The author proves that convolution with affine arclength measure on the curve parametrized by \(h(t) := (t, t^2, \dots, t^n)\) 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. A recent result of S. Dendrinos, N. Laghi and J. Wright [“Universal \(L^p\) improving for averages along polynomial curves in low dimensions, Preprint (2008), arXiv:0805.4344] establishes sharp Lebesgue space bounds(with an accompanying Lorentz space improvement) for convolution with affine arclength measure along polynormial curves in dimensions 2 and 3.