In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the \(L^p(\mathbb {S}^{d-1})\) to \(L^p(\mathbb {R}^d)\) Fourier extension estimates in the same range of exponents p for which they are global maximizers for the \(L^2(\mathbb {S}^{d-1})\) to \(L^p_{rad}L^2_{ang}(\mathbb {R}^d)\) mixed-norm Fourier extension inequalities. Moreover, in the case of low dimensions, we improve the range of exponents for which constant functions are known to be the unique global maximizers for the \(L^2(\mathbb {S}^{d-1})\) to \(L^p_{rad}L^2_{ang}(\mathbb {R}^d)\) mixed-norm Fourier extension estimate on the sphere, covering, for the case of dimensions \(d=2,3\) , the entire Stein–Tomas range. This is achieved by establishing novel hierarchies between certain weighted norms of Bessel functions.
