Uniform decoupling for convex curves

TL;DR

This work establishes a universal $\ell^2L^6$ decoupling inequality for general convex plane curves with minimal regularity assumptions. Central to the argument is a high/low decomposition combined with a multi-scale partition of the curve into admissible pieces and a carefully designed ideal partition that yields uniform control across all curves. The authors develop a robust framework of wave-packet pruning, spatial partitioning, and scale-persistent affine normalisations, culminating in a weak-type estimate that implies the main decoupling bound with a subpolynomial loss $C_\varepsilon R^\varepsilon$. The results extend the parabolic decoupling theory of Bourgain–Demeter and Guth–Maldague–Wang to a broad class of convex curves, with potential implications for restriction theory and geometric measure theory, while highlighting challenges in attaining uniform $(\log R)^c$ improvements in full generality.

Abstract

Using a high/low argument, we prove a universal $\ell^2L^6$ decoupling estimate with constant $C_εR^ε$ for general convex curves in the plane. These curves have no additional regularity assumptions, and the constant $C_ε$ is uniform across all such curves.

Figures (7)

• Figure 1: The high/low decomposition of $g_k$. The square function $g_k$ is essentially Fourier supported in a union of tubes that forms a bush centred at the origin. The low part $g_k^{\mathrm{lo}}$ is Fourier supported in $B(0,2\lambda_{k+1})$. The high part $g_k^{\mathrm{hi}}$ is essentially Fourier supported outside this ball, where these tubes have small overlap, as shown in this diagram. As a result, the terms $|f_{k+1,\tau_k}|^2*\omega_k$ are essentially orthogonal in this region.
• Figure 2: Sumsets of transversal boxs $\tau_k\in\mathop{\mathrm{\mathcal{T}}}\nolimits_k(\tau_1)$ and $\tau_k'\in\mathop{\mathrm{\mathcal{T}}}\nolimits_k(\tau_1')$.
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Theorems & Definitions (107)

• Theorem A
• Theorem B
• Definition 1.1: Affine dimension
• Definition 1.2: Vertical neighbourhoods
• Remark 1.3
• Definition 1.4: Ideal partition
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Example 1.8