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Decoupling for complex curves and improved decoupling for the cubic moment curve

Robert Schippa

Abstract

We prove sharp $\ell^2$-decoupling inequalities for non-degenerate complex curves via the bilinear argument due to Guo--Li--Yung--Zorin-Kranich, which in turn is inspired by the efficient congruencing argument of Wooley. Secondly, quantifying the iteration in the cubic case, we obtain a logarithmic refinement of the decoupling inequality for the cubic moment curve.

Decoupling for complex curves and improved decoupling for the cubic moment curve

Abstract

We prove sharp -decoupling inequalities for non-degenerate complex curves via the bilinear argument due to Guo--Li--Yung--Zorin-Kranich, which in turn is inspired by the efficient congruencing argument of Wooley. Secondly, quantifying the iteration in the cubic case, we obtain a logarithmic refinement of the decoupling inequality for the cubic moment curve.
Paper Structure (22 sections, 29 theorems, 163 equations)

This paper contains 22 sections, 29 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Decoupling inequalities for non-degenerate complex curves
  3. Logarithmic refinement in the cubic case
  4. Preliminaries
  5. Real and complex notations
  6. Normalized non-degenerate curves
  7. Decoupling constants
  8. Proof of Theorem \ref{['thm:DecouplingComplexCurves']}
  9. Reduction to bilinear decoupling
  10. Locally constant property and asymmetric decoupling
  11. Lower-dimensional decoupling via transversality
  12. Bilinear decoupling iteration
  13. Logarithmic improvement of decoupling for the cubic moment curve
  14. Preliminaries
  15. Notations
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $k \geq 1$ and $\gamma_{\mathbb{R}}:[0,1] \to \mathbb{R}^k$ denote the real moment curve. Let $f \in \mathcal{S}(\mathbb{R}^k)$ with $\text{supp}(\hat{f}) \subseteq \bigcup_{I \in \mathcal{I}_{\delta}} \theta_{I,\mathbb{R}}$. Then the following estimate holds:

Theorems & Definitions (47)

  • Theorem 1.1: $\ell^2$-decoupling for curves with torsion
  • Definition 1.2
  • Theorem 1.3: $\ell^2$-decoupling for non-degenerate complex curves
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Improved decoupling for the moment curve in three dimensions
  • Definition 2.1: Normalized non-degenerate curve
  • Lemma 2.2: Anisotropic Rescaling
  • Definition 3.1
  • ...and 37 more