Uniform bounds for convolution and restricted X-ray transforms along degenerate curves

TL;DR

This paper proves endpoint Lebesgue bounds for convolution and X-ray transforms along minimal-differentiability curves, analyzing their interpolants and extrapolants in affine and Euclidean contexts.

Contribution

It introduces new endpoint bounds for these transforms along degenerate curves with minimal smoothness assumptions, expanding understanding of their behavior.

Findings

Established endpoint Lebesgue space bounds for the transforms.

Analyzed the behavior of interpolants and extrapolants of these operators.

Extended results to curves with minimal differentiability hypotheses.

Abstract

We establish endpoint Lebesgue space bounds for convolution and restricted X-ray transforms along curves satisfying fairly minimal differentiability hypotheses, with affine and Euclidean arclengths. We also explore the behavior of certain natural interpolants and extrapolants of the affine and Euclidean versions of these operators.