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

Let $\Gamma$ be a convex curve in the plane and let $\mu \in M(\mathbb{R}^{2})$ be the arc-length measure of $\Gamma.$ Let us rotate $\Gamma$ by an angle $\theta$ and let $\mu_{\theta }$ be the corresponding measure. Let $T f(x,\theta)=f*\mu _{\theta }(x)$. Then 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 where $p$ and $p^{\prime }$ are conjugate exponents.