Amplitude dependent wave envelope estimates for the cone in $\mathbb{R}^3$

TL;DR

This paper develops amplitude-dependent wave envelope estimates for functions supported on a cone in three-dimensional space, leading to sharp decoupling and square function inequalities that improve understanding of Fourier analysis on conical surfaces.

Contribution

It introduces an $ ext{α}$-dependent wave envelope estimate for the cone, resulting in sharp decoupling inequalities and small cap decoupling results that refine previous bounds.

Findings

Established sharp square function inequalities for the cone.

Derived small cap decoupling estimates with refined subdivision of caps.

Extended wave envelope estimates to an amplitude-dependent framework.

Abstract

For functions $f$ with Fourier transform supported in the truncated cone, we bound superlevel sets $\{x\in\mathbb{R}^3:|f(x)|>\alpha\}$ using an $\alpha$-dependent version of the wave envelope estimate of Guth--Wang--Zhang. Our estimates imply both sharp square function and decoupling inequalities for the cone. We also obtain sharp small cap decoupling for the cone, where small caps $\gamma$ subdivide canonical $1\times R^{-1/2}\times R^{-1}$ planks into $R^{-\beta_2}\times R^{-\beta_1}\times R^{-1}$ sub-planks, for $\beta_1\in[\frac{1}{2},1]$ and $\beta_2\in[0,1]$.