McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t [xrarr ] (t, t2, …, tn), 0 < t < 1, maps Lp(ℝn) boundedly into Lp′(ℝn) if and only if 2n(n+1)/(n2+n+2) [les ] p [les ] 2. In proving this, a uniform estimate on damping oscillatory integrals with polynomial phase was crucial. In this paper, a remarkably simple proof of the same estimate on oscillatory integrals is presented. In addition, it is shown that the convolution operator with the affine arclength measure on any polynomial curve in ℝn maps Lp(ℝn) boundedly into Lp′(ℝn) if p = 2n(n+1)/(n2+n+2).