The author proves that convolution with affine arclength measure on the curve parametrized by
is a bounded operator from
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
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.