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 $γ$ 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 $γ$ for $d\ge 4$.

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