For convex analytic finite-type hypersurfaces, the square-root curvature damped Fourier transform decays at the optimal rate |ξ|^{-d/2} for d=2,3, and with a logarithmic loss for d=4.
$L^2$ affine Fourier restriction theorems for smooth surfaces in $\mathbb{R}^3$
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abstract
We prove sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for all surfaces defined by the graph of polynomials of degrees up to $d$ with bounded coefficients. The primary tool is a decoupling theorem for these surfaces.
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Damping oscillatory Integrals of convex analytic functions
For convex analytic finite-type hypersurfaces, the square-root curvature damped Fourier transform decays at the optimal rate |ξ|^{-d/2} for d=2,3, and with a logarithmic loss for d=4.