Some sharp restriction inequalities on the sphere

Abstract

In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\Vert \widehat{f\sigma }{\Vert }_{L^p({\mathbb{R}}^d)}\lesssim \Vert f{\Vert }_{L^q({\mathbb{S}}^{d-1},\sigma )}$$ in the cases $(d,p,q)=(d,2k,q)$ with $d,k\in \mathbb{N}$ and $q\in {\mathbb{R}}^{+}\cup \{\infty \}$ satisfying: (a) $k=2$, $q\ge 2$ and $3\le d\le 7$; (b) $k=2$, $q\ge 4$ and $d\ge 8$; (c) $k\ge 3$, $q\ge 2k$ and $d\ge 2$. We also prove a sharp multilinear weighted restriction inequality, with weight related to the $k$-fold convolution of the surface measure.

25 pages

References in corpus (2)

• A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions
• On Restricting Cauchy-Pexider Equations to Submanifolds

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