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arXiv · 2206.01093

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

Dominique Maldague·Larry Guth
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Abstract

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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Dominique Maldague, Larry Guth. 2022-08-13. Amplitude dependent wave envelope estimates for the cone in $\mathbb{R}^3$. https://arxiv.org/abs/2206.01093

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