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

For functions $f$ with Fourier transform supported in the truncated cone, we bound superlevel sets $\{x\in\mathbb{R}^3:|f(x)|>α\}$ using an $α$-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 $γ$ subdivide canonical $1\times R^{-1/2}\times R^{-1}$ planks into $R^{-β_2}\times R^{-β_1}\times R^{-1}$ sub-planks, for $β_1\in[\frac{1}{2},1]$ and $β_2\in[0,1]$.

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