Bounds on Oscillatory Integral Operators Based on Multilinear Estimates

By Jean Bourgain, Larry Guth. 2010

385 citations recorded by the source corpus.

We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved Lp estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hörmander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hörmander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3d933840949a358d3c34ed76f29061951058fd4

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Citations in the corpus

Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, methodology): “Unfortunately there is some loss in doing this which places some limit on the resulting progress; see [22].”
On the Schrödinger maximal function in higher dimension (2012, methodology): “ Hence an appropriate choice of K= Rδ permits to obtain (2.3) for some θ= θn<1 2. 3. A quantitativeestimate The proof of the second part of Theorem 1 depends on the more reﬁned analysis from §4 in [B-G] and its higher dimensional version. Before stating the relevant result from [B-G], we ﬁx some ”
Weighted restriction estimates and application to Falconer distance set problem (2018, methodology): “…k -broad estimates is that, it can be used as a formally weaker substitute for the (still open) k -linear restriction conjecture to obtain improved linear restriction estimate, following a fairly standard scheme originated in Bourgain-Guth [3] that converts multilinear estimates to linear ones.”
A restriction estimate using polynomial partitioning (2014, positive): “fk 1/13 L∞(S). We now brieﬂy explain how Theorem 0.3 implies Theorem 0.1. As an immediate corollary of Theorem 0.3 we get the estimate: (2) kBrK−ǫ EfkL3.25(B3 R ) ≤ CǫRǫkfkL∞(S). Following ideas from [BG], this estimate implies that (3) kEfkL3.25(B3 R ) ≤ CǫRǫkfkL∞(S) 4 LARRY GUTH Here is a quick sk”
Decouplings for curves and hypersurfaces with nonzero Gaussian curvature (2014, methodology): “The starting point in the argument is the observation in [2] that for each BK there exists a line L = L(BK) in the (ξ1, ξ2) plane such that if SL = {(ξ1, ξ2) : dist((ξ1, ξ2), L) ≤ C K } then for x ∈ BK |ĝdσ(x)| ≤ Cmax α |ĝαdσ(x)|+ (13)”
Restriction estimates using polynomial partitioning II (2016, methodology): “he estimate (1.2) if (1.3) p>2· 3n+1 3n−3 for nodd . (1.4) p>2· 3n+2 3n−2 for neven . The best previous estimates for the problem were proven by Tao [T2] for n= 4 and by Bourgain and the author [BG] for n≥ 5. For n= 4, the conjecture is that (1.2) holds for p>22 3. Theorem 1.1 gives the ran”
Strichartz estimates for Schrödinger equations on irrational tori (2013, methodology): “More recently, using multilinear restriction theory after [1, 8], Bourgain [7] improved the result (iii) for d 4 and showed that (1.”
Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds (2018, result): “This stands in contrast with the L-theory of such operators (see, for instance, [10, 21]).”
A sharp square function estimate for the cone in ℝ3 (2019, neutral): “(τ)= 1 K0 fτ. Now N1 K (Γ 1 K ) is the 1 K -neighborhood of an arc of a parabola of length 1, and each τ is the 1 K -neighborhood of an arc of the parabola of length 1 K0 . The Bourgain–Guth argument [4] says the following. At each point f(x) = P τ fτ(x). Let τ∗ satisfy maxτ |fτ|(x) = |fτ∗|(x). If |”
Improved Fourier restriction estimates in higher dimensions (2018, methodology): “Although many refinements were made since (including the work of Tao [34] which removed the O(n−2)-term with λ = 4), the linear coefficient was not improved for some thirty-five years when Bourgain and Guth [6] showed that it can be lowered to 3.”
Average decay of the Fourier transform of measures with applications (2015, methodology): “(4.3) For this we will use the Bourgain–Guth decomposition with λ = d−α2d−α−1 ; |TSg| .”
Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, methodology): “6, we will use a broad-narrow analysis, which has similar spirit as the techniques in the work of Bourgain-Guth [6], Bourgain [3], Bourgain-Demeter [5] and Guth [19].”
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means (2012, background): “n+2 n+3 , and by duality for p ≥ 2n+ n−1 . See Stein [61], p. 420. This was extended by Lee [40] to the case p < 2n+4 n+4 (or p >2n+4 n ) and recent improvements are proved by Bourgain and Guth [9]. The question whether Bochner-Riesz summability holds onLp(Rn) for all p and all δ>max{n|1 p ”
Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators (2019, neutral): “We refer to [18, 7, 49, 31, 8, 22] and references therein for details.”
On the Vinogradov mean value (2016, background): “3), which in some sense is a linear statement, to multi-linear expressions is crucial as it allows us to exploit transversality. This technique, which is basically simple, goes back to the joint work [BG] of L. Guth and the author. All available results on decoupling make use of this procedure. Cont”
The proof of the l 2 Decoupling Conjecture (2015, methodology): “Inequality (3) has been recently proved by the ﬁrst author for p = 2 n n − 1 in [11], using a variant of the induction on scales from [14] and the multilinear restriction Theorem 6.1.”
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (2016, background): “In particular one can mention milestone papers by Bourgain [3], Wolff [27], Tao [23], Bennett, Carbery, and Tao [2], Bourgain and Guth [4], and Guth [11] which also contain further important references.”