Global and local maximizers for some Fourier extension estimates on the sphere

Abstract: In this note we improve, for the case of low dimensions, the known range of exponents for which constant functions are the unique 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. Moreover, we show that in the same range of exponents for which constant functions are the unique maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{red}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension estimates they are also local maximizers for the $L^p(\mathbb{S}^{d-1})$ to $L^p(\mathbb{R}^d)$ Fourier extension estimates. As a by-product, we obtain that for the cases of dimensions $d=2,3$ constant functions are local maximizers for all $p\geq p_{\mathrm{st}}(d)$, where $p_{\mathrm{st}}$ denotes the Stein-Tomas endpoint, $p_{\mathrm{st}}(d):= 2(d+1)/(d-1)$.