Existence of extremizers for a model convolution operator

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^2,\mathrm{.}\mathrm{.}\mathrm{.},t^d)$ is a bounded operator from $L^p$ to $L^q$ if $(\frac{1}{p},\frac{1}{q})$ 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.