Spherical means and the restriction phenomenon

Let Gamma be a smooth compact convex planar curve with are length dm and le t 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 phenom enon. We obtain best possible indices for the above inequality when Gamma i s any convex curve and under various geometric assumptions.