Endpoint bounds for a generalized {R}adon transform

We prove that convolution with affine arclength measure on the curve parametrized by $h(t):=(t,t^2,\mathrm{.}\mathrm{.}\mathrm{.},t^n)$ is a bounded operator from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds.