Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

Strichartz, Robert S.

doi:10.1215/s0012-7094-77-04430-1

Cited by 1,008 publications

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“…The second assertion in each of (a) and (b) is [11], Lemma 2.1(ii). For the Stricharz estimates quoted in (a), see [21], [13]. We next establish the first part of (a).…”

Section: Lemma 21 (Group Estimates) Supposementioning

confidence: 81%

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

Colliander

Holmer

Tzirakis

2008

Trans. Amer. Math. Soc.

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Abstract. We prove low regularity global well-posedness for the 1d Zakharov system and the 3d Klein-Gordon-Schrödinger system, which are systems in two variables u :The Zakharov system is known to be locally well-posed in (u, n) ∈ L 2 × H −1/2 and the Klein-GordonSchrödinger system is known to be locally well-posed in (u, n) ∈ L 2 ×L 2 . Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the L 2 norm of u and controlling the growth of n via the estimates in the local theory.

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“…The second assertion in each of (a) and (b) is [11], Lemma 2.1(ii). For the Stricharz estimates quoted in (a), see [21], [13]. We next establish the first part of (a).…”

Section: Lemma 21 (Group Estimates) Supposementioning

confidence: 81%

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

Colliander

Holmer

Tzirakis

2008

Trans. Amer. Math. Soc.

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“…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 is…”

Section: Resultsmentioning

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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“…Somewhat similar connections exist between the cone (3) (in R × R n ) and solutions to the wave equation u tt − ∆u = 0, or between the sphere (1) and solutions to the Helmholtz equation ∆u + u = 0. We will not pursue these connections further here, but see for instance [49] and the (numerous) papers descended from that paper. (Some other connections between restriction estimates and PDEtype estimates are summarized in [62] and the references therein; for the Helmholtz equation, see for instance [4].…”

Section: Restriction Estimates and Extension Estimatesmentioning

confidence: 99%

“…This heuristic argument can be made rigorous by using orthogonality arguments such as the T T * method; see [63], [49], or [47]. In the particular cases of the sphere and paraboloid, the Tomas-Stein estimate yields R S ( 2(n+1) n+3 → 2); for the cone, it yields R S ( 2n n−2 → 2).…”

Section: Local Restriction Estimatesmentioning

confidence: 99%