Heat-flow monotonicity of Strichartz norms

Bennett, Jonathan; Bez, Neal; Carbery, Anthony; Hundertmark, Dirk

doi:10.2140/apde.2009.2.147

Cited by 50 publications

(80 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later on, in 2008, Carneiro [5] and Bennett, Bez, Carbery and Hundertmark [3] gave alternative proofs for these cases, including in addition the new case (p, q, d) = (4,8,1). All these proofs, although shedding new light at the problem via different angles, heavily rely on the crucial fact that p = 2k and q = 2kℓ for some integers k ≥ 2 and ℓ ≥ 1.…”

Section: Conjecture 1 a Function F (X) Maximizes (11) If And Only Imentioning

confidence: 99%

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Gonçalves

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

Abstract. In this paper we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension 2, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters. We use spherical harmonics and Gegenbauer polynomials to prove a sharpened weighted inequality for the Schrödinger equation that is maximized by radial functions.

show abstract

Section: Conjecture 1 a Function F (X) Maximizes (11) If And Only Imentioning

confidence: 99%

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Gonçalves

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…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].…”

mentioning

confidence: 99%

Maximizers for the Stein–Tomas Inequality

2016

View full text Add to dashboard Cite

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 ...

show abstract

“…We are interested in determining whether there exists a maximizing function u 0 with u 0 L 2 = 1 for which Bennett et al [2008] and Shao [2009]. We set…”

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

confidence: 99%

“…Recently, Carneiro [2008] proved a sharp Strichartz-type inequality by following the arguments in [Hundertmark and Zharnitsky 2006] and found its maximizers, which derives the same results in [Hundertmark and Zharnitsky 2006] as a corollary when d = 1, 2. Very recently, Bennett et al [2008] offered a new proof to determine the best constants by using the method of heat-flow. Shao [2009] showed that a maximizer exists for all nonendpoint Strichartz inequalities and in all dimensions by relying on the recent linear profile decomposition results for the Schrödinger equations.…”

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

confidence: 99%

Untitled

2009

APDE

View full text Add to dashboard Cite

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.

show abstract

Heat-flow monotonicity of Strichartz norms

Abstract: Our main result is that for d = 1, 2 the classical Strichartz norm e is f Lassociated to the free Schrödinger equation is nondecreasing as the initial datum f evolves under a certain quadratic heat flow.

Cited by 50 publications

References 17 publications

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Maximizers for the Stein–Tomas Inequality

Untitled

Contact Info

Product

Resources

About