Sharp smoothing properties of averages over curves

TL;DR

This paper establishes optimal smoothing and boundedness properties for averages over smooth nondegenerate curves in higher dimensions, resolving longstanding conjectures and advancing harmonic analysis understanding.

Contribution

It introduces a novel inductive approach to prove sharp $L^p$ Sobolev regularity estimates for averaging operators over curves in $ eal^d$, confirming a key conjecture.

Findings

Proved optimal $L^p$ Sobolev regularity estimates for averaging operators.

Established sharp local smoothing estimates in all dimensions.

First demonstration of nontrivial $L^p$ boundedness of maximal averages over dilations for $d extgreater 3$.

Abstract

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma$ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran-Guo-Hickman-Seeger. Besides, we show the sharp local smoothing estimates for every $d$. As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma$ for $d\ge 4$.