A counterexample for local smoothing for averages over curves

Source 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 R n \mathbb {R}^n for n ≥ 3 n \geq 3 . This demonstrates a limitation in the strength of local smoothing estimates towards establishing bounds for the corresponding maximal functions when n ≥ 5 n \geq 5 .