Revistes Catalanes amb Accés Obert (RACO)

Fourier restriction to convex surfaces of revolution in $\mathbb{R}^3$

F. Abi-Khuzam, B. Shayya
DOI: 10.5565/PUBLMAT_50106_04


If $\Gamma$ is a $C^3$ hypersurface in $\mathbb{R}^n$ and $d \sigma$ is induced Lebesgue measure on $\Gamma$, then it is well known that a Tomas-Stein Fourier restriction estimate on $\Gamma$ implies that $\Gamma$ has a nowhere vanishing Gaussian curvature. In a recent paper, Carbery and Ziesler observed that if induced Lebesgue measure is replaced by affine surface area, then a Tomas-Stein restriction estimate on $\Gamma$ implies that $\Gamma$ satisfies the affine isoperimetric inequality. Since the only property needed for a hypersurface to satisfy the affine isoperimetric inequality is convexity, this raised the question of whether a Tomas-Stein restriction estimate can be obtained for flat but convex hypersurfaces in $\mathbb{R}^n$ such as $\Gamma(x)=(x, e^{-1/|x|^m})$, $m= 1, 2, \dotsc$. We prove that this is indeed the case in dimension $n=3$.

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