Extremizability of Fourier restriction to the paraboloid. (English) Zbl 1429.42011
Summary: In this article, we prove that nearly all valid, scale-invariant Fourier restriction inequalities for the paraboloid in \(\mathbb{R}^{1 + d}\) 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.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
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