On the extremizers of an adjoint Fourier restriction inequality

Abstract

The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in <(S^2,\sigma )$ then $\widehat{f\sigma }\in L^4({\mathbb{R}}^3)$. We prove that all critical points $f$ of the functional $\norm{\widehat{fσ}}_{L^4}/\norm{f}_{\lt}$ are smooth; that any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix\cdot \xi }$ for some $\xi$; and that complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.

17 pages

References in corpus (3)

• Quasiextremals for a Radon-like transform
• On extremals for a Radon-like transform
• Heat-flow monotonicity of Strichartz norms

Cited by in corpus (2)

• Nonexistence of extremizers for certain convex curves
• Extremizers for Fourier restriction inequalities: convex arcs