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 $$\big\|\widehat{f \sigma}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim \|f\|_{L^{q}(\mathbb{S}^{d-1},\sigma)}$$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2014
- DOI:
- 10.48550/arXiv.1404.1106
- arXiv:
- arXiv:1404.1106
- Bibcode:
- 2014arXiv1404.1106C
- Keywords:
-
- Mathematics - Classical Analysis and ODEs
- E-Print:
- 25 pages