$L^p$ improving bounds for averages along curves

TL;DR

This paper investigates the boundedness of averaging operators along curves in the local $(L^p,L^q)$ setting, establishing sharp exponents for these bounds except at endpoints.

Contribution

It provides the first sharp local $(L^p,L^q)$ bounds for averages along curves, advancing understanding of their harmonic analysis properties.

Findings

Sharp local $(L^p,L^q)$ bounds established

Exponents are sharp except at endpoints

Advances in harmonic analysis of averaging operators

Abstract

We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.