Existence of extremizers for a model convolution operator

TL;DR

This paper proves the existence of extremizers for a convolution operator associated with the moment curve, showing precompactness of extremizing sequences and relating endpoint extremizers to X-ray transform extremizers.

Contribution

It establishes the existence and properties of extremizers for a specific convolution operator linked to the moment curve, extending previous understanding of such inequalities.

Findings

Existence of extremizers at non-end points.

Precompactness of extremizing sequences modulo symmetries.

Connection between endpoint extremizers and X-ray transform extremizers.

Abstract

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^{2},...,t^{d})$ 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.