A counterexample for local smoothing for averages over curves

TL;DR

This paper establishes a new necessary condition that limits the effectiveness of local smoothing estimates for averaging operators over smooth curves in higher dimensions, impacting the understanding of maximal function bounds.

Contribution

It introduces a counterexample showing the limitations of local smoothing estimates for averaging over curves in dimensions five and higher.

Findings

Identifies a necessary condition for local smoothing estimates

Demonstrates limitations in establishing bounds for maximal functions in high dimensions

Highlights the failure of certain smoothing estimates in dimensions n ≥ 5

Abstract

We provide a new necessary condition for local smoothing estimates for the averaging operator defined by convolution with a measure supported on a smooth non-degenerate curve in $\mathbb{R}^n$ for $n \geq 3$. This demonstrates a limitation in the strength of local smoothing estimates towards establishing bounds for the corresponding maximal functions when $n \geq 5$.