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 \[\|\widehat {f \sigma }\|_{L^{p}(\mathbb {R} ^{d})} \lesssim \|f\|_{L^{q}(\mathbb {S} ^{d-1},\sigma )}\] 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.