Sharp smoothing properties of averages over curves

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve in,. 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 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. As a result, we establish, for the first time, nontrivial boundedness of the maximal average over dilations of for.