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A decoupling proof of the Tomas restriction theorem (2112.04111v1)
Published 8 Dec 2021 in math.CA
Abstract: We give a new proof of a classic Fourier restriction theorem for the truncated paraboloid in $\mathbb{R}n$ based on the $l2$ decoupling theorem of Bourgain-Demeter. Focusing on the extension formulation of the restriction problem (dual to the original restriction formulation), we find that the $l2$ decoupling theorem directly implies a local variant of the desired extension estimate incurring an $\varepsilon$-loss. To upgrade this result to the desired global extension estimate, we employ some $\varepsilon$-removal techniques first introduced by Tao. By adhering to the extension formulation, we obtain a more natural proof of the required $\varepsilon$-removal result.