Abstract
We prove that constant functions are the unique real-valued maximizers for all L 2−L 2n adjoint Fourier restriction inequalities on the unit sphere S d−1⊂R d, d∈{3,4,5,6,7}, where n⩾3 is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler–Lagrange equation being smooth, a fact of independent interest which we establish in the companion paper [51]. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character e iξ⋅ω, for some ξ, thereby extending previous work of Christ & Shao [18] to arbitrary dimensions d⩾2 and general even exponents.
| Original language | English |
|---|---|
| Article number | 108825 |
| Journal | Journal of Functional Analysis |
| Volume | 280 |
| Issue number | 7 |
| Early online date | 12 Oct 2020 |
| DOIs | |
| Publication status | Published - 1 Apr 2021 |
Bibliographical note
Funding Information:The computer algebra systems Mathematica, Maxima and Octave were used to compute the entries of Tables 1 and 2 . D.O.S. is supported by the EPSRC New Investigator Award “Sharp Fourier Restriction Theory”, grant no. EP/T001364/1 , and is grateful to Jorge Vitória for a valuable discussion during the preparation of this work. The authors thank Pierpaolo Natalini for providing a copy of [48] , and the anonymous referee for carefully reading the manuscript and valuable suggestions.
Publisher Copyright:
© 2020 Elsevier Inc.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
Keywords
- Bessel integrals
- Convolution of singular measures
- Maximizers
- Sharp Fourier Restriction Theory
- Tomas–Stein inequality
ASJC Scopus subject areas
- Analysis