A counterexample for local smoothing for averages over curves

TL;DR

The paper addresses local smoothing for averages over curves and shows a new necessary condition when integrating over the dilation parameter $t$. By combining a decoupling-inspired, frequency-localized construction with stationary-phase analysis of the Fourier multiplier, the authors prove a time-averaged lower bound that forces the smoothing exponent to satisfy $\sigma(p,n) = \min\{ \tfrac{1}{n}, \tfrac{1}{n}(\tfrac{1}{2}+\tfrac{2}{p}), \tfrac{2}{p} \}$, with a piecewise dependence on $p$ and $n$. This yields a local-smoothing obstruction that rules out certain conjectured bounds for the associated geometric maximal operator in dimensions $n\ge5$, highlighting a fundamental limitation of the time-averaged local smoothing approach. The results refine the understanding of local smoothing conjectures for curve averages, connect to decoupling-type phenomena, and inform maximal operator theory for nondegenerate curves. The work thus narrows the path toward universal maximal-function bounds via local smoothing and clarifies the ranges where time-averaging cannot compensate for fixed-time smoothing losses.

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 $n \geq 5$.

Theorems & Definitions (8)

• Theorem 1.1
• Conjecture 1.2: Local smoothing conjecture for curve averages
• Conjecture 1.3
• Definition 2.1
• Proposition 2.2
• Proposition 3.1
• Lemma 3.2
• proof : Proof (of Proposition \ref{['necessity proposition']})