Spherical means and the restriction phenomenon

Let Gamma be a smooth compact convex planar curve with are length dm and let d sigma = psi dm where psi is a cutoff function. For Theta is an element of SO(2) set sigma (Theta)(E) = sigma(ThetaE) for any measurable planar set E. Then. for suitable functions f in R-2, the inequality {integral (SO(2)) [integral (R2) /(f) over cap(xi)/(2) d sigma (Theta)(xi)](s/2) d Theta}(1/s) less than or equal to c \\f\\(p) represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the above inequality when Gamma is any convex curve and under various geometric assumptions.

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 {∫SO(2) [∫ℝ2 |f̂(ξ)|2 dσΘ(ξ)]s/2 dΘ}1/s ≤ c∥f∥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.