Abstract
Let d≥2 be an integer and let 2d/(d−1)<q≤∞. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality ‖fσˆ‖Lrad qLang 2(Rd)≤Cd,q‖f‖L2(Sd−1,dσ), established by L. Vega in 1988. Letting Ad⊂(2d/(d−1),∞] be the set of exponents for which the constant functions on Sd−1 are the unique extremizers of this inequality, we show that: (i) Ad contains the even integers and ∞; (ii) Ad is an open set in the extended topology; (iii) Ad contains a neighborhood of infinity (q0(d),∞] with q0(d)≤([Formula presented]+o(1))dlogd. In low dimensions we show that q0(2)≤6.76;q0(3)≤5.45;q0(4)≤5.53;q0(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.
Original language | English |
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Pages (from-to) | 583-608 |
Number of pages | 26 |
Journal | Advances in Mathematics |
Volume | 341 |
DOIs | |
Publication status | Published - 7 Jan 2019 |
Bibliographical note
Funding Information:The software Mathematica was used to perform the numerical tasks described in this paper. The authors thank Dimitar Dimitrov for a careful reading of the manuscript and for having suggested the use of the quadratic approximation (4.2) in the argument of Section 4 . The authors are also thankful to Árpád Baricz, Doron Lubinsky, José Madrid, Fernando Rodriguez-Villegas and Krzysztof Stempak for helpful comments and discussions. E.C. acknowledges support from CNPq – Brazil ( CNPQ 305612/2014-0 ), FAPERJ – Brazil ( FAPERJ E-26/202.797/2015 ) and the Simons Associate Scheme from the International Centre for Theoretical Physics ( ICTP ) – Italy. D.O.S. was partially supported by the Hausdorff Center for Mathematics and DFG grant CRC 1060 . M.S. acknowledges support from FAPERJ – Brazil. Part of this work was carried out during a research visit to ICTP – Italy. The authors thank the warm hospitality of the Institute.
Publisher Copyright:
© 2018 Elsevier Inc.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
Keywords
- Bessel functions
- Delta calculus
- Extremizers
- Fourier restriction
- Mixed norm
- Optimal constants
ASJC Scopus subject areas
- Mathematics(all)