Sharp mixed norm spherical restriction

Let d >= 2 be an integer and let 2d/(d - 1) < q <= infinity. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality parallel to<(f sigma)over cap>parallel to(LradLang2)-L-n(R-d) <= C-d,C- q parallel to f parallel to L-2(Sd-1, d sigma), established by L. Vega in 1988. Letting A(d) subset of (2d/(d - 1), infinity] be the set of exponents for which the constant functions on Sd-1 are the unique extremizers of this inequality, we show that: (i) A(d) contains the even integers and infinity; (ii) A(d) is an open set in the extended topology; (iii) A(d) contains a neighborhood of infinity (qo (d), infinity] with qo (d) <= (1/2 + o(1)) d log d. In low dimensions we show that qo (2) <= 6.76; qo(3) <= 5.45; qo (4) <= 5.53; qo(5) <= 6.07. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions. (C) 2018 Elsevier Inc. All rights reserved.