Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

We prove that constant functions are the unique real-valued maximizers for all L2−L2n adjoint Fourier restriction inequalities on unit sphere Sd−1⊂Rd, d∈{3,4,5,6,7}, where n⩾3 is an integer. The proof uses tools from probability theory, Lie functional analysis, and theory of special functions. It also relies general solutions underlying Euler–Lagrange equation being smooth, a fact independent interest which we establish in companion paper [51]. further show complex-valued coincide with nonnegative multiplied by character eiξ⋅ω, some ξ, thereby extending previous work Christ & Shao [18] to arbitrary dimensions d⩾2 even exponents.