Some sharp restriction inequalities on the sphere

Emanuel Carneiro, Diogo Oliveira e Silva

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In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big\|\widehat{f \sigma}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim \|f\|_{L^{q}(\mathbb{S}^{d-1},\sigma)}$$ in the cases $(d,p,q) = (d,2k, q)$ with $d,k \in \mathbb{N}$ and $q\in \mathbb{R}^+ \cup \{\infty\}$ satisfying: (a) $k = 2$, $q \geq 2$ and $3 \leq d\leq 7$; (b) $k = 2$, $q \geq 4$ and $d \geq 8$; (c) $k \geq 3$, $q \geq 2k$ and $d \geq 2$. We also prove a sharp multilinear weighted restriction inequality, with weight related to the $k$-fold convolution of the surface measure.
Original languageEnglish
Pages (from-to)8233-8267
Number of pages35
JournalInternational Mathematics Research Notices
Issue number17
Early online date30 Oct 2014
Publication statusPublished - 1 Jan 2015

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25 pages


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