Geometric & Functional Analysis GAFA

, Volume 4, Issue 2, pp 213–235

C1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

  • V. Lebedev
  • A. Olevskiî
Article

DOI: 10.1007/BF01895838

Cite this article as:
Lebedev, V. & Olevskiî, A. Geometric and Functional Analysis (1994) 4: 213. doi:10.1007/BF01895838

Abstract

We prove that if aC1 smooth change of variable ϕ:ℝ→ℝ generates a bounded composition operatorff°ϕ in the spaceAp(ℝ)=Lp,p≠2, then φ is linear (affine).

We also prove that for a nonlinearC1 mapping φ, the norms of exponentialseiλϕ as Fourier multipliers inLp(ℝ) tend to infinity (λ∈ℝ,|λ|→∞). In both results the condition φ∈C1 is sharp, it cannot be replaced by the Lipschitz condition.

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • V. Lebedev
    • 1
  • A. Olevskiî
    • 1
  1. 1.School of Mathematical Sciences Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael