for the parabola and the half-cone to quadratic manifolds in higher dimensions and their conical extensions. To this end, we require transversality for the tangent spaces of the quadratic manifolds at separated points. This allows us to show biorthogonality for the associated system of quadratic equations. For the conical extensions we obtain a wave envelope estimate by High-Low arguments.