Improved Fourier restriction estimates in higher dimensions

Hickman, Jonathan; Rogers, Keith M.

doi:10.4310/cjm.2019.v7.n3.a1

Cited by 40 publications

(68 citation statements)

References 35 publications

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“…9 These estimates have an L 2 -norm appearing on the right-hand side. Relaxing L 2 to L 8 has led to further improvements on the Fourier restriction problem for the paraboloid [33,16]. Theorem 1.9.…”

Section: 2mentioning

confidence: 99%

“…This can be thought of as a dimensional reduction and, indeed, the proof of Theorem 1.9 will proceed by an induction on dimension. Polynomial partitioning has played an increasingly important rôle in the theory of oscillatory integral operators, beginning with the work on the restriction problem in [13,14] and, more recently, in [33,16]. 2) Non-concentration/transverse equidistribution.…”

Section: Key Features Of the Analysismentioning

confidence: 99%

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Sharp estimates for oscillatory integral operators via polynomial partitioning

2019

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The sharp range of L p -estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positivedefinite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments. The main result implies improved bounds for the Bochner-Riesz conjecture in dimensions n ě 4.3 Strictly speaking, in [9] weaker L 8´Lp bounds are proven, but the methods can be used to establish the L p´Lp strengthening: see, for instance, [14, §9] where the L p´Lp argument appears (although in a slightly disguised form). 4 In particular, Lee [19] proved that for positive-definite phases (1.4) holds for p ě 2¨n`2 n in all dimensions, extending the range in Theorem 1.1 when n is odd.

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Section: 2mentioning

confidence: 99%

Section: Key Features Of the Analysismentioning

confidence: 99%

Sharp estimates for oscillatory integral operators via polynomial partitioning

2019

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“…As of this writing, the above described restriction/extension conjecture is settled when d = 1, by Stein and Fefferman, and is open in all higher dimensions. More precisely, in higher dimensions, it is solved for q > 3.25 ([12, 24]; see also [31]) when d = 2, and for q > q(d), for some (explicit, yet complicated) q(d) < 2(d+3) d+1 when d ≥ 3 [13,14,28,29]. Classical symmetries of both the Fourier transform and the paraboloid lead to a wealth of symmetries for the extension and restriction operators.…”

Section: Introductionmentioning

confidence: 99%

Extremizability of Fourier restriction to the paraboloid

Stovall¹

2020

Advances in Mathematics

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In this article, we prove that nearly all valid, scale-invariant Fourier restriction inequalities for the paraboloid in R 1+d have extremizers and that L p -normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case q = 2. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.An equivalent formulation is that the extension operator

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“…Today it employs a variety of modern analytical tools, but the full range of exponents remains largely unsolved in dimensions d ≥ 3. The reader can consult the expository text by Tao [17] for an overview of developments prior to 2004, and the papers by Bourgain and Guth [3], Guth [7,8], and Hickman and Rogers [9] for just some of the numerous recent results.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%