Decoupling and near-optimal restriction estimates for Cantor sets

2016/07/28 by Izabella Łaba, Hong Wang, Laba, Izabella +1 · 1 citation
Computer Science · Mathematics · #28A80 #42B15 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1607.08302

openalex publication_date 2016/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any α∈(0,d), we construct Cantor sets in ℝ^d of Hausdorff dimension α such that the associated natural measure μ obeys the restriction estimate ‖ \widehatf dμ ‖_p ≤ C_p ‖ f ‖_L²(μ) for all p>2d/α. 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 α=d/k with k∈ℕ. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of Λ(p) sets.

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