Extremizers for the Airy–Strichartz inequality

Abstract

We identify the compactness threshold for optimizing sequences of the Airy-Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy-Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy-Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.

Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

“…Secondly, we study similar questions in the more general setting of the Fourier extension problem on the curve s " |y| p , for arbitrary p ą 1. We also consider odd curves s " y|y| p´1 , p ą 1, the case p " 3 relating to the Airy-Strichartz inequality [15,20,38]. Lastly, we study super-exponential decay and analyticity of the corresponding extremizers and their Fourier transform via a bootstrapping procedure.…”

“…Various results of a similar flavour to that of Theorem 1.3 have appeared in the recent literature. They are typically derived from a sophisticated application of concentrationcompactness techniques [9,39], a full profile decomposition [24,25,38], or the missing mass method as in [19,20]. We introduce a new variant which follows the spirit of the celebrated works of Lieb [4,27] and Lions [28,29].…”

Extremizers for Fourier restriction on hyperboloids

“…Main theorem. The first result to address the sharp form of (1.5) is due to Quilodrán [12], in which he computes the exact values of H d,p in the endpoint cases (d, p) = (2, 4), (2,6) and (3,4), and establishes the non-existence of extremizers in these cases. 2 A crucial element of his proof is the fact that the Lebesgue exponents p under consideration are even integers, a fact that allows one to use the convolution structure of the problem via an application of Plancherel's theorem.…”

“…The precursor [3] of the present work contains two main results. The first result [3, Theorem 1] is the explicit computation of the optimal constant H d,p in the case (d, p) = (1,6) and the proof that extremizers do not exist in this case. The second result [3,Theorem 2] establishes the existence of extremizers in all non-endpoint cases of (1.5) in dimensions d ∈ {1, 2}.…”

“…Several authors have investigated the interface between bilinear restriction theory and these extremal questions, both from the restriction side and the partial differential equations point of view. Here we mention the works [1,2,5,6,7,10,14], all of which deal with these connections. Many other authors have contributed to the development of the area, and we refer the reader to [3] for an exposition of related literature on sharp Fourier restriction theory.…”