Sharp smoothing properties of averages over curves

Abstract

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 [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$ .