“…Because the sharp L p (dx) → L q (λ γ dt) estimates for the restriction operator are completely invariant under affine transformations of R d and reparametrizations of γ, there has been considerable interest (such as [3,2,4,13,16,18,19,27]) in the question of whether such bounds hold uniformly over certain large classes of curves. This is part of a broader program ( [9,11,14,26,24] and many others) to determine whether curvature-dependent bounds for various operators arising in harmonic analysis can be generalized, uniformly, by the addition of appropriate affine arclength or surface measures.…”