Endpoint Lebesgue estimates for weighted averages on polynomial curves

TL;DR

This paper proves optimal Lebesgue bounds for weighted averages along polynomial curves, with estimates independent of the specific curves and robust under diffeomorphisms, advancing understanding of polynomial Radon transforms.

Contribution

It introduces uniform Lebesgue estimates for polynomial curve averages with optimal weights, independent of the curves and dependent only on polynomial degree.

Findings

Established optimal Lebesgue estimates for weighted polynomial curve averages.

Proved estimates are uniform and independent of specific curves.

Demonstrated strong invariance properties under diffeomorphisms.

Abstract

We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying a natural nilpotency hypothesis.