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Square function estimates for cones over quadratic manifolds

Robert Schippa

TL;DR

The paper develops $L^4$-square function estimates for quadratic manifolds and their conical extensions under a quantitative transversality condition on tangent spaces, extending known results for parabolic and complex-analytic models to higher dimensions. It combines a biorthogonality framework with High–Low and Kakeya-based wave-envelope methods to prove sharp-like square function bounds and leverages these estimates to derive a multi-parameter local smoothing result for averages over quadratic surfaces. The innovations include canonical coverings of $\delta$-neighborhoods, a detailed Kakeya-type analysis for cones, and an induction-on-scales argument that transfers base-manifold estimates to conical extensions, together with oscillatory integral representations that place the problems in a Fourier integral operator setting. These results advance understanding of how transversality and curvature interact to control multilinear interactions in frequency space, with potential implications for Kakeya-type phenomena and local smoothing in higher-dimensional quadratic settings.

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.

Square function estimates for cones over quadratic manifolds

TL;DR

The paper develops -square function estimates for quadratic manifolds and their conical extensions under a quantitative transversality condition on tangent spaces, extending known results for parabolic and complex-analytic models to higher dimensions. It combines a biorthogonality framework with High–Low and Kakeya-based wave-envelope methods to prove sharp-like square function bounds and leverages these estimates to derive a multi-parameter local smoothing result for averages over quadratic surfaces. The innovations include canonical coverings of -neighborhoods, a detailed Kakeya-type analysis for cones, and an induction-on-scales argument that transfers base-manifold estimates to conical extensions, together with oscillatory integral representations that place the problems in a Fourier integral operator setting. These results advance understanding of how transversality and curvature interact to control multilinear interactions in frequency space, with potential implications for Kakeya-type phenomena and local smoothing in higher-dimensional quadratic settings.

Abstract

We extend the -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.

Paper Structure

This paper contains 16 sections, 21 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Square function estimates for quadratic manifolds
  3. Square function estimates for cones over quadratic manifolds
  4. Local smoothing estimates for multi-parameter averages
  5. Preliminaries
  6. Square function estimates for quadratic manifolds
  7. Proof of Theorem \ref{['thm:SquareFunctionQuadraticManifold']}
  8. Examples
  9. Square function estimates for the conical extension
  10. Kakeya estimates for cones over quadratic manifolds
  11. Induction on scales
  12. Local smoothing estimates for multi-parameter averages
  13. Oscillatory integral representation
  14. Applying the square function estimate
  15. Conclusion via maximal function estimates
  16. ...and 1 more sections

Key Result

Theorem 1.2

Suppose that the quadratic manifold $M_Q$ is given by eq:ParametrizationQuadraticManifold and for the generators $A_i = \partial^2 q_i$ of the quadratic forms Assumption ass:SpanCondition holds true. Then, for $f \in \mathcal{S}(\mathbb{R}^{d+l})$ with $\text{supp}(\hat{f}) \subseteq \mathcal{N}_{\d

Theorems & Definitions (35)

  • Theorem 1.2: Square function estimates for quadratic manifolds
  • Remark 1.3
  • Theorem 1.4: Square function estimates for conical extensions of quadratic manifolds
  • Theorem 1.5: Local smoothing estimates for multi-parameter averages
  • Lemma 3.1: Double-mean value theorem
  • proof : Proof of Theorem \ref{['thm:SquareFunctionQuadraticManifold']}
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 25 more