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

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 $\beta_2\in[0,1]$.