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arXiv · 1607.08302

Decoupling and near-optimal restriction estimates for Cantor sets

Izabella Laba·Hong Wang
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Abstract

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for all $p>2d/\alpha$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $\alpha=d/k$ with $k\in\mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $\Lambda(p)$ sets.

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Izabella Laba, Hong Wang. 2016-07-28. Decoupling and near-optimal restriction estimates for Cantor sets. https://arxiv.org/abs/1607.08302

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