On extremizing sequences for the adjoint restriction inequality on the cone

Abstract

Abstract. It is known that extremizers for the L 2 to L 6 adjoint Fourier restriction inequality on the cone in Ê 3 exist. Here, we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the cone. If we use the knowledge of the exact form of the extremizers, as found by Carneiro, then we can show that nonnegative extremizing sequences converge, after the application of symmetries.

Extremizers for the Airy–Strichartz inequality

“…A second group of works examines the behaviour of extremizing sequences from the point of view of concentration-compactness. This originated in work by Kunze [34] for the problem (1.6) in one space dimension with p = q = 6 and by Shao [45] in higher dimensions (see also [41] when the paraboloid is replaced by the cone, and [28,27] for a fourth order version). For the Stein-Tomas problem on the sphere, this was carried out by Christ and Shao [14] in dimension two and by Shao [46] in dimension one.…”

Extremizers for Fourier restriction inequalities: Convex arcs

“…Following previous work from [18] and [17], Quilodrán proved in [21,Proposition 4.5] that, for the same range of α and value of p, there exists a constant β > 0 such that…”

“…Other results on (non-)existence of extremizers and/or computation of sharp constants for Fourier restriction operators and Strichartz inequalities can be found in [4,9,11,12,14,21,22].…”

“…Using the analysis of a certain bilinear form from [17,21], we establish the following refinement: (4) fσ L 6 (R 2 ) f 1−β/2 L 2 (σ) sup C⊂ |C| −1/4…”

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.…”