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The adjoint Fourier restriction inequality on the d-dimensional hyperboloid holds provided , if , and , if . Quilodrán  recently found the values of the optimal constants in the endpoint cases and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed in , namely: (i) we find the explicit value of the optimal constant in the endpoint case (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 . This completes the qualitative description of this problem in low dimensions.
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Mateus Sousa, Emanuel Carneiro, Diogo Oliveira e Silva, Extremizers for Fourier restriction on hyperboloids. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 389–415DOI 10.1016/J.ANIHPC.2018.06.001