A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

Fourier restriction for smooth hyperbolic 2-surfaces

“…In a similar way as in [BMVp20b], Theorem 1.2 will be a consequence of the following local Fourier extension estimate: Theorem 1.3. Assume that 3.25 ≥ q > 2.6.Then, for every ε > 0, there is a sufficiently large M (ε) ∈ N such that for any φ ∈ Hyp M(ε) the following holds true: there is a constant C ε such that for any R ≥ 1…”

“…Indeed, a simple interpolation argument as in [BMVp20b] shows that the estimate in Theorem 1.3 implies the following:…”

“…Improvements over the results for the saddle by means of an adaptation of the polynomial partitioning method from Guth's articles [Gu16] were achieved by C. H. Cho and J. Lee [ChL17], and J. Kim [K17]. Moreover, for a particular class of one-variate perturbations of the hyperbolic paraboloid, an analogue of Guth's result had been proved by the authors in [BMVp20b], and more lately were reported by S. Guo and C. Oh for compact subsets of general polynomial surfaces with negative Gaussian curvature in [GO20]. Moreover, by making use of Lorentzian symmetries, B. Bruce [Br20a] has established analogous results for compact subsets of the one-sheeted hyperboloid.…”

A restriction estimate for surfaces with negative Gaussian curvatures

“…Buschenhenke, Müller, and Vargas first studied the problem of restriction estimates for perturbed hyperbolic paraboloids in one variable, given by h(ξ, η) = ξη + g(η) for some smooth function g. A typical and model example is g(η) = η 3 . They obtained the restriction estimate (1.3) for this typical example in the range p > 10/3 in [BMV20a] and in the range p > 3.25 in [BMV20c]; for functions g that are of finite types in the range p > 10/3 in [BMV19]; and for flat functions g with g ′′′ monotone in the range p > 10/3 in [BMV20b].…”

Factorisation in Restriction theory and near extremisers

“…Clearly (1.5) implies (1.4), while the converse uses certain orthogonality arguments. A detailed proof can be found in Lemma 5.1 in [BMV20].…”