Sharp smoothing properties of averages over curves

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 $d\ge 4$.

31 pages, the endpoint $L^p$ Sobolev regularity results are included, and the paper is revised substantially

References in corpus (2)

• Sharp $L^p$ bounds for the helical maximal function
• $L^p$ Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities