A counterexample for local smoothing for averages over curves

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 $n \geq 5$.