Extremizability of Fourier restriction to the paraboloid

Abstract

In this article, we prove that nearly all valid, scale-invariant Fourier restriction inequalities for the paraboloid in R 1+d have extremizers and that L p -normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case q = 2. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.An equivalent formulation is that the extension operator

Extremals for $α$-Strichartz inequalities

“…We should remark that in the proof of Lemma 3.3, by the construction, we know that the Fourier supports of f β n and q N n are mutually disjoint. This crucial fact also implies the conclusion (39). On the other hand, define operators [ Gβ n ] on the Fourier side by…”

“…There may be some papers providing slightly different procedures by using similar ingredients such as [42, Appendix A] and [26,Theorem 4.26]. We refer to [39] for a brief discussion on the L 2 -based linear profile decomposition and a generalization in the L p setting, see also [4] for some recent results on the L p -generalization.…”

Sharp Restriction Theory

“…Sharp restriction theory is becoming increasingly more popular, as shown by the large body of work that appeared in the last decade, and in particular in the last few years. We mention a few interesting works that deal with sharp restriction theory on conics (see Figure 1), namely spheres [4,8,47,48,66,69,112,124], paraboloids [9,11,37,46,72,73,75,132,139,140,146], cones [18,19,33,118,121], and hyperboloids [40,54,90,114,119].…”

Maximisers for Strichartz inequalities on the torus

“…As just three examples of the many other recent papers dealing with other choices of M, we mention the work of Stovall [39], Carneiro et al [7], and Frank and Sabin [18], for the cases in which M is a paraboloid, a hyperboloid, and a cubic curve, respectively.…”