JournalsaihpcVol. 36, No. 2pp. 389–415

Extremizers for Fourier restriction on hyperboloids

  • Mateus Sousa

    IMPA – Instituto de Matemática Pura e Aplicada, Rio de Janeiro – RJ, 22460-320, Brazil
  • Emanuel Carneiro

    IMPA – Instituto de Matemática Pura e Aplicada, Rio de Janeiro – RJ, 22460-320, Brazil
  • Diogo Oliveira e Silva

    Hausdorff Center for Mathematics, 53115 Bonn, Germany
Extremizers for Fourier restriction on hyperboloids cover
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Abstract

The L2LpL^{2}\rightarrow L^{p} adjoint Fourier restriction inequality on the d-dimensional hyperboloid HdRd+1\mathbb{H}^{d} \subset \mathbb{R}^{d + 1} holds provided 6p<6 \leq p < \infty , if d=1d = 1, and 2(d+2)/dp2(d+1)/(d1)2(d + 2)/ d \leq p \leq 2(d + 1)/ (d−1), if d2d \geq 2. Quilodrán [35] recently found the values of the optimal constants in the endpoint cases (d,p){(2,4),(2,6),(3,4)}(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)(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{1,2}d \in \{1,2\}. This completes the qualitative description of this problem in low dimensions.

Cite this article

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. 389–415

DOI 10.1016/J.ANIHPC.2018.06.001