Spherical means and the restriction phenomenon

  • Luca Brandolini
  • Alex Iosevich
  • Giancarlo Travaglini


Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2) set σΘ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ2, the inequality.
$$\left\{ {\int_{SO(2)} {\left[ {\int_{\mathbb{R}^2 } {\left| {\hat f(\xi )} \right|^2 d\sigma _\Theta (\xi )} } \right]^{s/2} d\Theta } } \right\}^{1/s} \leqslant c\left\| f \right\|_p$$
represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the above inequality when Γ is any convex curve and under various geometric assumptions.

Math Subject Classifications


Keywords and Phrases

spherical means restriction of the Fourier transform 

Copyright information

© Birkhäuser Boston 2000

Authors and Affiliations

  • Luca Brandolini
    • 1
  • Alex Iosevich
    • 2
  • Giancarlo Travaglini
    • 3
  1. 1.Dipartimento di IngegneriaUniversità degli Studi di BergamoDalmine (BG)Italy
  2. 2.Department of MathematicsUniversity of MissouriColumbia
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly

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