On the extremizers of an adjoint Fourier restriction inequality. (English) Zbl 1258.35007
Let \(S^2\) denote the unit sphere in \(\mathbb R^3\), equipped with surface measure \(\sigma\). The adjoint Fourier restriction inequlity states that there exists \(C < \infty\) such that
\[
\| \widehat{f \sigma}\|_{L^4(\mathbb R^3)} \leq C \| f \|_{L^2(S^2, \sigma)}
\]
for all \(f \in L^2(S^2)\).
In this paper, the authors prove that all critical points of the functional \(\| \widehat{f \sigma}\|_{L^4(\mathbb R^3)} / \| f \|_{L^2(S^2, \sigma)}\) are smooth. The authors also characterize general complex-valued extremizers in terms of positive ones, and show that the precompactness does continue to hold for complex-valued extremizing sequences, modulo the action of a natural noncompact symmetry group of the inequality.
In this paper, the authors prove that all critical points of the functional \(\| \widehat{f \sigma}\|_{L^4(\mathbb R^3)} / \| f \|_{L^2(S^2, \sigma)}\) are smooth. The authors also characterize general complex-valued extremizers in terms of positive ones, and show that the precompactness does continue to hold for complex-valued extremizing sequences, modulo the action of a natural noncompact symmetry group of the inequality.
Reviewer: Lijing Sun (Milwaukee)
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
References:
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