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

Abstract:

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 \begin{equation*}\|Tf\|_{L^{3}(\mathbb {T}\times \mathbb {R}^{2})}\leq c\|f\| _{L^{3/2}(\mathbb {R}^{2})}. \end{equation*} 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 \begin{equation*}\left \| Tf \right \| _{L^{s}\left (\mathbb {T} ,L^{p^{\prime }}\left (\mathbb {R}^{2}\right )\right )}\leq c\left \| f\right \| _{L^{p}\left (\mathbb {R}^{2}\right )} \end{equation*} where $p$ and $p^{\prime }$ are conjugate exponents.