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We apply recent circle tangency estimates due to Pramanik--Yang--Zahl toprove sharp weighted Fourier extension estimates for the cone in $\mathbb{R}^3$and $1$-dimensional weights. The idea of using circle tangency estimates tostudy Fourier extension of the cone is originally due to Tom Wolff, who used itin part to prove the first decoupling estimates. We make an improvement to thebest known Mizohata--Takeuchi-type estimates for the cone in $\mathbb{R}^3$ andthe $1$-dimensional weights as a corollary of our main theorem, where thepreviously best known bound follows as a corollary of refined decouplingestimates.

A sharp weighted Fourier extension estimate for the cone in $\mathbb{R}^3$ based on circle tangencies