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 ${\beta }_2\in [0,1]$.