Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty \right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$, and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$, for some integer $m\,\ge \,2$. Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$, be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$, let
$$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$
An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$.