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Small cap decoupling for the parabola with logarithmic constant

Ben Johnsrude

TL;DR

The paper proves a logarithmic-loss version of small-cap decoupling for functions with Fourier support near the truncated parabola $\mathbb{P}^1$. It develops an amplitude-dependent wave envelope framework, grounded in Gevrey-class wave packets, to control interference and prune high-amplitude components, enabling a multiscale argument that yields a bound of the form $D_{p,q}(R;\beta) \lesssim (\log R)^{18+3p}\big(R^{\beta(p-\frac{p}{q}-1)-1}+R^{p\beta(\frac{1}{2}-\frac{1}{q})}\big)$. A central technical achievement is Theorem $\['squarefunction'$)$, a wave-envelope–driven square-function bound that replaces the subpolynomial factors with explicit logarithmic losses, which in turn implies the small-cap result (Theorem $\['smallcap'$)$ via a careful adaptation of known decoupling arguments. The work extends Guth–Maldague’s wave-envelope approach from the cone to the parabola, providing a clear mechanism for achieving log-scale control in decoupling, and clarifies the role of amplitude-pruning and multiscale decomposition in managing the broad/narrow contributions.

Abstract

We note that the subpolynomial factor for the $\ell^qL^p$ small cap decoupling constants for the truncated parabola $\mathbb{P}^1=\{(t,t^2):|t|\leq 1\}$ may be controlled by a suitable power of $\log R$. This is achieved by considering a suitable amplitude-dependent wave envelope estimate, as was introduced in a recent paper of Guth and Maldague to demonstrate a small cap decoupling for the $(2+1)$ cone; we demonstrate that the version for $\mathbb{P}^1$ may be taken with a loss controlled by a power of $\log R$ as well.

Small cap decoupling for the parabola with logarithmic constant

TL;DR

The paper proves a logarithmic-loss version of small-cap decoupling for functions with Fourier support near the truncated parabola . It develops an amplitude-dependent wave envelope framework, grounded in Gevrey-class wave packets, to control interference and prune high-amplitude components, enabling a multiscale argument that yields a bound of the form . A central technical achievement is Theorem )\['smallcap' via a careful adaptation of known decoupling arguments. The work extends Guth–Maldague’s wave-envelope approach from the cone to the parabola, providing a clear mechanism for achieving log-scale control in decoupling, and clarifies the role of amplitude-pruning and multiscale decomposition in managing the broad/narrow contributions.

Abstract

We note that the subpolynomial factor for the small cap decoupling constants for the truncated parabola may be controlled by a suitable power of . This is achieved by considering a suitable amplitude-dependent wave envelope estimate, as was introduced in a recent paper of Guth and Maldague to demonstrate a small cap decoupling for the cone; we demonstrate that the version for may be taken with a loss controlled by a power of as well.
Paper Structure (14 sections, 27 theorems, 208 equations)

This paper contains 14 sections, 27 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Infrastructure for proving Theorem \ref{['squarefunction']}
  4. Overview of the argument
  5. Construction of the rapidly-decaying wave packets
  6. Initial notation-setting
  7. Pruning
  8. Square functions
  9. Proof of Theorem \ref{['squarefunction']}
  10. Bounding the broad sets
  11. Broad/narrow analysis
  12. Reduction to local estimates
  13. Proof of Theorem \ref{['smallcap']}
  14. Appendix: Proofs of square function lemmas

Key Result

Theorem 1.1

Let $p,q\geq 1$ satisfy $\frac{3}{p}+\frac{1}{q}\leq 1$, $R>2$, and $\beta\in[\frac{1}{2},1]$. Then the small cap decoupling constant satisfies

Theorems & Definitions (56)

  • Theorem 1.1: Small cap decoupling with logarithmic losses
  • Theorem 1.2: Wave envelope estimate
  • Theorem 2.1: Theorem 1.2(i) of HR
  • Definition 2.2: Sufficiently rapid cutoffs
  • Proposition 2.3: Existence of a Gevrey-class partition of unity
  • proof
  • Definition 2.4
  • Lemma 2.5: Pruning lemmas
  • proof
  • Lemma 2.6: Pointwise local constancy lemmas
  • ...and 46 more