Geodesic
Sharp smoothing properties of averages over curves
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

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 $d\ge 4$. 
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Hyerim Ko; Sanghyuk Lee; Sewook Oh. Sharp smoothing properties of averages over curves. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.2

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