Some remarks on Fourier restriction estimates

Original · EN

We provide Lᵖ → Lq refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let S⊂ R³ be a compact C∞ surface with strictly positive second fundamental form. We derive sharp Lᵖ(S) → Lq(R³) estimates for the associated Fourier extension operator for q> 3.25 and q≥ 2p' from an estimate of Guth that was used to obtain L∞(S) → Lq(R³) bounds for q>3.25. We present a slightly weaker result when S is the hyperbolic paraboloid in R³ based on the work of Cho and Lee. Finally, we give some refinements for the truncated paraboloid in higher dimensions.