  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 $d\ge 4$.
