Extremizers for Fourier restriction on hyperboloids

Abstract

The L2→Lp adjoint Fourier restriction inequality on the d-dimensional hyperboloid Hd⊂Rd+1 holds provided 6≤p<∞, if d=1, and 2(d+2)/d≤p≤2(d+1)/(d−1), if d≥2. Quilodrán [35] recently found the values of the optimal constants in the endpoint cases (d,p)∈{(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∈{1,2}. This completes the qualitative description of this problem in low dimensions.