Abstract
In their work Ikromov and Müller (Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra. Princeton University Press, Princeton, 2016) proved the full range L^p-L^2 Fourier restriction estimates for a very general class of hypersurfaces in {mathbb {R}}^3 which includes the class of real analytic hypersurfaces. In this article we partly extend their results to the mixed norm case where the coordinates are split in two directions, one tangential and the other normal to the surface at a fixed given point. In particular, we resolve completely the adapted case and partly the non-adapted case. In the non-adapted case the case when the linear height h_text {lin}(phi ) is below two is settled completely.
Highlights
For a given smooth hypersurface S in Rn, its surface measure dσ, and a smooth compactly supported function ρ ≥ 0, ρ ∈ C0∞(S), the associated Fourier restriction problem asks for which p, q ∈ [1, ∞] the estimate 1/q | f |q ρdσ≤ C p,q f L p(Rn), f ∈ S(Rn ), (1.1)holds true
The three dimensional case, as of yet, is far from being completely understood even when S is the sphere, and there has been a lot of deep work in the direction of understanding L p − Lq estimates for surfaces with both vanishing and non-vanishing Gaussian curvature
If either (a) φ is adapted in its original coordinates, or (b) φ is non-adapted, hlin(φ) < 2, and φ is real analytic, Fig. 3 Necessary conditions in the (1/ p1, 1/ p3)-plane the estimate (1.2) holds true for all (1/ p1, 1/ p3) as determined by Theorem 1.2, except for the point (1/ p1, 1/ p3) = (0, 1/(2h(φ))) where it is false if ρ(0) = 0 and either h(φ) = 1 or ν(φ) = 1
Summary
For a given smooth hypersurface S in Rn, its surface measure dσ , and a smooth compactly supported function ρ ≥ 0, ρ ∈ C0∞(S), the associated Fourier restriction problem asks for which p, q ∈ [1, ∞] the estimate. 1.2 we state the main results of this paper, namely Theorem 1.2 which states the necessary conditions, and Theorem 1.3 which gives us the mixed norm Fourier restriction estimates in the adapted case and the case hlin(φ) < 2. 4, Proposition 4.2, we deal with the adapted case, i.e., we prove that if φ is adapted in its original coordinates, the estimate (1.2) holds for all p’s determined by the necessary conditions, except occasionally for a certain endpoint. A further notational convention adopted from [16] is the use of symbols χ0 and χ1 in denoting certain nonnegative smooth compactly supported functions on R. These cutoff functions χ0 and χ1 may vary from line to line, and sometimes, when several χ0 and χ1 appear within the same formula, they may even designate different functions
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