Extremizers for Fourier restriction on hyperboloids

Carneiro, Emanuel; Oliveira e Silva, Diogo; Sousa, Mateus

doi:10.1016/j.anihpc.2018.06.001

Carneiro, Emanuel ; Oliveira e Silva, Diogo ; Sousa, Mateus

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 389-415.

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Résumé

The $L^{2} \to L^{p}$ adjoint Fourier restriction inequality on the d-dimensional hyperboloid $H^{d} \subset R^{d + 1}$ holds provided $6 \leq p < \infty$ , if $d = 1$ , and $2 (d + 2) / d \leq p \leq 2 (d + 1) / (d - 1)$ , if $d \geq 2$ . Quilodrán [35] recently found the values of the optimal constants in the endpoint cases $(d, p) \in {(2, 4), (2, 6), (3, 4)}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed in [35], namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d, p) = (1, 6)$ (the remaining endpoint for which p is an even integer) and show that there are no extremizers in this case; and (ii) we establish the existence of extremizers in all non-endpoint cases in dimensions $d \in {1, 2}$ . This completes the qualitative description of this problem in low dimensions.

DOI : 10.1016/j.anihpc.2018.06.001

Classification : 42B10
Mots-clés : Fourier restriction, Extremizers, Hyperboloid, Klein–Gordon equation

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     title = {Extremizers for {Fourier} restriction on hyperboloids},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {389--415},
     publisher = {Elsevier},
     volume = {36},
     number = {2},
     year = {2019},
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     mrnumber = {3913191},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2018.06.001/}
}

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Carneiro, Emanuel; Oliveira e Silva, Diogo; Sousa, Mateus. Extremizers for Fourier restriction on hyperboloids. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 389-415. doi : 10.1016/j.anihpc.2018.06.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2018.06.001/

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