Sharp local smoothing estimates for curve averages

TL;DR

This paper establishes sharp local smoothing estimates for curve averages across all dimensions, leading to new bounds for the helical maximal operator and Bochner-Riesz estimates, advancing understanding of Fourier decay and wave envelopes.

Contribution

It introduces novel Fourier decay and wave envelope estimates for nondegenerate curves, enabling sharp smoothing bounds and extending known results to higher dimensions.

Findings

Proved sharp local smoothing estimates for curve averages in all dimensions.

Established the sharp $L^p$ boundedness of the helical maximal operator in $\

Derived Bochner-Riesz estimates for nondegenerate curves in all dimensions.

Abstract

We prove sharp local smoothing estimates for curve averages in all dimensions. As a corollary, we prove the sharp $L^p$ boundedness of the helical maximal operator in $\mathbb{R}^4$, which was previously known only for $\mathbb{R}^2$ and $\mathbb{R}^3$. We also improve previously known results in higher dimensions. There are new ingredients in the proof: Fourier decay estimates and wave envelope estimates for nondegenerate curves in $\mathbb{R}^n$. As a byproduct, we prove Bochner-Riesz estimates for nondegenerate curves in all dimensions.