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

TL;DR

This paper proves endpoint $L^p o L^q$ bounds for convolution operators along polynomial curves in higher dimensions, with bounds depending only on the dimension and polynomial degree.

Contribution

It establishes strong-type endpoint bounds for polynomial curve convolution operators in dimensions four and higher, a significant extension in harmonic analysis.

Findings

Endpoint bounds depend only on dimension and polynomial degree

Results apply to affine arclength measure on polynomial curves

Bounds are valid for dimensions $d \,\geq\, 4$

Abstract

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 $d \geq 4$. The bounds established depend only on the dimension $d$ and the degree of the polynomial.