Extremizers for Fourier restriction on hyperboloids

Abstract

We prove that in dimensions d ≥ 3, the non-endpoint, Lorentz-invariant L 2 → L p adjoint Fourier restriction inequality on the d-dimensional hyperboloid H d ⊆ R d+1 possesses maximizers. The analogous result had been previously established in dimensions d = 1, 2 using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.2010 Mathematics Subject Classification. 42B10. A simple rescaling argument transfers all the results of this paper to the hyperboloids H d s = (ξ, τ ) ∈ R d × R : τ = (s 2 + |ξ| 2 ) 1 2 , s > 0.

Extremizers for the Airy–Strichartz inequality

“…On the other hand, they give universal informations about optimizing sequences. Studying optimizing sequences may also lead to the non-existence of optimizers [42,43,12].…”

Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

“…In the non-compact setting, it is in general non-trivial to verify condition (iv) of [13, Proposition 1.1]. To overcome this difficulty, various arguments using Sobolev embeddings and the Rellich-Kondrachov compactness theorem have been employed in [7,14,34,35]. In our case, it is not clear how such an argument would go.…”

Sharp mixed norm spherical restriction

“…The existence of extremizers for (1.7) in all dimensions was established in [23] for the paraboloid, and in [21] for the cone. For the hyperboloid, the works [8,20] settle the endpoint cases where q is even, i.e. (d, q) = (2, 6), (3,4), (3,6), (4,4), finding the sharp constant C in (1.7) and showing that there are no extremizers in these cases.…”

“…(d, q) = (2, 2m), with m ≥ 4, is still unknown. The existence of extremizers in all non-endpoint cases in dimensions d = 2 and d = 3 was established in [8].…”