Square function estimates for cones over quadratic manifolds, and sharp $\mkern-0.05mu$multiparameter $\mkern-0.05mu$local $\mkern-0.05mu$smoothing $\mkern-0.05mu$estimates

Robert Schippa

Vol. 345 (2026), No. 1, 127–168

DOI: 10.2140/pjm.2026.345.127

Abstract

We prove sharp $L^4$-square function estimates for 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. As an application we obtain a sharp local smoothing estimate for a multiparameter average over surfaces satisfying nondegeneracy and transversality assumptions.

Keywords

square function estimates, wave envelope estimates, local smoothing estimates

Mathematical Subject Classification

Primary: 42B20, 42B25

Milestones

Received: 8 February 2026

Revised: 27 June 2026

Accepted: 2 July 2026

Published: 25 September 2026