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Carneiro, Emanuel; Oliveira e Silva, Diogo; Sousa, Mateus (2019): Sharp mixed norm spherical restriction. In: Advances in Mathematics, Vol. 341: pp. 583-608
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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.