Sharp mixed norm spherical restriction

Emanuel Carneiro, Diogo Oliveira e Silva*, Mateus Sousa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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))dlog⁡d. 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 languageEnglish
Pages (from-to)583-608
Number of pages26
JournalAdvances in Mathematics
Volume341
DOIs
Publication statusPublished - 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)

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