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Spherical means and the restriction phenomenon

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Abstract

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.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Bergamo, Viale Marconi 5, 24044, Dalmine (BG), Italy

    Luca Brandolini

  2. Department of Mathematics, University of Missouri, Columbia

    Alex Iosevich

  3. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126, Milano, Italy

    Giancarlo Travaglini

Authors
  1. Luca Brandolini
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  2. Alex Iosevich
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  3. Giancarlo Travaglini
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Communicated by Fulvio Ricci

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Brandolini, L., Iosevich, A. & Travaglini, G. Spherical means and the restriction phenomenon. The Journal of Fourier Analysis and Applications 7, 359–372 (2001). https://doi.org/10.1007/BF02514502

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  • DOI: https://doi.org/10.1007/BF02514502

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