Convex curves, radon transforms and convolution operators defined by singular measures

Let Gamma be a convex curve in the plane and let mu is an element of M(R-2) be the arc-length measure of Gamma. Let us rotate Gamma by an angle theta and let mu (theta) be the corresponding measure. Let Tf(x, theta) = f *mu ( theta) (x). Then parallel to Tf parallel to (L3(TxR2)) less than or equal to c parallel tof parallel to (L3/2(R2)). This is optimal for an arbitrary Gamma. Depending on the curvature of Gamma , this estimate can be improved by introducing mixed-norm estimates of the form parallel to Tf parallel to (Ls(T,Lp'(R2))) less than or equal to c parallel tof parallel to (Lp(R2)) where p and p' are conjugate exponents.