LAUSR

The operator T, defined by convolution with the affine arc length measure on the moment curve parametrized by $$h(t)=(t,t^{2},\ldots ,t^{d})$$h(t)=(t,t2,…,td) is a bounded operator from $$L^{p}$$Lp to $$L^{q}$$Lq if $$(\frac{1}{p}, \frac{1}{q})$$(1p,1q) lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of T. We also establish a relation between extremizers for T at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.