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We prove that constant functions are the unique real-valued maximizers forall $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere$\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ isan integer. The proof uses tools from probability theory, Lie theory,functional analysis, and the theory of special functions. It also relies ongeneral solutions of the underlying Euler-Lagrange equation being smooth, afact of independent interest which we establish in a companion paper. Wefurther show that complex-valued maximizers coincide with nonnegativemaximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$,thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq2$ and general even exponents.

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres