Endpoint bounds for a generalized {R}adon transform

TL;DR

This paper proves the boundedness of a generalized Radon transform involving affine arclength measure on polynomial curves, establishing the full conjectured range of exponents and nearly sharp Lorentz space bounds.

Contribution

It extends previous results by establishing endpoint bounds for the Radon transform on polynomial curves, improving the known range of exponents.

Findings

Boundedness of the transform for the full conjectured exponent range

Nearly sharp Lorentz space bounds achieved

Improvement over previous results by M. Christ

Abstract

We prove that convolution with affine arclength measure on the curve parametrized by $h(t) := (t,t^2,...,t^n)$ is a bounded operator from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds.