Extremizers for Fourier restriction inequalities: Convex arcs

Silva, Diogo Oliveira e

doi:10.1007/s11854-014-0035-4

Cited by 15 publications

(21 citation statements)

References 23 publications

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“…In retrospect, the works of Kunze [19], Foschi [11] and Hundertmark-Zharnitsky [15] were the pioneers on the existence and classification of extremizers for adjoint restriction inequalities (over the paraboloid and cone) in low dimensions. This line of research flourished and these papers were followed by a pool of very interesting works in the interface of extremal analysis and differential equations, see for instance [1,3,9,10,14,20,21,22,24,25,26], in addition to the ones previously cited in this introduction.…”

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confidence: 96%

Some Sharp Restriction Inequalities on the Sphere

Carneiro

Silva

2014

Int Math Res Notices

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In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the spherein the cases (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying: (a) k = 2, q ≥ 2 and 3 ≤ d ≤ 7;(b) k = 2, q ≥ 4 and d ≥ 8; (c) k ≥ 3, q ≥ 2k and d ≥ 2. We also prove a sharp multilinear weighted restriction inequality, with weight related to the k-fold convolution of the surface measure.Building up on the work of Christ and Shao [6, 7], Foschi [12] recently obtained the sharp form of (1.2) in the Stein-Tomas endpoint case (d, p, q) = (3, 4, 2), showing that the constant functions are global extremizers.Here we extend this paradigm to other suitable triples (d, p, q). In fact, defining, our first result is the following:Theorem 1. Let (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying:

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Section: )mentioning

confidence: 96%

Some Sharp Restriction Inequalities on the Sphere

Carneiro

Silva

2014

Int Math Res Notices

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“…The literature on sharp Fourier restriction inequalities related to the paraboloid and cone is extensive and we highlight the works [1,4,6,14,19,21,26]. Other interesting works on sharp Strichartz-type estimates and on the existence of extremizers for other Fourier restriction estimates include [2,3,5,12,13,16,18,20,23,25,27,28,29].…”

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confidence: 99%

A sharp trilinear inequality related to Fourier restriction on the circle

Carneiro¹,

Foschi²,

Silva³

et al. 2017

Rev. Mat. Iberoam.

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Abstract. In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the L 2 − L 6 Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper [24].We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program.

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Section: Rupert L Frank Elliott H Lieb and Julien Sabinmentioning

confidence: 99%

“…In this case the role of antipodal points is played by pairs of points with opposite normal vectors. For earlier results in the case of general curves (N = 2), but with pairs of points with opposite normal vectors excluded, we refer to [27].The mechanism of antipodal concentration was discovered by Christ and Shao in [12]. In their analysis, however, the fact that q is even plays a major role.…”

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confidence: 99%

Maximizers for the Stein–Tomas Inequality

2016

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Abstract. We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a wellknown conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension. Main resultA fundamental result in harmonic analysis is the Stein-Tomas theorem [30,36], which states that if f ∈ L 2 (S N −1 ), N ≥ 2, then the inverse Fourier transformf of f dω, with dω the surface measure on S N −1 , that is, and its L q (R N ) norm is bounded by a constant times the L 2 (S N −1 ) norm of f . Moreover, it is well known that the exponent q is optimal (smallest possible) for this to hold for any f ∈ L 2 (S N −1 ). In this paper we are interested in the optimal Stein-Tomas constant,where · denotes the norm in L 2 (S N −1 ). The value of R N and optimizing functions are only known in dimension N = 3 due to a remarkable work of Foschi [15]; see [11] for partial progress in N = 2. Our main concern here is whether the supremum defining R N is attained and, more generally, the description of maximizing sequences for R N . These questions were recently considered in fundamental papers by Christ and Shao, where the existence of a maximizer for N = 3 [12] and N = 2 [29] was shown, as well as a precompactness result for maximizing sequences for N = 3 [13]. What makes dimensions N = 2 and 3 special is that the exponent q in (1.1) is an even integer, so that one can multiply out |f | q . Our results will be valid in any dimension.c 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Christ and Shao discovered that for the problem of existence of an maximizer for R N a key role is played by the Strichartz inequality [32]. The optimal constant in this inequality isThe overall factor (2π) −(d+2)/d and the factor 1/2 in front of the Laplacian are not important, but simplify some formulas below. We say that a sequence (The following is our main result.up to modulations and, in particular, there is a maximizer for R N .Clearly, the optimization problem for R N is invariant under modulations, so precompactness up to modulations is the best one can expect. Our theorem says that assumption (1.2) is sufficient for this. In fact, it is easy to see that (1.2) is also necessary for the precompactness modulo modulations of all maximizing sequences. We will comment on this in Remark 2.5, where we will also see that (1.2) holds with ≥ instead of >.As we will argue below, in dimensions N = 2 and N = 3, the strict inequality (1.2) holds and so we recover the Christ-Shao results on the existence of optimizers [12,29] and precompactness in N = 3 [13] and we obtain, for the first time, precompactness of maximizing sequences for N = 2.We believe, but cannot prove, that the strict inequality (1.2) holds in any dimension. To verify it, it seems natural to first compute S N −1 and then to use a perturbation argument to establish (1.2). In ...

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Extremizers for Fourier restriction inequalities: Convex arcs

Cited by 15 publications

References 23 publications

Some Sharp Restriction Inequalities on the Sphere

Some Sharp Restriction Inequalities on the Sphere

A sharp trilinear inequality related to Fourier restriction on the circle

Maximizers for the Stein–Tomas Inequality

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