Maximizers for the Strichartz Inequality

Foschi, Damiano

doi:10.4171/jems/95

Cited by 122 publications

(258 citation statements)

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“…To verify it, it seems natural to first compute S N −1 and then to use a perturbation argument to establish (1.2). In fact, by a remarkable work of Foschi [14] (see also [21,5]), the value of S N −1 is known for N = 2 and N = 3. We cite the following conjecture from [14]; see also [21].…”

Section: )mentioning

confidence: 99%

“…In fact, by a remarkable work of Foschi [14] (see also [21,5]), the value of S N −1 is known for N = 2 and N = 3. We cite the following conjecture from [14]; see also [21].Assuming that this conjecture is true we can generalize an argument from [12,29] and obtain existence of a maximizer for the R d+1 problem. In connection with Conjecture 1.2 we would like to mention that the existence and precompactness problem for the optimization corresponding to S d was solved by Kunze [23] in d = 1 and by Shao [28] in d ≥ 1.…”

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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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confidence: 99%

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confidence: 99%

Maximizers for the Stein–Tomas Inequality

2016

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“…For the problem of existence of such optimal S ‫ރ‬ schr and explicitly characterizing the maximizers, Kunze [2003] treated the d = 1 case and showed that maximizers exist by an elaborate concentration-compactness method. Foschi [2007] explicitly determined the best constants when d = 1, 2, and showed that the only maximizers are Gaussians up to the natural symmetries associated to the Strichartz inequality by using the sharp Cauchy-Schwarz inequality and the space-time Fourier transform. Hundertmark and Zharnitsky [2006] independently obtained this result by an interesting representation formula of the Strichartz inequalities in lower dimensions.…”

Section: The Linear Profile Decomposition For the Airy Equation 87mentioning

confidence: 99%

“…x Re(e i(·)a n φ) L ‫ޒ‬ airy , the explicit φ had been uniquely determined by Foschi [2007] and Hundertmark and Zharnitsky [2006] independently: they are Gaussians up to the natural symmetries enjoyed by the Strichartz inequality for the Schrödinger equation.…”

Section: The Linear Profile Decomposition For the Airy Equation 87mentioning

confidence: 99%

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2009

APDE

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UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS ENNO LENZMANNWe prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation,in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

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“…Sharp constants and maximisers have been calculated for a number of space-time estimates for dispersive equations (see for example [1,9,13,15,22,27]). Proving the existence of such maximisers has also received attention (see for example [4,7,8,10,11,19,24,26]).…”

Section: Introductionmentioning

confidence: 99%