Endpoint $L^p\to L^q$ bounds for integration along certain polynomial curves

We establish strong-type endpoint $L^p({\mathbb{R}}^d)\to L^q({\mathbb{R}}^d)$ bounds for the operator given by convolution with affine arclength measure on polynomial curves for $d\ge 4$. The bounds established depend only on the dimension $d$ and the degree of the polynomial.