Restriction for homogeneous polynomial surfaces in $\mathbb {R}^3$

Abstract

We prove an optimal restriction theorem for an arbitrary homogeneous polynomial hypersurface (of degree at least 2) in R 3 , with affine curvature introduced as a mitigating factor.

Linear and bilinear restriction to certain rotationally symmetric hypersurfaces

“…This completes the proof of Theorem 1.1, modulo the proof of Theorem 2.1. We note that related applications of square functions (albeit more complex ones) have also appeared in the work [9,10] of Carbery-Kenig-Ziesler.…”

Uniform estimates for Fourier restriction to polynomial curves in ℝ d

“…Because the sharp L p (dx) → L q (λ γ dt) estimates for the restriction operator are completely invariant under affine transformations of R d and reparametrizations of γ, there has been considerable interest (such as [3,2,4,13,16,18,19,27]) in the question of whether such bounds hold uniformly over certain large classes of curves. This is part of a broader program ( [9,11,14,26,24] and many others) to determine whether curvature-dependent bounds for various operators arising in harmonic analysis can be generalized, uniformly, by the addition of appropriate affine arclength or surface measures.…”

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

“…It was shown in [2] (using techniques and results from Carbery, Kenig and Ziesler [8], and Molinet and Pilod [30]) that if E = 0, then equation (1.1) is locally well-posed in H s = H s (R 2 ), s > 1 2 . In this case, the symbol of the equation is a cubic polynomial in two dimensions.…”

“…In the proof of Theorem 1.1 we follow the ideas of Bourgain, Saut and Tzvetkov [38], and extended by Molinet and Pilod in [30] for the study of the Zakharov-Kuznetsov (ZK) equation and applied in [2] for NV in the case E = 0. The proofs in the above mentioned works are essentially based on the use of a sharp L 4 Strichartz smoothing estimate, previously obtained by Carbery, Kenig and Ziesler [8], in the case of linear dynamics arising from polynomial symbols. In the case of NV equation with E = 0, however, the symbol is not a polynomial, but a rational function, bounded but no longer smooth, much in the spirit of KP equations.…”

“…If |t| is large enough, we will have for w ∈ B(0, 1), |P n (t, z 0 (t) + w)| 2 |t| Now we deal with the derivative term. We have 8) in the case where the rootẑ 0 is not degenerate. The remaining case, where we haveP n−1 (ẑ 0 ) = 0, goes as follows.…”