Some remarks on Fourier restriction estimates

Abstract

We provide $L^p\to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset {\mathbb{R}}^3$ be a compact $C^{\infty }$ surface with strictly positive second fundamental form. We derive sharp $L^p(S)\to L^q({\mathbb{R}}^3)$ estimates for the associated Fourier extension operator for $q>3.25$ and $q\ge 2p^{\prime }$ from an estimate of Guth that was used to obtain $L^{\infty }(S)\to L^q({\mathbb{R}}^3)$ bounds for $q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic paraboloid in ${\mathbb{R}}^3$ based on the work of Cho and Lee. Finally, we give some refinements for the truncated paraboloid in higher dimensions.

Remark was added after Theorem 1.2. After the submission of the previous version, we learned that Bassam Shayya proved the extension operator estimate in Theorem 1.2 up to the sharp line in arXiv:1512.03238. This note will remain unpublished

References in corpus (3)

• Weighted restriction estimates using polynomial partitioning
• Improved restriction estimate for hyperbolic surfaces in R3
• On space-time estimates for the Schroedinger operator

Cited by in corpus (7)

• On Fourier restriction for finite-type perturbations of the hyperboloid
• Restriction estimates for hyperboloids in higher dimensions via bilinear estimates
• Fourier restriction in low fractal dimensions
• A sharp Lp Lq Fourier restriction theorem for a conical surface of finite type
• A restriction estimate for surfaces with negative Gaussian curvatures
• Factorisation in Restriction theory and near extremisers
• Fourier restriction estimates for surfaces of co-dimension two in ${\mathbb{R}}^5$