Linear and bilinear restriction to certain rotationally symmetric hypersurfaces

Abstract

Abstract. Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has non-negative coefficients. In particular, we obtain uniform-depending only on the dimension and polynomial degree-estimates for restriction with affine surface measure, slightly beyond the bilinear range. The main step in the proof of our linear result is an (unconditional) bilinear adjoint restriction estimate for pieces at different scales.

Multi-scale bilinear restriction estimates for general phases

“…In particular, if Φ 1 and Φ 2 are elliptic phases with curvature Financial support by the DFG through the CRC "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is acknowledged. 1 at different scales, then (1.1) was obtained by Stovall [20] with an almost sharp dependence on the scale. If n = 2, and Φ 1 = Φ 2 = ξ m1 1 + ξ m2 2 are surfaces of finite type with rectangular sets Λ j ⊂ {0 < ξ 1 , ξ 2 < 1}, then the essentially optimal dependence of the constant C on the rectangles Λ j and parameters m j 2 has been obtained in recent work of Buschenhenke-Müller-Vargas [3].…”

“…Equivalently, the eigenvalues of ∇ 2 Φ all have the same sign, and are essentially of the size ∇ 2 Φ j L ∞ (Λ) . A typical example of an elliptic phase is the Schrodinger phase Φ = 1 2 |ξ| 2 , or the Klein-Gordon phase (m 2 + |ξ| 2 ) 1 2 in the region |ξ| ≪ m. Bilinear restriction estimates for elliptic phases was recently exploited by Stovall [20] to deduce new results for the linear restriction problem. Applying Theorem 1.4 to the case of elliptic phases, gives the following improvement to [20, Theorem 2.1].…”

“…Secondly, bilinear restriction estimates of the form (1.1) have played an important role in the classical linear restriction problem in harmonic analysis, see for instance [15,20]. In particular, understanding the dependence of the constant C on the phases Φ j , has led to improved linear restriction estimates for surfaces with curvature that may degenerate in certain directions [3,20].…”

“…The proof of Theorem 1.2 follows the strategy used by Tao in the proof of the endpoint bilinear restriction estimate for the cone [21]. In particular, we replace the combinatorial Kakeya type arguments used by Wolff in the seminal paper [27] and further exploited in, for instance, [22,14,4,20], with energy estimates across transverse hypersurfaces. A similar argument was used by Bejenaru [1] to obtain a bilinear restriction estimate with q = r for general phases at unit scale.…”

Optimal Bilinear Restriction Estimates for General Hypersurfaces and the Role of the Shape Operator

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It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow from the corresponding linear estimates. This subject was extensively studied for conic and parabolic surfaces with sharp results proved by Wolff [28] and Tao [23, 24], and with later generalizations [16,17,8,20]. In this paper we provide a unified theory for general hypersurfaces and clarify the role of curvature in this problem, by making statements in terms of the shape operators of the hypersurfaces involved.2010 Mathematics Subject Classification.

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